\(\int \cot ^4(c+d x) (a+b \tan (c+d x))^{5/2} \, dx\) [526]

Optimal result

Integrand size = 23, antiderivative size = 237 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\frac {5 b \left (8 a^2-b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 \sqrt {a} d}-\frac {i (a-i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {i (a+i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {\left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 d}-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 d}-\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d} \] Output:

5/8*b*(8*a^2-b^2)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/a^(1/2)/d-I*(a-I 
*b)^(5/2)*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/d+I*(a+I*b)^(5/2)* 
arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))/d+1/8*(8*a^2-11*b^2)*cot(d*x 
+c)*(a+b*tan(d*x+c))^(1/2)/d-13/12*a*b*cot(d*x+c)^2*(a+b*tan(d*x+c))^(1/2) 
/d-1/3*a^2*cot(d*x+c)^3*(a+b*tan(d*x+c))^(1/2)/d
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 7.63 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.33 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=-\frac {\frac {48 b \operatorname {Hypergeometric2F1}\left (2,\frac {7}{2},\frac {9}{2},1+\frac {b \tan (c+d x)}{a}\right ) (a+b \tan (c+d x))^{7/2}}{a^2}-56 (i a+b) \left (-3 (a-i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )+\sqrt {a+b \tan (c+d x)} (4 a-3 i b+b \tan (c+d x))\right )+56 (i a-b) \left (-3 (a+i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+\sqrt {a+b \tan (c+d x)} (4 a+3 i b+b \tan (c+d x))\right )+\frac {7 \left (33 b^3+59 a b^2 \cot (c+d x)+34 a^2 b \cot ^2(c+d x)+8 a^3 \cot ^3(c+d x)+15 b^3 \text {arctanh}\left (\sqrt {1+\frac {b \tan (c+d x)}{a}}\right ) \sqrt {1+\frac {b \tan (c+d x)}{a}}\right )}{\sqrt {a+b \tan (c+d x)}}}{168 d} \] Input:

Integrate[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^(5/2),x]
 

Output:

-1/168*((48*b*Hypergeometric2F1[2, 7/2, 9/2, 1 + (b*Tan[c + d*x])/a]*(a + 
b*Tan[c + d*x])^(7/2))/a^2 - 56*(I*a + b)*(-3*(a - I*b)^(3/2)*ArcTanh[Sqrt 
[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + Sqrt[a + b*Tan[c + d*x]]*(4*a - (3*I 
)*b + b*Tan[c + d*x])) + 56*(I*a - b)*(-3*(a + I*b)^(3/2)*ArcTanh[Sqrt[a + 
 b*Tan[c + d*x]]/Sqrt[a + I*b]] + Sqrt[a + b*Tan[c + d*x]]*(4*a + (3*I)*b 
+ b*Tan[c + d*x])) + (7*(33*b^3 + 59*a*b^2*Cot[c + d*x] + 34*a^2*b*Cot[c + 
 d*x]^2 + 8*a^3*Cot[c + d*x]^3 + 15*b^3*ArcTanh[Sqrt[1 + (b*Tan[c + d*x])/ 
a]]*Sqrt[1 + (b*Tan[c + d*x])/a]))/Sqrt[a + b*Tan[c + d*x]])/d
 

Rubi [A] (warning: unable to verify)

Time = 2.20 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.02, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.957, Rules used = {3042, 4048, 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.

\(\displaystyle \int \cot ^4(c+d x) (a+b \tan (c+d x))^{5/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+b \tan (c+d x))^{5/2}}{\tan (c+d x)^4}dx\)

\(\Big \downarrow \) 4048

\(\displaystyle \frac {1}{3} \int \frac {\cot ^3(c+d x) \left (13 b a^2-6 \left (a^2-3 b^2\right ) \tan (c+d x) a-b \left (5 a^2-6 b^2\right ) \tan ^2(c+d x)\right )}{2 \sqrt {a+b \tan (c+d x)}}dx-\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \int \frac {\cot ^3(c+d x) \left (13 b a^2-6 \left (a^2-3 b^2\right ) \tan (c+d x) a-b \left (5 a^2-6 b^2\right ) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}}dx-\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \int \frac {13 b a^2-6 \left (a^2-3 b^2\right ) \tan (c+d x) a-b \left (5 a^2-6 b^2\right ) \tan (c+d x)^2}{\tan (c+d x)^3 \sqrt {a+b \tan (c+d x)}}dx-\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\)

\(\Big \downarrow \) 4132

\(\displaystyle \frac {1}{6} \left (-\frac {\int \frac {3 \cot ^2(c+d x) \left (13 b^2 \tan ^2(c+d x) a^2+\left (8 a^2-11 b^2\right ) a^2+8 b \left (3 a^2-b^2\right ) \tan (c+d x) a\right )}{2 \sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\right )-\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (-\frac {3 \int \frac {\cot ^2(c+d x) \left (13 b^2 \tan ^2(c+d x) a^2+\left (8 a^2-11 b^2\right ) a^2+8 b \left (3 a^2-b^2\right ) \tan (c+d x) a\right )}{\sqrt {a+b \tan (c+d x)}}dx}{4 a}-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\right )-\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (-\frac {3 \int \frac {13 b^2 \tan (c+d x)^2 a^2+\left (8 a^2-11 b^2\right ) a^2+8 b \left (3 a^2-b^2\right ) \tan (c+d x) a}{\tan (c+d x)^2 \sqrt {a+b \tan (c+d x)}}dx}{4 a}-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\right )-\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\)

\(\Big \downarrow \) 4132

\(\displaystyle \frac {1}{6} \left (-\frac {3 \left (-\frac {\int -\frac {\cot (c+d x) \left (-16 \left (a^2-3 b^2\right ) \tan (c+d x) a^3-b \left (8 a^2-11 b^2\right ) \tan ^2(c+d x) a^2+5 b \left (8 a^2-b^2\right ) a^2\right )}{2 \sqrt {a+b \tan (c+d x)}}dx}{a}-\frac {a \left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )}{4 a}-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\right )-\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (-\frac {3 \left (\frac {\int \frac {\cot (c+d x) \left (-16 \left (a^2-3 b^2\right ) \tan (c+d x) a^3-b \left (8 a^2-11 b^2\right ) \tan ^2(c+d x) a^2+5 b \left (8 a^2-b^2\right ) a^2\right )}{\sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {a \left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )}{4 a}-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\right )-\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (-\frac {3 \left (\frac {\int \frac {-16 \left (a^2-3 b^2\right ) \tan (c+d x) a^3-b \left (8 a^2-11 b^2\right ) \tan (c+d x)^2 a^2+5 b \left (8 a^2-b^2\right ) a^2}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {a \left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )}{4 a}-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\right )-\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\)

\(\Big \downarrow \) 4136

\(\displaystyle \frac {1}{6} \left (-\frac {3 \left (\frac {5 a^2 b \left (8 a^2-b^2\right ) \int \frac {\cot (c+d x) \left (\tan ^2(c+d x)+1\right )}{\sqrt {a+b \tan (c+d x)}}dx+\int -\frac {16 \left (\left (a^2-3 b^2\right ) a^3+b \left (3 a^2-b^2\right ) \tan (c+d x) a^2\right )}{\sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {a \left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )}{4 a}-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\right )-\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (-\frac {3 \left (\frac {5 a^2 b \left (8 a^2-b^2\right ) \int \frac {\cot (c+d x) \left (\tan ^2(c+d x)+1\right )}{\sqrt {a+b \tan (c+d x)}}dx-16 \int \frac {\left (a^2-3 b^2\right ) a^3+b \left (3 a^2-b^2\right ) \tan (c+d x) a^2}{\sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {a \left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )}{4 a}-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\right )-\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (-\frac {3 \left (\frac {5 a^2 b \left (8 a^2-b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-16 \int \frac {\left (a^2-3 b^2\right ) a^3+b \left (3 a^2-b^2\right ) \tan (c+d x) a^2}{\sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {a \left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )}{4 a}-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\right )-\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\)

\(\Big \downarrow \) 4022

\(\displaystyle -\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {a \left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {5 a^2 b \left (8 a^2-b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-16 \left (\frac {1}{2} a^2 (a-i b)^3 \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} a^2 (a+i b)^3 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx\right )}{2 a}\right )}{4 a}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {a \left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {5 a^2 b \left (8 a^2-b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-16 \left (\frac {1}{2} a^2 (a-i b)^3 \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} a^2 (a+i b)^3 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx\right )}{2 a}\right )}{4 a}\right )\)

\(\Big \downarrow \) 4020

\(\displaystyle -\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {a \left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {5 a^2 b \left (8 a^2-b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-16 \left (\frac {i a^2 (a-i b)^3 \int -\frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}-\frac {i a^2 (a+i b)^3 \int -\frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}\right )}{2 a}\right )}{4 a}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {a \left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {5 a^2 b \left (8 a^2-b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-16 \left (\frac {i a^2 (a+i b)^3 \int \frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}-\frac {i a^2 (a-i b)^3 \int \frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}\right )}{2 a}\right )}{4 a}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {a \left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {5 a^2 b \left (8 a^2-b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-16 \left (\frac {a^2 (a-i b)^3 \int \frac {1}{\frac {i \tan ^2(c+d x)}{b}+\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}+\frac {a^2 (a+i b)^3 \int \frac {1}{-\frac {i \tan ^2(c+d x)}{b}-\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}\right )}{2 a}\right )}{4 a}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {a \left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {5 a^2 b \left (8 a^2-b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-16 \left (\frac {a^2 (a-i b)^{5/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}+\frac {a^2 (a+i b)^{5/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\right )}{2 a}\right )}{4 a}\right )\)

\(\Big \downarrow \) 4117

\(\displaystyle -\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {a \left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {\frac {5 a^2 b \left (8 a^2-b^2\right ) \int \frac {\cot (c+d x)}{\sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{d}-16 \left (\frac {a^2 (a-i b)^{5/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}+\frac {a^2 (a+i b)^{5/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\right )}{2 a}\right )}{4 a}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {a \left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {\frac {10 a^2 \left (8 a^2-b^2\right ) \int \frac {1}{\frac {a+b \tan (c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \tan (c+d x)}}{d}-16 \left (\frac {a^2 (a-i b)^{5/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}+\frac {a^2 (a+i b)^{5/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\right )}{2 a}\right )}{4 a}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{6} \left (-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}-\frac {3 \left (-\frac {a \left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {-\frac {10 a^{3/2} b \left (8 a^2-b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}-16 \left (\frac {a^2 (a-i b)^{5/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}+\frac {a^2 (a+i b)^{5/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\right )}{2 a}\right )}{4 a}\right )\)

Input:

Int[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^(5/2),x]
 

Output:

-1/3*(a^2*Cot[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]])/d + ((-13*a*b*Cot[c + d 
*x]^2*Sqrt[a + b*Tan[c + d*x]])/(2*d) - (3*((-16*((a^2*(a - I*b)^(5/2)*Arc 
Tan[Tan[c + d*x]/Sqrt[a - I*b]])/d + (a^2*(a + I*b)^(5/2)*ArcTan[Tan[c + d 
*x]/Sqrt[a + I*b]])/d) - (10*a^(3/2)*b*(8*a^2 - b^2)*ArcTanh[Sqrt[a + b*Ta 
n[c + d*x]]/Sqrt[a]])/d)/(2*a) - (a*(8*a^2 - 11*b^2)*Cot[c + d*x]*Sqrt[a + 
 b*Tan[c + d*x]])/d))/(4*a))/6
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

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]
 

rule 221

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4020

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]
 

rule 4022

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]
 

rule 4048

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m 
 - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Simp[1 
/(d*(n + 1)*(c^2 + d^2))   Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + 
f*x])^(n + 1)*Simp[a^2*d*(b*d*(m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c 
*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d) 
*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*( 
n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[ 
b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 2] && LtQ 
[n, -1] && IntegerQ[2*m]
 

rule 4117

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]
 

rule 4132

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])))
 

rule 4136

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]
 

Maple [B] (warning: unable to verify)

result has leaf size over 500,000. Avoiding possible recursion issues.

Time = 48.35 (sec) , antiderivative size = 1543287, normalized size of antiderivative = 6511.76

\[\text {output too large to display}\]

Input:

int(cot(d*x+c)^4*(a+b*tan(d*x+c))^(5/2),x)
                                                                                    
                                                                                    
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1242 vs. \(2 (193) = 386\).

Time = 0.31 (sec) , antiderivative size = 2504, normalized size of antiderivative = 10.57 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Too large to display} \] Input:

integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^(5/2),x, algorithm="fricas")
 

Output:

[-1/48*(24*a*d*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 + d^2*sqrt(-(25*a^8*b^2 - 
 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))/d^2)*log((5*a^8*b - 
14*a^4*b^5 - 8*a^2*b^7 + b^9)*sqrt(b*tan(d*x + c) + a) + ((a^2 - b^2)*d^3* 
sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) + 
2*(5*a^5*b^2 - 10*a^3*b^4 + a*b^6)*d)*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 + 
d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4 
))/d^2))*tan(d*x + c)^3 - 24*a*d*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 + d^2*s 
qrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))/d^ 
2)*log((5*a^8*b - 14*a^4*b^5 - 8*a^2*b^7 + b^9)*sqrt(b*tan(d*x + c) + a) - 
 ((a^2 - b^2)*d^3*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b 
^8 + b^10)/d^4) + 2*(5*a^5*b^2 - 10*a^3*b^4 + a*b^6)*d)*sqrt(-(a^5 - 10*a^ 
3*b^2 + 5*a*b^4 + d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a 
^2*b^8 + b^10)/d^4))/d^2))*tan(d*x + c)^3 - 24*a*d*sqrt(-(a^5 - 10*a^3*b^2 
 + 5*a*b^4 - d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^ 
8 + b^10)/d^4))/d^2)*log((5*a^8*b - 14*a^4*b^5 - 8*a^2*b^7 + b^9)*sqrt(b*t 
an(d*x + c) + a) + ((a^2 - b^2)*d^3*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110* 
a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) - 2*(5*a^5*b^2 - 10*a^3*b^4 + a*b^6)*d)* 
sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 - d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 
110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))/d^2))*tan(d*x + c)^3 + 24*a*d*sqrt( 
-(a^5 - 10*a^3*b^2 + 5*a*b^4 - d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 11...
 

Sympy [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)**4*(a+b*tan(d*x+c))**(5/2),x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^(5/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F(-2)]

Exception generated. \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^(5/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 8.63 (sec) , antiderivative size = 4455, normalized size of antiderivative = 18.80 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Too large to display} \] Input:

int(cot(c + d*x)^4*(a + b*tan(c + d*x))^(5/2),x)
 

Output:

(((5*a*b^3)/3 - 2*a^3*b)*(a + b*tan(c + d*x))^(3/2) + (a^4*b - (5*a^2*b^3) 
/8)*(a + b*tan(c + d*x))^(1/2) + (a^2*b - (11*b^3)/8)*(a + b*tan(c + d*x)) 
^(5/2))/(d*(a + b*tan(c + d*x))^3 - a^3*d - 3*a*d*(a + b*tan(c + d*x))^2 + 
 3*a^2*d*(a + b*tan(c + d*x))) - log((5*b^9*(a^2 + b^2)^3*(11*b^8 - 128*a^ 
8 + 15*a^2*b^6 - 896*a^4*b^4 + 592*a^6*b^2))/(8*d^5) - ((-((-b^2*d^4*(5*a^ 
4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d 
^4)^(1/2)*(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5 
*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a 
^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*(((- 
((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 1 
0*a^3*b^2*d^2)/d^4)^(1/2)*((32*b^9*(32*a^4 - 5*b^4 + 27*a^2*b^2))/d - 128* 
b^8*(-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d 
^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*(3*a^2 + 2*b^2)*(a + b*tan(c + d*x))^(1/2) 
))/2 - (a*b^8*(a + b*tan(c + d*x))^(1/2)*(320*a^6 + 1191*b^6 + 80*a^2*b^4 
- 4864*a^4*b^2))/d^2))/2 - (a*b^9*(407*b^8 - 736*a^8 - 3225*a^2*b^6 + 1088 
*a^4*b^4 + 3984*a^6*b^2))/d^3))/2 - (b^8*(a + b*tan(c + d*x))^(1/2)*(128*a 
^12 + 153*b^12 - 7*a^2*b^10 + 9895*a^4*b^8 - 27465*a^6*b^6 + 26320*a^8*b^4 
 - 832*a^10*b^2))/(4*d^4)))/2)*(-((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6 
*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 1 
0*a^3*b^2*d^2)/(4*d^4))^(1/2) - log((5*b^9*(a^2 + b^2)^3*(11*b^8 - 128*...
 

Reduce [F]

\[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\int \cot \left (d x +c \right )^{4} \left (a +\tan \left (d x +c \right ) b \right )^{\frac {5}{2}}d x \] Input:

int(cot(d*x+c)^4*(a+b*tan(d*x+c))^(5/2),x)
 

Output:

int(cot(d*x+c)^4*(a+b*tan(d*x+c))^(5/2),x)