Integrand size = 33, antiderivative size = 601 \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {b^{3/2} \left (63 a^4 A b+46 a^2 A b^3+15 A b^5-35 a^5 B-6 a^3 b^2 B-3 a b^4 B\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {8 a^4 A+31 a^2 A b^2+15 A b^4-11 a^3 b B-3 a b^3 B}{4 a^3 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b (A b-a B)}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b \left (13 a^2 A b+5 A b^3-9 a^3 B-a b^2 B\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \]
-1/4*b^(3/2)*(63*A*a^4*b+46*A*a^2*b^3+15*A*b^5-35*B*a^5-6*B*a^3*b^2-3*B*a* b^4)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))/a^(7/2)/(a^2+b^2)^3/d-1/2*(a ^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*arctan(-1+2^(1/2)*tan(d*x+ c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)-1/2*(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B) -b^3*(A+B))*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)+1/4*( 3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*ln(1-2^(1/2)*tan(d*x+c)^( 1/2)+tan(d*x+c))/(a^2+b^2)^3/d*2^(1/2)-1/4*(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A +B)+3*a*b^2*(A+B))*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)^3/d *2^(1/2)+1/4*(-8*A*a^4-31*A*a^2*b^2-15*A*b^4+11*B*a^3*b+3*B*a*b^3)/a^3/(a^ 2+b^2)^2/d/tan(d*x+c)^(1/2)+1/2*b*(A*b-B*a)/a/(a^2+b^2)/d/tan(d*x+c)^(1/2) /(a+b*tan(d*x+c))^2+1/4*b*(13*A*a^2*b+5*A*b^3-9*B*a^3-B*a*b^2)/a^2/(a^2+b^ 2)^2/d/tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))
Result contains complex when optimal does not.
Time = 6.35 (sec) , antiderivative size = 585, normalized size of antiderivative = 0.97 \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\frac {b (A b-a B)}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {\frac {-\frac {2 \left (\frac {2 \left (-a^4 b \left (a^2 A-A b^2+2 a b B\right )+\frac {1}{8} a^2 b \left (8 a^4 A+31 a^2 A b^2+15 A b^4-11 a^3 b B-3 a b^3 B\right )+\frac {1}{8} b^2 \left (24 a^4 A b+31 a^2 A b^3+15 A b^5-8 a^5 B-3 a^3 b^2 B-3 a b^4 B\right )\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \left (a^2+b^2\right ) d}+\frac {-\frac {\sqrt [4]{-1} \left (a^3 \left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right )-i a^3 \left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right )\right ) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}-\frac {\sqrt [4]{-1} \left (a^3 \left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right )+i a^3 \left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right )\right ) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}}{a^2+b^2}\right )}{a}-\frac {8 a^4 A+31 a^2 A b^2+15 A b^4-11 a^3 b B-3 a b^3 B}{2 a d \sqrt {\tan (c+d x)}}}{a \left (a^2+b^2\right )}+\frac {\frac {9}{2} a^2 b (A b-a B)+\frac {1}{2} b^2 \left (4 a^2 A+5 A b^2-a b B\right )}{a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{2 a \left (a^2+b^2\right )} \]
(b*(A*b - a*B))/(2*a*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x]) ^2) + (((-2*((2*(-(a^4*b*(a^2*A - A*b^2 + 2*a*b*B)) + (a^2*b*(8*a^4*A + 31 *a^2*A*b^2 + 15*A*b^4 - 11*a^3*b*B - 3*a*b^3*B))/8 + (b^2*(24*a^4*A*b + 31 *a^2*A*b^3 + 15*A*b^5 - 8*a^5*B - 3*a^3*b^2*B - 3*a*b^4*B))/8)*ArcTan[(Sqr t[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*(a^2 + b^2)*d) + (-((( -1)^(1/4)*(a^3*(3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B) - I*a^3*(a^3*A - 3* a*A*b^2 + 3*a^2*b*B - b^3*B))*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d) - ((-1)^(1/4)*(a^3*(3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B) + I*a^3*(a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B))*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d) /(a^2 + b^2)))/a - (8*a^4*A + 31*a^2*A*b^2 + 15*A*b^4 - 11*a^3*b*B - 3*a*b ^3*B)/(2*a*d*Sqrt[Tan[c + d*x]]))/(a*(a^2 + b^2)) + ((9*a^2*b*(A*b - a*B)) /2 + (b^2*(4*a^2*A + 5*A*b^2 - a*b*B))/2)/(a*(a^2 + b^2)*d*Sqrt[Tan[c + d* x]]*(a + b*Tan[c + d*x])))/(2*a*(a^2 + b^2))
Time = 3.01 (sec) , antiderivative size = 528, normalized size of antiderivative = 0.88, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.788, Rules used = {3042, 4092, 27, 3042, 4132, 27, 3042, 4132, 27, 3042, 4136, 27, 3042, 4017, 27, 1482, 1476, 1082, 217, 1479, 25, 27, 1103, 4117, 73, 218}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \tan (c+d x)}{\tan (c+d x)^{3/2} (a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4092 |
| \(\displaystyle \frac {\int \frac {4 A a^2-b B a-4 (A b-a B) \tan (c+d x) a+5 A b^2+5 b (A b-a B) \tan ^2(c+d x)}{2 \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}dx}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {4 A a^2-b B a-4 (A b-a B) \tan (c+d x) a+5 A b^2+5 b (A b-a B) \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}dx}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {4 A a^2-b B a-4 (A b-a B) \tan (c+d x) a+5 A b^2+5 b (A b-a B) \tan (c+d x)^2}{\tan (c+d x)^{3/2} (a+b \tan (c+d x))^2}dx}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {\frac {\int \frac {8 A a^4-11 b B a^3+31 A b^2 a^2-8 \left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x) a^2-3 b^3 B a+15 A b^4+3 b \left (-9 B a^3+13 A b a^2-b^2 B a+5 A b^3\right ) \tan ^2(c+d x)}{2 \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}dx}{a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {8 A a^4-11 b B a^3+31 A b^2 a^2-8 \left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x) a^2-3 b^3 B a+15 A b^4+3 b \left (-9 B a^3+13 A b a^2-b^2 B a+5 A b^3\right ) \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}dx}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {8 A a^4-11 b B a^3+31 A b^2 a^2-8 \left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x) a^2-3 b^3 B a+15 A b^4+3 b \left (-9 B a^3+13 A b a^2-b^2 B a+5 A b^3\right ) \tan (c+d x)^2}{\tan (c+d x)^{3/2} (a+b \tan (c+d x))}dx}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {\frac {-\frac {2 \int \frac {-8 B a^5+24 A b a^4-3 b^2 B a^3+8 \left (A a^2+2 b B a-A b^2\right ) \tan (c+d x) a^3+31 A b^3 a^2-3 b^4 B a+15 A b^5+b \left (8 A a^4-11 b B a^3+31 A b^2 a^2-3 b^3 B a+15 A b^4\right ) \tan ^2(c+d x)}{2 \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {-8 B a^5+24 A b a^4-3 b^2 B a^3+8 \left (A a^2+2 b B a-A b^2\right ) \tan (c+d x) a^3+31 A b^3 a^2-3 b^4 B a+15 A b^5+b \left (8 A a^4-11 b B a^3+31 A b^2 a^2-3 b^3 B a+15 A b^4\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {-8 B a^5+24 A b a^4-3 b^2 B a^3+8 \left (A a^2+2 b B a-A b^2\right ) \tan (c+d x) a^3+31 A b^3 a^2-3 b^4 B a+15 A b^5+b \left (8 A a^4-11 b B a^3+31 A b^2 a^2-3 b^3 B a+15 A b^4\right ) \tan (c+d x)^2}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\int \frac {8 \left (\left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) a^3+\left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right ) \tan (c+d x) a^3\right )}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}+\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan ^2(c+d x)+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\frac {8 \int \frac {\left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) a^3+\left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right ) \tan (c+d x) a^3}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}+\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan ^2(c+d x)+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {8 \int \frac {\left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) a^3+\left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right ) \tan (c+d x) a^3}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}+\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {\frac {-\frac {\frac {16 \int \frac {a^3 \left (-B a^3+3 A b a^2+3 b^2 B a-A b^3+\left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right ) \tan (c+d x)\right )}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}+\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\frac {16 a^3 \int \frac {-B a^3+3 A b a^2+3 b^2 B a-A b^3+\left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right ) \tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}+\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\frac {-\frac {\frac {16 a^3 \left (\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {-\frac {\frac {16 a^3 \left (\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {-\frac {\frac {16 a^3 \left (\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {-\frac {\frac {16 a^3 \left (\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {-\frac {\frac {16 a^3 \left (\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {-\frac {\frac {16 a^3 \left (\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\frac {16 a^3 \left (\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )+\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}+\frac {16 a^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}d\tan (c+d x)}{d \left (a^2+b^2\right )}+\frac {16 a^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {-\frac {\frac {2 b^2 \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \int \frac {1}{a+b \tan (c+d x)}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}+\frac {16 a^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}+\frac {-\frac {2 \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right )}{a d \sqrt {\tan (c+d x)}}-\frac {\frac {16 a^3 \left (\frac {1}{2} \left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {2 b^{3/2} \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d \left (a^2+b^2\right )}}{a}}{2 a \left (a^2+b^2\right )}}{4 a \left (a^2+b^2\right )}\) |
(b*(A*b - a*B))/(2*a*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x]) ^2) + ((-(((2*b^(3/2)*(63*a^4*A*b + 46*a^2*A*b^3 + 15*A*b^5 - 35*a^5*B - 6 *a^3*b^2*B - 3*a*b^4*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqr t[a]*(a^2 + b^2)*d) + (16*a^3*(((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*( A + B) - b^3*(A + B))*(-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2]))/2 + ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*(-1/2*Log[1 - Sqrt[2]*Sqrt[Ta n[c + d*x]] + Tan[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2])))/2))/((a^2 + b^2)*d))/a) - (2*(8*a^4*A + 31*a^ 2*A*b^2 + 15*A*b^4 - 11*a^3*b*B - 3*a*b^3*B))/(a*d*Sqrt[Tan[c + d*x]]))/(2 *a*(a^2 + b^2)) + (b*(13*a^2*A*b + 5*A*b^3 - 9*a^3*B - a*b^2*B))/(a*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])))/(4*a*(a^2 + b^2))
3.5.15.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(d*q - a*e)/(2*a*c) Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a , c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- a)*c]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[2/f Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq rt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] & & NeQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[b*(A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1) /(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^ 2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b* B*(b*c*(m + 1) + a*d*(n + 1)) + A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2 )) - (A*b - a*B)*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/f Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ [b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^ n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Simp[ (A*b^2 - a*b*B + a^2*C)/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^n*((1 + Tan[ e + f*x]^2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] & & !GtQ[n, 0] && !LeQ[n, -1]
Time = 0.06 (sec) , antiderivative size = 466, normalized size of antiderivative = 0.78
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 b^{2} \left (\frac {\left (\frac {15}{8} A \,a^{4} b^{2}+\frac {11}{4} A \,a^{2} b^{4}+\frac {7}{8} A \,b^{6}-\frac {11}{8} B \,a^{5} b -\frac {7}{4} B \,a^{3} b^{3}-\frac {3}{8} B a \,b^{5}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )+\frac {a \left (17 A \,a^{4} b +26 A \,a^{2} b^{3}+9 A \,b^{5}-13 B \,a^{5}-18 B \,a^{3} b^{2}-5 B a \,b^{4}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{8}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\left (63 A \,a^{4} b +46 A \,a^{2} b^{3}+15 A \,b^{5}-35 B \,a^{5}-6 B \,a^{3} b^{2}-3 B a \,b^{4}\right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{3} \left (a^{2}+b^{2}\right )^{3}}-\frac {2 A}{a^{3} \sqrt {\tan \left (d x +c \right )}}+\frac {\frac {\left (-3 A \,a^{2} b +A \,b^{3}+B \,a^{3}-3 B a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-A \,a^{3}+3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(466\) |
| default | \(\frac {-\frac {2 b^{2} \left (\frac {\left (\frac {15}{8} A \,a^{4} b^{2}+\frac {11}{4} A \,a^{2} b^{4}+\frac {7}{8} A \,b^{6}-\frac {11}{8} B \,a^{5} b -\frac {7}{4} B \,a^{3} b^{3}-\frac {3}{8} B a \,b^{5}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )+\frac {a \left (17 A \,a^{4} b +26 A \,a^{2} b^{3}+9 A \,b^{5}-13 B \,a^{5}-18 B \,a^{3} b^{2}-5 B a \,b^{4}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{8}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\left (63 A \,a^{4} b +46 A \,a^{2} b^{3}+15 A \,b^{5}-35 B \,a^{5}-6 B \,a^{3} b^{2}-3 B a \,b^{4}\right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{3} \left (a^{2}+b^{2}\right )^{3}}-\frac {2 A}{a^{3} \sqrt {\tan \left (d x +c \right )}}+\frac {\frac {\left (-3 A \,a^{2} b +A \,b^{3}+B \,a^{3}-3 B a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-A \,a^{3}+3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(466\) |
1/d*(-2*b^2/a^3/(a^2+b^2)^3*(((15/8*A*a^4*b^2+11/4*A*a^2*b^4+7/8*A*b^6-11/ 8*B*a^5*b-7/4*B*a^3*b^3-3/8*B*a*b^5)*tan(d*x+c)^(3/2)+1/8*a*(17*A*a^4*b+26 *A*a^2*b^3+9*A*b^5-13*B*a^5-18*B*a^3*b^2-5*B*a*b^4)*tan(d*x+c)^(1/2))/(a+b *tan(d*x+c))^2+1/8*(63*A*a^4*b+46*A*a^2*b^3+15*A*b^5-35*B*a^5-6*B*a^3*b^2- 3*B*a*b^4)/(a*b)^(1/2)*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2)))-2/a^3*A/tan (d*x+c)^(1/2)+2/(a^2+b^2)^3*(1/8*(-3*A*a^2*b+A*b^3+B*a^3-3*B*a*b^2)*2^(1/2 )*(ln((1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1-2^(1/2)*tan(d*x+c)^(1/2)+ tan(d*x+c)))+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*tan( d*x+c)^(1/2)))+1/8*(-A*a^3+3*A*a*b^2-3*B*a^2*b+B*b^3)*2^(1/2)*(ln((1-2^(1/ 2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))+2 *arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))) ))
Leaf count of result is larger than twice the leaf count of optimal. 9032 vs. \(2 (544) = 1088\).
Time = 189.65 (sec) , antiderivative size = 18091, normalized size of antiderivative = 30.10 \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\text {Timed out} \]
Time = 0.36 (sec) , antiderivative size = 607, normalized size of antiderivative = 1.01 \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\frac {\frac {{\left (35 \, B a^{5} b^{2} - 63 \, A a^{4} b^{3} + 6 \, B a^{3} b^{4} - 46 \, A a^{2} b^{5} + 3 \, B a b^{6} - 15 \, A b^{7}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6}\right )} \sqrt {a b}} - \frac {8 \, A a^{6} + 16 \, A a^{4} b^{2} + 8 \, A a^{2} b^{4} + {\left (8 \, A a^{4} b^{2} - 11 \, B a^{3} b^{3} + 31 \, A a^{2} b^{4} - 3 \, B a b^{5} + 15 \, A b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left (16 \, A a^{5} b - 13 \, B a^{4} b^{2} + 49 \, A a^{3} b^{3} - 5 \, B a^{2} b^{4} + 25 \, A a b^{5}\right )} \tan \left (d x + c\right )}{{\left (a^{7} b^{2} + 2 \, a^{5} b^{4} + a^{3} b^{6}\right )} \tan \left (d x + c\right )^{\frac {5}{2}} + 2 \, {\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + {\left (a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} \sqrt {\tan \left (d x + c\right )}} - \frac {2 \, \sqrt {2} {\left ({\left (A - B\right )} a^{3} + 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} - {\left (A + B\right )} b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left ({\left (A - B\right )} a^{3} + 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} - {\left (A + B\right )} b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} {\left ({\left (A + B\right )} a^{3} - 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} + {\left (A - B\right )} b^{3}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left ({\left (A + B\right )} a^{3} - 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} + {\left (A - B\right )} b^{3}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}}}{4 \, d} \]
1/4*((35*B*a^5*b^2 - 63*A*a^4*b^3 + 6*B*a^3*b^4 - 46*A*a^2*b^5 + 3*B*a*b^6 - 15*A*b^7)*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^9 + 3*a^7*b^2 + 3* a^5*b^4 + a^3*b^6)*sqrt(a*b)) - (8*A*a^6 + 16*A*a^4*b^2 + 8*A*a^2*b^4 + (8 *A*a^4*b^2 - 11*B*a^3*b^3 + 31*A*a^2*b^4 - 3*B*a*b^5 + 15*A*b^6)*tan(d*x + c)^2 + (16*A*a^5*b - 13*B*a^4*b^2 + 49*A*a^3*b^3 - 5*B*a^2*b^4 + 25*A*a*b ^5)*tan(d*x + c))/((a^7*b^2 + 2*a^5*b^4 + a^3*b^6)*tan(d*x + c)^(5/2) + 2* (a^8*b + 2*a^6*b^3 + a^4*b^5)*tan(d*x + c)^(3/2) + (a^9 + 2*a^7*b^2 + a^5* b^4)*sqrt(tan(d*x + c))) - (2*sqrt(2)*((A - B)*a^3 + 3*(A + B)*a^2*b - 3*( A - B)*a*b^2 - (A + B)*b^3)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*((A - B)*a^3 + 3*(A + B)*a^2*b - 3*(A - B)*a*b^2 - (A + B)*b^3)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) - sqrt(2)*( (A + B)*a^3 - 3*(A - B)*a^2*b - 3*(A + B)*a*b^2 + (A - B)*b^3)*log(sqrt(2) *sqrt(tan(d*x + c)) + tan(d*x + c) + 1) + sqrt(2)*((A + B)*a^3 - 3*(A - B) *a^2*b - 3*(A + B)*a*b^2 + (A - B)*b^3)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6))/d
Timed out. \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\text {Timed out} \]
Time = 57.04 (sec) , antiderivative size = 35300, normalized size of antiderivative = 58.74 \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \]
((B*tan(c + d*x)^(1/2)*(5*b^4 + 13*a^2*b^2))/(4*a*(a^4 + b^4 + 2*a^2*b^2)) + (B*b*tan(c + d*x)^(3/2)*(3*b^4 + 11*a^2*b^2))/(4*a^2*(a^4 + b^4 + 2*a^2 *b^2)))/(a^2*d + b^2*d*tan(c + d*x)^2 + 2*a*b*d*tan(c + d*x)) - ((2*A)/a + (A*tan(c + d*x)^2*(15*b^6 + 31*a^2*b^4 + 8*a^4*b^2))/(4*a^3*(a^4 + b^4 + 2*a^2*b^2)) + (A*tan(c + d*x)*(16*a^4*b + 25*b^5 + 49*a^2*b^3))/(4*a^2*(a^ 4 + b^4 + 2*a^2*b^2)))/(a^2*d*tan(c + d*x)^(1/2) + b^2*d*tan(c + d*x)^(5/2 ) + 2*a*b*d*tan(c + d*x)^(3/2)) + (log(29491200*A^5*a^22*b^35*d^4 - ((tan( c + d*x)^(1/2)*(7610564608*A^4*a^27*b^33*d^5 - 597688320*A^4*a^23*b^37*d^5 - 1671430144*A^4*a^25*b^35*d^5 - 58982400*A^4*a^21*b^39*d^5 + 85774565376 *A^4*a^29*b^31*d^5 + 385487994880*A^4*a^31*b^29*d^5 + 1104303620096*A^4*a^ 33*b^27*d^5 + 2240523796480*A^4*a^35*b^25*d^5 + 3345249468416*A^4*a^37*b^2 3*d^5 + 3717287903232*A^4*a^39*b^21*d^5 + 3053967114240*A^4*a^41*b^19*d^5 + 1807474491392*A^4*a^43*b^17*d^5 + 726513221632*A^4*a^45*b^15*d^5 + 17076 8990208*A^4*a^47*b^13*d^5 + 10492051456*A^4*a^49*b^11*d^5 - 4917821440*A^4 *a^51*b^9*d^5 - 923009024*A^4*a^53*b^7*d^5 + 8388608*A^4*a^55*b^5*d^5) + ( (((480*A^4*a^2*b^10*d^4 - 16*A^4*b^12*d^4 - 16*A^4*a^12*d^4 - 4080*A^4*a^4 *b^8*d^4 + 7232*A^4*a^6*b^6*d^4 - 4080*A^4*a^8*b^4*d^4 + 480*A^4*a^10*b^2* d^4)^(1/2) + 80*A^2*a^3*b^3*d^2 - 24*A^2*a*b^5*d^2 - 24*A^2*a^5*b*d^2)/(a^ 12*d^4 + b^12*d^4 + 6*a^2*b^10*d^4 + 15*a^4*b^8*d^4 + 20*a^6*b^6*d^4 + 15* a^8*b^4*d^4 + 6*a^10*b^2*d^4))^(1/2)*(((((((480*A^4*a^2*b^10*d^4 - 16*A...