Integrand size = 33, antiderivative size = 493 \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=-\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {b^{5/2} \left (9 a^2 A b+5 A b^3-7 a^3 B-3 a b^2 B\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{7/2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {2 a^2 A+5 A b^2-3 a b B}{3 a^2 \left (a^2+b^2\right ) d \tan ^{\frac {3}{2}}(c+d x)}+\frac {4 a^2 A b+5 A b^3-2 a^3 B-3 a b^2 B}{a^3 \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)}}+\frac {b (A b-a B)}{a \left (a^2+b^2\right ) d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))} \]
b^(5/2)*(9*A*a^2*b+5*A*b^3-7*B*a^3-3*B*a*b^2)*arctan(b^(1/2)*tan(d*x+c)^(1 /2)/a^(1/2))/a^(7/2)/(a^2+b^2)^2/d+1/2*(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*a rctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^2/d*2^(1/2)+1/2*(2*a*b*(A-B)- a^2*(A+B)+b^2*(A+B))*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^2/d*2^(1 /2)+1/4*(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+ta n(d*x+c))/(a^2+b^2)^2/d*2^(1/2)-1/4*(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*ln(1 +2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)^2/d*2^(1/2)+(4*A*a^2*b+5*A *b^3-2*B*a^3-3*B*a*b^2)/a^3/(a^2+b^2)/d/tan(d*x+c)^(1/2)+1/3*(-2*A*a^2-5*A *b^2+3*B*a*b)/a^2/(a^2+b^2)/d/tan(d*x+c)^(3/2)+b*(A*b-B*a)/a/(a^2+b^2)/d/t an(d*x+c)^(3/2)/(a+b*tan(d*x+c))
Result contains complex when optimal does not.
Time = 4.19 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.58 \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\frac {\frac {3 \left (\sqrt [4]{-1} a^{7/2} (a+i b)^2 (A-i B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+b^{5/2} \left (9 a^2 A b+5 A b^3-7 a^3 B-3 a b^2 B\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )+\sqrt [4]{-1} a^{7/2} (a-i b)^2 (A+i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )\right )}{a^{5/2} \left (a^2+b^2\right )}+\frac {-2 a^2 A-5 A b^2+3 a b B}{a \tan ^{\frac {3}{2}}(c+d x)}+\frac {3 \left (4 a^2 A b+5 A b^3-2 a^3 B-3 a b^2 B\right )}{a^2 \sqrt {\tan (c+d x)}}+\frac {3 b (A b-a B)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{3 a \left (a^2+b^2\right ) d} \]
((3*((-1)^(1/4)*a^(7/2)*(a + I*b)^2*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + b^(5/2)*(9*a^2*A*b + 5*A*b^3 - 7*a^3*B - 3*a*b^2*B)*ArcTan[(Sq rt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] + (-1)^(1/4)*a^(7/2)*(a - I*b)^2*(A + I *B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]))/(a^(5/2)*(a^2 + b^2)) + (-2*a ^2*A - 5*A*b^2 + 3*a*b*B)/(a*Tan[c + d*x]^(3/2)) + (3*(4*a^2*A*b + 5*A*b^3 - 2*a^3*B - 3*a*b^2*B))/(a^2*Sqrt[Tan[c + d*x]]) + (3*b*(A*b - a*B))/(Tan [c + d*x]^(3/2)*(a + b*Tan[c + d*x])))/(3*a*(a^2 + b^2)*d)
Time = 2.50 (sec) , antiderivative size = 433, 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.
| \(\displaystyle \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \tan (c+d x)}{\tan (c+d x)^{5/2} (a+b \tan (c+d x))^2}dx\) |
| \(\Big \downarrow \) 4092 |
| \(\displaystyle \frac {\int \frac {2 A a^2-3 b B a-2 (A b-a B) \tan (c+d x) a+5 A b^2+5 b (A b-a B) \tan ^2(c+d x)}{2 \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}dx}{a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {2 A a^2-3 b B a-2 (A b-a B) \tan (c+d x) a+5 A b^2+5 b (A b-a B) \tan ^2(c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))}dx}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {2 A a^2-3 b B a-2 (A b-a B) \tan (c+d x) a+5 A b^2+5 b (A b-a B) \tan (c+d x)^2}{\tan (c+d x)^{5/2} (a+b \tan (c+d x))}dx}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {-\frac {2 \int \frac {3 \left (-2 B a^3+4 A b a^2+2 (a A+b B) \tan (c+d x) a^2-3 b^2 B a+5 A b^3+b \left (2 A a^2-3 b B a+5 A b^2\right ) \tan ^2(c+d x)\right )}{2 \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}dx}{3 a}-\frac {2 \left (2 a^2 A-3 a b B+5 A b^2\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {-2 B a^3+4 A b a^2+2 (a A+b B) \tan (c+d x) a^2-3 b^2 B a+5 A b^3+b \left (2 A a^2-3 b B a+5 A b^2\right ) \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}dx}{a}-\frac {2 \left (2 a^2 A-3 a b B+5 A b^2\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {-2 B a^3+4 A b a^2+2 (a A+b B) \tan (c+d x) a^2-3 b^2 B a+5 A b^3+b \left (2 A a^2-3 b B a+5 A b^2\right ) \tan (c+d x)^2}{\tan (c+d x)^{3/2} (a+b \tan (c+d x))}dx}{a}-\frac {2 \left (2 a^2 A-3 a b B+5 A b^2\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {-\frac {-\frac {2 \int -\frac {2 A a^4+4 b B a^3-2 (A b-a B) \tan (c+d x) a^3-4 A b^2 a^2+3 b^3 B a-5 A b^4-b \left (-2 B a^3+4 A b a^2-3 b^2 B a+5 A b^3\right ) \tan ^2(c+d x)}{2 \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a}-\frac {2 \left (-2 a^3 B+4 a^2 A b-3 a b^2 B+5 A b^3\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (2 a^2 A-3 a b B+5 A b^2\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {2 A a^4+4 b B a^3-2 (A b-a B) \tan (c+d x) a^3-4 A b^2 a^2+3 b^3 B a-5 A b^4-b \left (-2 B a^3+4 A b a^2-3 b^2 B a+5 A b^3\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a}-\frac {2 \left (-2 a^3 B+4 a^2 A b-3 a b^2 B+5 A b^3\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (2 a^2 A-3 a b B+5 A b^2\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {2 A a^4+4 b B a^3-2 (A b-a B) \tan (c+d x) a^3-4 A b^2 a^2+3 b^3 B a-5 A b^4-b \left (-2 B a^3+4 A b a^2-3 b^2 B a+5 A b^3\right ) \tan (c+d x)^2}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a}-\frac {2 \left (-2 a^3 B+4 a^2 A b-3 a b^2 B+5 A b^3\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (2 a^2 A-3 a b B+5 A b^2\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\int \frac {2 \left (a^3 \left (A a^2+2 b B a-A b^2\right )-a^3 \left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x)\right )}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}-\frac {b^3 \left (-7 a^3 B+9 a^2 A b-3 a b^2 B+5 A b^3\right ) \int \frac {\tan ^2(c+d x)+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (-2 a^3 B+4 a^2 A b-3 a b^2 B+5 A b^3\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (2 a^2 A-3 a b B+5 A b^2\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {\frac {2 \int \frac {a^3 \left (A a^2+2 b B a-A b^2\right )-a^3 \left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}-\frac {b^3 \left (-7 a^3 B+9 a^2 A b-3 a b^2 B+5 A b^3\right ) \int \frac {\tan ^2(c+d x)+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (-2 a^3 B+4 a^2 A b-3 a b^2 B+5 A b^3\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (2 a^2 A-3 a b B+5 A b^2\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\frac {2 \int \frac {a^3 \left (A a^2+2 b B a-A b^2\right )-a^3 \left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}-\frac {b^3 \left (-7 a^3 B+9 a^2 A b-3 a b^2 B+5 A b^3\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (-2 a^3 B+4 a^2 A b-3 a b^2 B+5 A b^3\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (2 a^2 A-3 a b B+5 A b^2\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {-\frac {\frac {\frac {4 \int \frac {a^3 \left (A a^2+2 b B a-A b^2-\left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x)\right )}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}-\frac {b^3 \left (-7 a^3 B+9 a^2 A b-3 a b^2 B+5 A b^3\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (-2 a^3 B+4 a^2 A b-3 a b^2 B+5 A b^3\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (2 a^2 A-3 a b B+5 A b^2\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {\frac {4 a^3 \int \frac {A a^2+2 b B a-A b^2-\left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}-\frac {b^3 \left (-7 a^3 B+9 a^2 A b-3 a b^2 B+5 A b^3\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (-2 a^3 B+4 a^2 A b-3 a b^2 B+5 A b^3\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (2 a^2 A-3 a b B+5 A b^2\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {-\frac {\frac {\frac {4 a^3 \left (\frac {1}{2} \left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} \left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d \left (a^2+b^2\right )}-\frac {b^3 \left (-7 a^3 B+9 a^2 A b-3 a b^2 B+5 A b^3\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (-2 a^3 B+4 a^2 A b-3 a b^2 B+5 A b^3\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (2 a^2 A-3 a b B+5 A b^2\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {-\frac {\frac {\frac {4 a^3 \left (\frac {1}{2} \left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} \left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}-\frac {b^3 \left (-7 a^3 B+9 a^2 A b-3 a b^2 B+5 A b^3\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (-2 a^3 B+4 a^2 A b-3 a b^2 B+5 A b^3\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (2 a^2 A-3 a b B+5 A b^2\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {-\frac {\frac {\frac {4 a^3 \left (\frac {1}{2} \left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} \left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {b^3 \left (-7 a^3 B+9 a^2 A b-3 a b^2 B+5 A b^3\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (-2 a^3 B+4 a^2 A b-3 a b^2 B+5 A b^3\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (2 a^2 A-3 a b B+5 A b^2\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {\frac {\frac {4 a^3 \left (\frac {1}{2} \left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} \left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {b^3 \left (-7 a^3 B+9 a^2 A b-3 a b^2 B+5 A b^3\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (-2 a^3 B+4 a^2 A b-3 a b^2 B+5 A b^3\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (2 a^2 A-3 a b B+5 A b^2\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {-\frac {\frac {\frac {4 a^3 \left (\frac {1}{2} \left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {b^3 \left (-7 a^3 B+9 a^2 A b-3 a b^2 B+5 A b^3\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (-2 a^3 B+4 a^2 A b-3 a b^2 B+5 A b^3\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (2 a^2 A-3 a b B+5 A b^2\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\frac {4 a^3 \left (\frac {1}{2} \left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {b^3 \left (-7 a^3 B+9 a^2 A b-3 a b^2 B+5 A b^3\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (-2 a^3 B+4 a^2 A b-3 a b^2 B+5 A b^3\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (2 a^2 A-3 a b B+5 A b^2\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {\frac {4 a^3 \left (\frac {1}{2} \left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )-\frac {1}{2} \left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {b^3 \left (-7 a^3 B+9 a^2 A b-3 a b^2 B+5 A b^3\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (-2 a^3 B+4 a^2 A b-3 a b^2 B+5 A b^3\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (2 a^2 A-3 a b B+5 A b^2\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-\frac {\frac {\frac {4 a^3 \left (\frac {1}{2} \left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {b^3 \left (-7 a^3 B+9 a^2 A b-3 a b^2 B+5 A b^3\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (-2 a^3 B+4 a^2 A b-3 a b^2 B+5 A b^3\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (2 a^2 A-3 a b B+5 A b^2\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {-\frac {\frac {\frac {4 a^3 \left (\frac {1}{2} \left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {b^3 \left (-7 a^3 B+9 a^2 A b-3 a b^2 B+5 A b^3\right ) \int \frac {1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}d\tan (c+d x)}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (-2 a^3 B+4 a^2 A b-3 a b^2 B+5 A b^3\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (2 a^2 A-3 a b B+5 A b^2\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {-\frac {\frac {\frac {4 a^3 \left (\frac {1}{2} \left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {2 b^3 \left (-7 a^3 B+9 a^2 A b-3 a b^2 B+5 A b^3\right ) \int \frac {1}{a+b \tan (c+d x)}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (-2 a^3 B+4 a^2 A b-3 a b^2 B+5 A b^3\right )}{a d \sqrt {\tan (c+d x)}}}{a}-\frac {2 \left (2 a^2 A-3 a b B+5 A b^2\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}+\frac {-\frac {2 \left (2 a^2 A-3 a b B+5 A b^2\right )}{3 a d \tan ^{\frac {3}{2}}(c+d x)}-\frac {\frac {\frac {4 a^3 \left (\frac {1}{2} \left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} \left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {2 b^{5/2} \left (-7 a^3 B+9 a^2 A b-3 a b^2 B+5 A b^3\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d \left (a^2+b^2\right )}}{a}-\frac {2 \left (-2 a^3 B+4 a^2 A b-3 a b^2 B+5 A b^3\right )}{a d \sqrt {\tan (c+d x)}}}{a}}{2 a \left (a^2+b^2\right )}\) |
(-((((-2*b^(5/2)*(9*a^2*A*b + 5*A*b^3 - 7*a^3*B - 3*a*b^2*B)*ArcTan[(Sqrt[ b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*(a^2 + b^2)*d) + (4*a^3*(-1/2*(( 2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*(-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[ c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2])) + ( (a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*(-1/2*Log[1 - Sqrt[2]*Sqrt[Tan [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*(4*a^2*A*b + 5*A*b ^3 - 2*a^3*B - 3*a*b^2*B))/(a*d*Sqrt[Tan[c + d*x]]))/a) - (2*(2*a^2*A + 5* A*b^2 - 3*a*b*B))/(3*a*d*Tan[c + d*x]^(3/2)))/(2*a*(a^2 + b^2)) + (b*(A*b - a*B))/(a*(a^2 + b^2)*d*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x]))
3.5.9.3.1 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 = 373, normalized size of antiderivative = 0.76
| method | result | size |
| derivativedivides | \(\frac {\frac {2 b^{3} \left (\frac {\left (\frac {1}{2} A \,a^{2} b +\frac {1}{2} A \,b^{3}-\frac {1}{2} B \,a^{3}-\frac {1}{2} B a \,b^{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{a +b \tan \left (d x +c \right )}+\frac {\left (9 A \,a^{2} b +5 A \,b^{3}-7 B \,a^{3}-3 B a \,b^{2}\right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{3} \left (a^{2}+b^{2}\right )^{2}}-\frac {2 A}{3 a^{2} \tan \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 \left (-2 A b +B a \right )}{a^{3} \sqrt {\tan \left (d x +c \right )}}+\frac {\frac {\left (-A \,a^{2}+A \,b^{2}-2 B a b \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (2 A a b -B \,a^{2}+B \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(373\) |
| default | \(\frac {\frac {2 b^{3} \left (\frac {\left (\frac {1}{2} A \,a^{2} b +\frac {1}{2} A \,b^{3}-\frac {1}{2} B \,a^{3}-\frac {1}{2} B a \,b^{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{a +b \tan \left (d x +c \right )}+\frac {\left (9 A \,a^{2} b +5 A \,b^{3}-7 B \,a^{3}-3 B a \,b^{2}\right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{3} \left (a^{2}+b^{2}\right )^{2}}-\frac {2 A}{3 a^{2} \tan \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 \left (-2 A b +B a \right )}{a^{3} \sqrt {\tan \left (d x +c \right )}}+\frac {\frac {\left (-A \,a^{2}+A \,b^{2}-2 B a b \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (2 A a b -B \,a^{2}+B \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(373\) |
1/d*(2*b^3/a^3/(a^2+b^2)^2*((1/2*A*a^2*b+1/2*A*b^3-1/2*B*a^3-1/2*B*a*b^2)* tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))+1/2*(9*A*a^2*b+5*A*b^3-7*B*a^3-3*B*a*b^2 )/(a*b)^(1/2)*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2)))-2/3/a^2*A/tan(d*x+c) ^(3/2)-2*(-2*A*b+B*a)/a^3/tan(d*x+c)^(1/2)+2/(a^2+b^2)^2*(1/8*(-A*a^2+A*b^ 2-2*B*a*b)*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*arcta n(-1+2^(1/2)*tan(d*x+c)^(1/2)))+1/8*(2*A*a*b-B*a^2+B*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)))))
Leaf count of result is larger than twice the leaf count of optimal. 6095 vs. \(2 (450) = 900\).
Time = 75.10 (sec) , antiderivative size = 12216, normalized size of antiderivative = 24.78 \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\text {Timed out} \]
Time = 0.35 (sec) , antiderivative size = 449, normalized size of antiderivative = 0.91 \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=-\frac {\frac {12 \, {\left (7 \, B a^{3} b^{3} - 9 \, A a^{2} b^{4} + 3 \, B a b^{5} - 5 \, A b^{6}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \sqrt {a b}} + \frac {4 \, {\left (2 \, A a^{4} + 2 \, A a^{2} b^{2} + 3 \, {\left (2 \, B a^{3} b - 4 \, A a^{2} b^{2} + 3 \, B a b^{3} - 5 \, A b^{4}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (3 \, B a^{4} - 5 \, A a^{3} b + 3 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} \tan \left (d x + c\right )\right )}}{{\left (a^{5} b + a^{3} b^{3}\right )} \tan \left (d x + c\right )^{\frac {5}{2}} + {\left (a^{6} + a^{4} b^{2}\right )} \tan \left (d x + c\right )^{\frac {3}{2}}} + \frac {3 \, {\left (2 \, \sqrt {2} {\left ({\left (A + B\right )} a^{2} - 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left ({\left (A + B\right )} a^{2} - 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left ({\left (A - B\right )} a^{2} + 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left ({\left (A - B\right )} a^{2} + 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}}}{12 \, d} \]
-1/12*(12*(7*B*a^3*b^3 - 9*A*a^2*b^4 + 3*B*a*b^5 - 5*A*b^6)*arctan(b*sqrt( tan(d*x + c))/sqrt(a*b))/((a^7 + 2*a^5*b^2 + a^3*b^4)*sqrt(a*b)) + 4*(2*A* a^4 + 2*A*a^2*b^2 + 3*(2*B*a^3*b - 4*A*a^2*b^2 + 3*B*a*b^3 - 5*A*b^4)*tan( d*x + c)^2 + 2*(3*B*a^4 - 5*A*a^3*b + 3*B*a^2*b^2 - 5*A*a*b^3)*tan(d*x + c ))/((a^5*b + a^3*b^3)*tan(d*x + c)^(5/2) + (a^6 + a^4*b^2)*tan(d*x + c)^(3 /2)) + 3*(2*sqrt(2)*((A + B)*a^2 - 2*(A - B)*a*b - (A + B)*b^2)*arctan(1/2 *sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*((A + B)*a^2 - 2*(A - B)*a*b - (A + B)*b^2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c )))) + sqrt(2)*((A - B)*a^2 + 2*(A + B)*a*b - (A - B)*b^2)*log(sqrt(2)*sqr t(tan(d*x + c)) + tan(d*x + c) + 1) - sqrt(2)*((A - B)*a^2 + 2*(A + B)*a*b - (A - B)*b^2)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1))/(a^4 + 2*a^2*b^2 + b^4))/d
Timed out. \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\text {Timed out} \]
Time = 27.53 (sec) , antiderivative size = 24620, normalized size of antiderivative = 49.94 \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]
(log(80*A^5*a^24*b^20*d^4 - ((((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16* A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a* b^3*d^2 + 16*A^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4 *d^4 + 4*a^6*b^2*d^4))^(1/2)*(tan(c + d*x)^(1/2)*(9472*A^4*a^31*b^15*d^5 - 3040*A^4*a^23*b^23*d^5 - 9056*A^4*a^25*b^21*d^5 - 12352*A^4*a^27*b^19*d^5 - 4256*A^4*a^29*b^17*d^5 - 400*A^4*a^21*b^25*d^5 + 13760*A^4*a^33*b^13*d^ 5 + 7744*A^4*a^35*b^11*d^5 + 1968*A^4*a^37*b^9*d^5 + 224*A^4*a^39*b^7*d^5 + 32*A^4*a^41*b^5*d^5) + ((((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4 *a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a*b^3 *d^2 + 16*A^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^ 4 + 4*a^6*b^2*d^4))^(1/2)*(12928*A^3*a^25*b^23*d^6 - 800*A^3*a^21*b^27*d^6 - 2080*A^3*a^23*b^25*d^6 - ((tan(c + d*x)^(1/2)*(3200*A^2*a^22*b^28*d^7 + 33920*A^2*a^24*b^26*d^7 + 158208*A^2*a^26*b^24*d^7 + 425536*A^2*a^28*b^22 *d^7 + 727296*A^2*a^30*b^20*d^7 + 820672*A^2*a^32*b^18*d^7 + 615936*A^2*a^ 34*b^16*d^7 + 304256*A^2*a^36*b^14*d^7 + 98432*A^2*a^38*b^12*d^7 + 22016*A ^2*a^40*b^10*d^7 + 3072*A^2*a^42*b^8*d^7 - 704*A^2*a^44*b^6*d^7 - 512*A^2* a^46*b^4*d^7 - 64*A^2*a^48*b^2*d^7) + ((((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8 *d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2)*(1280*A*a^24*b^28*d^8 - (tan(c +...