Integrand size = 33, antiderivative size = 364 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\frac {\left (8 a^2 A-35 A b^2+20 a b B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{9/2} d}-\frac {(A-i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{5/2} d}-\frac {(A+i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{5/2} d}+\frac {b \left (27 a^2 A b+35 A b^3-12 a^3 B-20 a b^2 B\right )}{12 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac {(7 A b-4 a B) \cot (c+d x)}{4 a^2 d (a+b \tan (c+d x))^{3/2}}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}+\frac {b \left (11 a^4 A b+62 a^2 A b^3+35 A b^5-4 a^5 B-40 a^3 b^2 B-20 a b^4 B\right )}{4 a^4 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}} \]
1/4*(8*A*a^2-35*A*b^2+20*B*a*b)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/a^ (9/2)/d-(A-I*B)*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/(a-I*b)^(5/2 )/d-(A+I*B)*arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))/(a+I*b)^(5/2)/d+ 1/4*b*(11*A*a^4*b+62*A*a^2*b^3+35*A*b^5-4*B*a^5-40*B*a^3*b^2-20*B*a*b^4)/a ^4/(a^2+b^2)^2/d/(a+b*tan(d*x+c))^(1/2)+1/12*b*(27*A*a^2*b+35*A*b^3-12*B*a ^3-20*B*a*b^2)/a^3/(a^2+b^2)/d/(a+b*tan(d*x+c))^(3/2)+1/4*(7*A*b-4*B*a)*co t(d*x+c)/a^2/d/(a+b*tan(d*x+c))^(3/2)-1/2*A*cot(d*x+c)^2/a/d/(a+b*tan(d*x+ c))^(3/2)
Time = 6.36 (sec) , antiderivative size = 593, normalized size of antiderivative = 1.63 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {(7 A b-4 a B) \cot (c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}-\frac {\frac {2 \left (\frac {1}{4} b^2 \left (-8 a^2 A+35 A b^2-20 a b B\right )-a \left (-2 a^2 b B-\frac {5}{4} a b (7 A b-4 a B)\right )\right )}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac {2 \left (\frac {2 \left (\frac {3 \left (a^2+b^2\right )^2 \left (8 a^2 A-35 A b^2+20 a b B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 \sqrt {a} d}+\frac {i \sqrt {a-i b} \left (-\frac {3}{2} i a^4 \left (a^2 A-A b^2+2 a b B\right )+\frac {3}{2} a^4 \left (2 a A b-a^2 B+b^2 B\right )\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(-a+i b) d}-\frac {i \sqrt {a+i b} \left (\frac {3}{2} i a^4 \left (a^2 A-A b^2+2 a b B\right )+\frac {3}{2} a^4 \left (2 a A b-a^2 B+b^2 B\right )\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(-a-i b) d}\right )}{a \left (a^2+b^2\right )}+\frac {2 \left (-\frac {3}{8} b^2 \left (a^2+b^2\right ) \left (8 a^2 A-35 A b^2+20 a b B\right )-a \left (3 a^3 b (A b-a B)-\frac {3}{8} a b \left (27 a^2 A b+35 A b^3-12 a^3 B-20 a b^2 B\right )\right )\right )}{a \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}\right )}{3 a \left (a^2+b^2\right )}}{a}}{2 a} \]
-1/2*(A*Cot[c + d*x]^2)/(a*d*(a + b*Tan[c + d*x])^(3/2)) - (-1/2*((7*A*b - 4*a*B)*Cot[c + d*x])/(a*d*(a + b*Tan[c + d*x])^(3/2)) - ((2*((b^2*(-8*a^2 *A + 35*A*b^2 - 20*a*b*B))/4 - a*(-2*a^2*b*B - (5*a*b*(7*A*b - 4*a*B))/4)) )/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) + (2*((2*((3*(a^2 + b^2)^ 2*(8*a^2*A - 35*A*b^2 + 20*a*b*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a] ])/(8*Sqrt[a]*d) + (I*Sqrt[a - I*b]*(((-3*I)/2)*a^4*(a^2*A - A*b^2 + 2*a*b *B) + (3*a^4*(2*a*A*b - a^2*B + b^2*B))/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x] ]/Sqrt[a - I*b]])/((-a + I*b)*d) - (I*Sqrt[a + I*b]*(((3*I)/2)*a^4*(a^2*A - A*b^2 + 2*a*b*B) + (3*a^4*(2*a*A*b - a^2*B + b^2*B))/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((-a - I*b)*d)))/(a*(a^2 + b^2)) + (2*((- 3*b^2*(a^2 + b^2)*(8*a^2*A - 35*A*b^2 + 20*a*b*B))/8 - a*(3*a^3*b*(A*b - a *B) - (3*a*b*(27*a^2*A*b + 35*A*b^3 - 12*a^3*B - 20*a*b^2*B))/8)))/(a*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])))/(3*a*(a^2 + b^2)))/a)/(2*a)
Time = 3.08 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.16, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.758, Rules used = {3042, 4092, 27, 3042, 4132, 27, 3042, 4132, 27, 3042, 4132, 27, 3042, 4136, 27, 3042, 4022, 3042, 4020, 25, 73, 221, 4117, 73, 221}
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.
| \(\displaystyle \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \tan (c+d x)}{\tan (c+d x)^3 (a+b \tan (c+d x))^{5/2}}dx\) |
| \(\Big \downarrow \) 4092 |
| \(\displaystyle -\frac {\int \frac {\cot ^2(c+d x) \left (7 A b \tan ^2(c+d x)+4 a A \tan (c+d x)+7 A b-4 a B\right )}{2 (a+b \tan (c+d x))^{5/2}}dx}{2 a}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\cot ^2(c+d x) \left (7 A b \tan ^2(c+d x)+4 a A \tan (c+d x)+7 A b-4 a B\right )}{(a+b \tan (c+d x))^{5/2}}dx}{4 a}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {7 A b \tan (c+d x)^2+4 a A \tan (c+d x)+7 A b-4 a B}{\tan (c+d x)^2 (a+b \tan (c+d x))^{5/2}}dx}{4 a}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle -\frac {-\frac {\int -\frac {\cot (c+d x) \left (8 A a^2+8 B \tan (c+d x) a^2+20 b B a-35 A b^2-5 b (7 A b-4 a B) \tan ^2(c+d x)\right )}{2 (a+b \tan (c+d x))^{5/2}}dx}{a}-\frac {(7 A b-4 a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}}{4 a}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {\cot (c+d x) \left (8 A a^2+8 B \tan (c+d x) a^2+20 b B a-35 A b^2-5 b (7 A b-4 a B) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^{5/2}}dx}{2 a}-\frac {(7 A b-4 a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}}{4 a}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {8 A a^2+8 B \tan (c+d x) a^2+20 b B a-35 A b^2-5 b (7 A b-4 a B) \tan (c+d x)^2}{\tan (c+d x) (a+b \tan (c+d x))^{5/2}}dx}{2 a}-\frac {(7 A b-4 a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}}{4 a}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle -\frac {\frac {\frac {2 \int \frac {3 \cot (c+d x) \left (-8 (A b-a B) \tan (c+d x) a^3-b \left (-12 B a^3+27 A b a^2-20 b^2 B a+35 A b^3\right ) \tan ^2(c+d x)+\left (a^2+b^2\right ) \left (8 A a^2+20 b B a-35 A b^2\right )\right )}{2 (a+b \tan (c+d x))^{3/2}}dx}{3 a \left (a^2+b^2\right )}-\frac {2 b \left (-12 a^3 B+27 a^2 A b-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}}{2 a}-\frac {(7 A b-4 a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}}{4 a}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {\cot (c+d x) \left (-8 (A b-a B) \tan (c+d x) a^3-b \left (-12 B a^3+27 A b a^2-20 b^2 B a+35 A b^3\right ) \tan ^2(c+d x)+\left (a^2+b^2\right ) \left (8 A a^2+20 b B a-35 A b^2\right )\right )}{(a+b \tan (c+d x))^{3/2}}dx}{a \left (a^2+b^2\right )}-\frac {2 b \left (-12 a^3 B+27 a^2 A b-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}}{2 a}-\frac {(7 A b-4 a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}}{4 a}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {-8 (A b-a B) \tan (c+d x) a^3-b \left (-12 B a^3+27 A b a^2-20 b^2 B a+35 A b^3\right ) \tan (c+d x)^2+\left (a^2+b^2\right ) \left (8 A a^2+20 b B a-35 A b^2\right )}{\tan (c+d x) (a+b \tan (c+d x))^{3/2}}dx}{a \left (a^2+b^2\right )}-\frac {2 b \left (-12 a^3 B+27 a^2 A b-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}}{2 a}-\frac {(7 A b-4 a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}}{4 a}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle -\frac {\frac {\frac {\frac {2 \int \frac {\cot (c+d x) \left (-8 \left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x) a^4-b \left (-4 B a^5+11 A b a^4-40 b^2 B a^3+62 A b^3 a^2-20 b^4 B a+35 A b^5\right ) \tan ^2(c+d x)+\left (a^2+b^2\right )^2 \left (8 A a^2+20 b B a-35 A b^2\right )\right )}{2 \sqrt {a+b \tan (c+d x)}}dx}{a \left (a^2+b^2\right )}-\frac {2 b \left (-4 a^5 B+11 a^4 A b-40 a^3 b^2 B+62 a^2 A b^3-20 a b^4 B+35 A b^5\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{a \left (a^2+b^2\right )}-\frac {2 b \left (-12 a^3 B+27 a^2 A b-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}}{2 a}-\frac {(7 A b-4 a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}}{4 a}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\int \frac {\cot (c+d x) \left (-8 \left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x) a^4-b \left (-4 B a^5+11 A b a^4-40 b^2 B a^3+62 A b^3 a^2-20 b^4 B a+35 A b^5\right ) \tan ^2(c+d x)+\left (a^2+b^2\right )^2 \left (8 A a^2+20 b B a-35 A b^2\right )\right )}{\sqrt {a+b \tan (c+d x)}}dx}{a \left (a^2+b^2\right )}-\frac {2 b \left (-4 a^5 B+11 a^4 A b-40 a^3 b^2 B+62 a^2 A b^3-20 a b^4 B+35 A b^5\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{a \left (a^2+b^2\right )}-\frac {2 b \left (-12 a^3 B+27 a^2 A b-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}}{2 a}-\frac {(7 A b-4 a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}}{4 a}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\int \frac {-8 \left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x) a^4-b \left (-4 B a^5+11 A b a^4-40 b^2 B a^3+62 A b^3 a^2-20 b^4 B a+35 A b^5\right ) \tan (c+d x)^2+\left (a^2+b^2\right )^2 \left (8 A a^2+20 b B a-35 A b^2\right )}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx}{a \left (a^2+b^2\right )}-\frac {2 b \left (-4 a^5 B+11 a^4 A b-40 a^3 b^2 B+62 a^2 A b^3-20 a b^4 B+35 A b^5\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{a \left (a^2+b^2\right )}-\frac {2 b \left (-12 a^3 B+27 a^2 A b-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}}{2 a}-\frac {(7 A b-4 a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}}{4 a}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\left (a^2+b^2\right )^2 \left (8 a^2 A+20 a b B-35 A b^2\right ) \int \frac {\cot (c+d x) \left (\tan ^2(c+d x)+1\right )}{\sqrt {a+b \tan (c+d x)}}dx+\int -\frac {8 \left (\left (-B a^2+2 A b a+b^2 B\right ) a^4+\left (A a^2+2 b B a-A b^2\right ) \tan (c+d x) a^4\right )}{\sqrt {a+b \tan (c+d x)}}dx}{a \left (a^2+b^2\right )}-\frac {2 b \left (-4 a^5 B+11 a^4 A b-40 a^3 b^2 B+62 a^2 A b^3-20 a b^4 B+35 A b^5\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{a \left (a^2+b^2\right )}-\frac {2 b \left (-12 a^3 B+27 a^2 A b-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}}{2 a}-\frac {(7 A b-4 a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}}{4 a}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\left (a^2+b^2\right )^2 \left (8 a^2 A+20 a b B-35 A b^2\right ) \int \frac {\cot (c+d x) \left (\tan ^2(c+d x)+1\right )}{\sqrt {a+b \tan (c+d x)}}dx-8 \int \frac {\left (-B a^2+2 A b a+b^2 B\right ) a^4+\left (A a^2+2 b B a-A b^2\right ) \tan (c+d x) a^4}{\sqrt {a+b \tan (c+d x)}}dx}{a \left (a^2+b^2\right )}-\frac {2 b \left (-4 a^5 B+11 a^4 A b-40 a^3 b^2 B+62 a^2 A b^3-20 a b^4 B+35 A b^5\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{a \left (a^2+b^2\right )}-\frac {2 b \left (-12 a^3 B+27 a^2 A b-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}}{2 a}-\frac {(7 A b-4 a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}}{4 a}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\left (a^2+b^2\right )^2 \left (8 a^2 A+20 a b B-35 A b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-8 \int \frac {\left (-B a^2+2 A b a+b^2 B\right ) a^4+\left (A a^2+2 b B a-A b^2\right ) \tan (c+d x) a^4}{\sqrt {a+b \tan (c+d x)}}dx}{a \left (a^2+b^2\right )}-\frac {2 b \left (-4 a^5 B+11 a^4 A b-40 a^3 b^2 B+62 a^2 A b^3-20 a b^4 B+35 A b^5\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{a \left (a^2+b^2\right )}-\frac {2 b \left (-12 a^3 B+27 a^2 A b-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}}{2 a}-\frac {(7 A b-4 a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}}{4 a}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle -\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {(7 A b-4 a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b \left (-12 a^3 B+27 a^2 A b-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b \left (-4 a^5 B+11 a^4 A b-40 a^3 b^2 B+62 a^2 A b^3-20 a b^4 B+35 A b^5\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (a^2+b^2\right )^2 \left (8 a^2 A+20 a b B-35 A b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-8 \left (\frac {1}{2} a^4 (a-i b)^2 (-B+i A) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} a^4 (a+i b)^2 (B+i A) \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx\right )}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {(7 A b-4 a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b \left (-12 a^3 B+27 a^2 A b-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b \left (-4 a^5 B+11 a^4 A b-40 a^3 b^2 B+62 a^2 A b^3-20 a b^4 B+35 A b^5\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (a^2+b^2\right )^2 \left (8 a^2 A+20 a b B-35 A b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-8 \left (\frac {1}{2} a^4 (a-i b)^2 (-B+i A) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} a^4 (a+i b)^2 (B+i A) \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx\right )}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle -\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {(7 A b-4 a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b \left (-12 a^3 B+27 a^2 A b-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b \left (-4 a^5 B+11 a^4 A b-40 a^3 b^2 B+62 a^2 A b^3-20 a b^4 B+35 A b^5\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (a^2+b^2\right )^2 \left (8 a^2 A+20 a b B-35 A b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-8 \left (-\frac {i a^4 (a+i b)^2 (B+i A) \int -\frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}-\frac {i a^4 (a-i b)^2 (-B+i A) \int -\frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}\right )}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {(7 A b-4 a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b \left (-12 a^3 B+27 a^2 A b-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b \left (-4 a^5 B+11 a^4 A b-40 a^3 b^2 B+62 a^2 A b^3-20 a b^4 B+35 A b^5\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (a^2+b^2\right )^2 \left (8 a^2 A+20 a b B-35 A b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-8 \left (\frac {i a^4 (a+i b)^2 (B+i A) \int \frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}+\frac {i a^4 (a-i b)^2 (-B+i A) \int \frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}\right )}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {(7 A b-4 a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b \left (-12 a^3 B+27 a^2 A b-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b \left (-4 a^5 B+11 a^4 A b-40 a^3 b^2 B+62 a^2 A b^3-20 a b^4 B+35 A b^5\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (a^2+b^2\right )^2 \left (8 a^2 A+20 a b B-35 A b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-8 \left (\frac {a^4 (a-i b)^2 (-B+i A) \int \frac {1}{-\frac {i \tan ^2(c+d x)}{b}-\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}-\frac {a^4 (a+i b)^2 (B+i A) \int \frac {1}{\frac {i \tan ^2(c+d x)}{b}+\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}\right )}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {(7 A b-4 a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b \left (-12 a^3 B+27 a^2 A b-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b \left (-4 a^5 B+11 a^4 A b-40 a^3 b^2 B+62 a^2 A b^3-20 a b^4 B+35 A b^5\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (a^2+b^2\right )^2 \left (8 a^2 A+20 a b B-35 A b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-8 \left (\frac {a^4 (a-i b)^2 (-B+i A) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}-\frac {a^4 (a+i b)^2 (B+i A) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}\right )}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle -\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {(7 A b-4 a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b \left (-12 a^3 B+27 a^2 A b-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b \left (-4 a^5 B+11 a^4 A b-40 a^3 b^2 B+62 a^2 A b^3-20 a b^4 B+35 A b^5\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {\left (a^2+b^2\right )^2 \left (8 a^2 A+20 a b B-35 A b^2\right ) \int \frac {\cot (c+d x)}{\sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{d}-8 \left (\frac {a^4 (a-i b)^2 (-B+i A) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}-\frac {a^4 (a+i b)^2 (B+i A) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}\right )}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {(7 A b-4 a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b \left (-12 a^3 B+27 a^2 A b-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b \left (-4 a^5 B+11 a^4 A b-40 a^3 b^2 B+62 a^2 A b^3-20 a b^4 B+35 A b^5\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {2 \left (a^2+b^2\right )^2 \left (8 a^2 A+20 a b B-35 A b^2\right ) \int \frac {1}{\frac {a+b \tan (c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \tan (c+d x)}}{b d}-8 \left (\frac {a^4 (a-i b)^2 (-B+i A) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}-\frac {a^4 (a+i b)^2 (B+i A) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}\right )}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {(7 A b-4 a B) \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b \left (-12 a^3 B+27 a^2 A b-20 a b^2 B+35 A b^3\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b \left (-4 a^5 B+11 a^4 A b-40 a^3 b^2 B+62 a^2 A b^3-20 a b^4 B+35 A b^5\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {-\frac {2 \left (a^2+b^2\right )^2 \left (8 a^2 A+20 a b B-35 A b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}-8 \left (\frac {a^4 (a-i b)^2 (-B+i A) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}-\frac {a^4 (a+i b)^2 (B+i A) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}\right )}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{2 a}}{4 a}\) |
-1/2*(A*Cot[c + d*x]^2)/(a*d*(a + b*Tan[c + d*x])^(3/2)) - (-(((7*A*b - 4* a*B)*Cot[c + d*x])/(a*d*(a + b*Tan[c + d*x])^(3/2))) + ((-2*b*(27*a^2*A*b + 35*A*b^3 - 12*a^3*B - 20*a*b^2*B))/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x ])^(3/2)) + ((-8*(-((a^4*(a + I*b)^2*(I*A + B)*ArcTan[Tan[c + d*x]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d)) + (a^4*(a - I*b)^2*(I*A - B)*ArcTan[Tan[c + d* x]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d)) - (2*(a^2 + b^2)^2*(8*a^2*A - 35*A*b ^2 + 20*a*b*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d))/(a* (a^2 + b^2)) - (2*b*(11*a^4*A*b + 62*a^2*A*b^3 + 35*A*b^5 - 4*a^5*B - 40*a ^3*b^2*B - 20*a*b^4*B))/(a*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]]))/(a*(a^ 2 + b^2)))/(2*a))/(4*a)
3.4.64.3.1 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/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f) Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2 Int[(a + b*Tan[e + f*x])^m*( 1 - I*Tan[e + f*x]), x], x] + Simp[(c - I*d)/2 Int[(a + b*Tan[e + f*x])^m *(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !IntegerQ[m]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(13091\) vs. \(2(322)=644\).
Time = 0.27 (sec) , antiderivative size = 13092, normalized size of antiderivative = 35.97
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(13092\) |
| default | \(\text {Expression too large to display}\) | \(13092\) |
Leaf count of result is larger than twice the leaf count of optimal. 7654 vs. \(2 (316) = 632\).
Time = 111.45 (sec) , antiderivative size = 15325, normalized size of antiderivative = 42.10 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
\[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \cot ^{3}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
Timed out. \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Timed out} \]
Time = 14.67 (sec) , antiderivative size = 71314, normalized size of antiderivative = 195.92 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
(((a + b*tan(c + d*x))^3*(35*A*b^6 + 62*A*a^2*b^4 + 11*A*a^4*b^2 - 40*B*a^ 3*b^3 - 20*B*a*b^5 - 4*B*a^5*b))/(4*(a^8 + a^4*b^4 + 2*a^6*b^2)) - ((a + b *tan(c + d*x))^2*(175*A*b^6 + 310*A*a^2*b^4 + 39*A*a^4*b^2 - 208*B*a^3*b^3 - 100*B*a*b^5 - 12*B*a^5*b))/(12*(a^7 + a^3*b^4 + 2*a^5*b^2)) + (2*(A*b^4 - B*a*b^3))/(3*a*(a^2 + b^2)) + (2*(a + b*tan(c + d*x))*(7*A*b^6 + 13*A*a ^2*b^4 - 10*B*a^3*b^3 - 4*B*a*b^5))/(3*a^2*(a^4 + b^4 + 2*a^2*b^2)))/(d*(a + b*tan(c + d*x))^(7/2) - 2*a*d*(a + b*tan(c + d*x))^(5/2) + a^2*d*(a + b *tan(c + d*x))^(3/2)) + atan((((a + b*tan(c + d*x))^(1/2)*(321126400*A^4*a ^28*b^44*d^5 + 2422210560*A^4*a^30*b^42*d^5 + 1411383296*A^4*a^32*b^40*d^5 - 54989422592*A^4*a^34*b^38*d^5 - 325864914944*A^4*a^36*b^36*d^5 - 101129 4928896*A^4*a^38*b^34*d^5 - 2054783238144*A^4*a^40*b^32*d^5 - 292349070540 8*A^4*a^42*b^30*d^5 - 2962565365760*A^4*a^44*b^28*d^5 - 2094150975488*A^4* a^46*b^26*d^5 - 943762440192*A^4*a^48*b^24*d^5 - 175655354368*A^4*a^50*b^2 2*d^5 + 74523344896*A^4*a^52*b^20*d^5 + 62081990656*A^4*a^54*b^18*d^5 + 17 307795456*A^4*a^56*b^16*d^5 + 1629487104*A^4*a^58*b^14*d^5 + 44302336*A^4* a^60*b^12*d^5 + 104857600*A^4*a^62*b^10*d^5 + 25165824*A^4*a^64*b^8*d^5 - 104857600*B^4*a^30*b^42*d^5 - 838860800*B^4*a^32*b^40*d^5 - 838860800*B^4* a^34*b^38*d^5 + 17624465408*B^4*a^36*b^36*d^5 + 114621939712*B^4*a^38*b^34 *d^5 + 382445027328*B^4*a^40*b^32*d^5 + 842753114112*B^4*a^42*b^30*d^5 + 1 327925035008*B^4*a^44*b^28*d^5 + 1546246946816*B^4*a^46*b^26*d^5 + 1344...