\(\int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx\) [594]

Optimal result

Integrand size = 33, antiderivative size = 236 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=-\frac {(a (A-B)+b (A+B)) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a (A-B)+b (A+B)) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {2 b^{3/2} (A b-a B) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{3/2} \left (a^2+b^2\right ) d}-\frac {(b (A-B)-a (A+B)) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {2 A \sqrt {\cot (c+d x)}}{a d} \] Output:

1/2*(a*(A-B)+b*(A+B))*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/(a^2+b^2 
)/d+1/2*(a*(A-B)+b*(A+B))*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/(a^2+ 
b^2)/d+2*b^(3/2)*(A*b-B*a)*arctan(a^(1/2)*cot(d*x+c)^(1/2)/b^(1/2))/a^(3/2 
)/(a^2+b^2)/d-1/2*(b*(A-B)-a*(A+B))*arctanh(2^(1/2)*cot(d*x+c)^(1/2)/(1+co 
t(d*x+c)))*2^(1/2)/(a^2+b^2)/d-2*A*cot(d*x+c)^(1/2)/a/d
 

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.06 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {\sqrt {\cot (c+d x)} \left (\frac {2 \sqrt {2} (a (A-B)+b (A+B)) \left (\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )\right )}{a^2+b^2}+\frac {8 b^{3/2} (-A b+a B) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} \left (a^2+b^2\right )}-\frac {\sqrt {2} (b (-A+B)+a (A+B)) \left (\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )}{a^2+b^2}-\frac {8 A}{a \sqrt {\tan (c+d x)}}\right ) \sqrt {\tan (c+d x)}}{4 d} \] Input:

Integrate[(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x 
]
 

Output:

(Sqrt[Cot[c + d*x]]*((2*Sqrt[2]*(a*(A - B) + b*(A + B))*(ArcTan[1 - Sqrt[2 
]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/(a^2 + b^ 
2) + (8*b^(3/2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a] 
])/(a^(3/2)*(a^2 + b^2)) - (Sqrt[2]*(b*(-A + B) + a*(A + B))*(Log[1 - Sqrt 
[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x] 
] + Tan[c + d*x]]))/(a^2 + b^2) - (8*A)/(a*Sqrt[Tan[c + d*x]]))*Sqrt[Tan[c 
 + d*x]])/(4*d)
 

Rubi [A] (warning: unable to verify)

Time = 1.42 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.06, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 4064, 3042, 4090, 27, 3042, 4136, 3042, 4017, 25, 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.

\(\displaystyle \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cot (c+d x)^{3/2} (A+B \tan (c+d x))}{a+b \tan (c+d x)}dx\)

\(\Big \downarrow \) 4064

\(\displaystyle \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A \cot (c+d x)+B)}{a \cot (c+d x)+b}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (B-A \tan \left (c+d x+\frac {\pi }{2}\right )\right )}{b-a \tan \left (c+d x+\frac {\pi }{2}\right )}dx\)

\(\Big \downarrow \) 4090

\(\displaystyle -\frac {2 \int \frac {(A b-a B) \cot ^2(c+d x)+a A \cot (c+d x)+A b}{2 \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a}-\frac {2 A \sqrt {\cot (c+d x)}}{a d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {(A b-a B) \cot ^2(c+d x)+a A \cot (c+d x)+A b}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a}-\frac {2 A \sqrt {\cot (c+d x)}}{a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \frac {(A b-a B) \tan \left (c+d x+\frac {\pi }{2}\right )^2-a A \tan \left (c+d x+\frac {\pi }{2}\right )+A b}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}-\frac {2 A \sqrt {\cot (c+d x)}}{a d}\)

\(\Big \downarrow \) 4136

\(\displaystyle -\frac {\frac {b^2 (A b-a B) \int \frac {\cot ^2(c+d x)+1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a^2+b^2}+\frac {\int \frac {a (a A+b B)+a (A b-a B) \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx}{a^2+b^2}}{a}-\frac {2 A \sqrt {\cot (c+d x)}}{a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {b^2 (A b-a B) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}+\frac {\int \frac {a (a A+b B)-a (A b-a B) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}}{a}-\frac {2 A \sqrt {\cot (c+d x)}}{a d}\)

\(\Big \downarrow \) 4017

\(\displaystyle -\frac {\frac {b^2 (A b-a B) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}+\frac {2 \int -\frac {a (a A+b B+(A b-a B) \cot (c+d x))}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{a}-\frac {2 A \sqrt {\cot (c+d x)}}{a d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {b^2 (A b-a B) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 \int \frac {a (a A+b B+(A b-a B) \cot (c+d x))}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{a}-\frac {2 A \sqrt {\cot (c+d x)}}{a d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {b^2 (A b-a B) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 a \int \frac {a A+b B+(A b-a B) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{a}-\frac {2 A \sqrt {\cot (c+d x)}}{a d}\)

\(\Big \downarrow \) 1482

\(\displaystyle -\frac {\frac {b^2 (A b-a B) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 a \left (\frac {1}{2} (a (A-B)+b (A+B)) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (b (A-B)-a (A+B)) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 A \sqrt {\cot (c+d x)}}{a d}\)

\(\Big \downarrow \) 1476

\(\displaystyle -\frac {\frac {b^2 (A b-a B) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 a \left (\frac {1}{2} (a (A-B)+b (A+B)) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\frac {1}{2} (b (A-B)-a (A+B)) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 A \sqrt {\cot (c+d x)}}{a d}\)

\(\Big \downarrow \) 1082

\(\displaystyle -\frac {\frac {b^2 (A b-a B) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 a \left (\frac {1}{2} (a (A-B)+b (A+B)) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} (b (A-B)-a (A+B)) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 A \sqrt {\cot (c+d x)}}{a d}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {\frac {b^2 (A b-a B) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 a \left (\frac {1}{2} (a (A-B)+b (A+B)) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (b (A-B)-a (A+B)) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 A \sqrt {\cot (c+d x)}}{a d}\)

\(\Big \downarrow \) 1479

\(\displaystyle -\frac {\frac {b^2 (A b-a B) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 a \left (\frac {1}{2} (a (A-B)+b (A+B)) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (b (A-B)-a (A+B)) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 A \sqrt {\cot (c+d x)}}{a d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {b^2 (A b-a B) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 a \left (\frac {1}{2} (a (A-B)+b (A+B)) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (b (A-B)-a (A+B)) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 A \sqrt {\cot (c+d x)}}{a d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {b^2 (A b-a B) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 a \left (\frac {1}{2} (a (A-B)+b (A+B)) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (b (A-B)-a (A+B)) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 A \sqrt {\cot (c+d x)}}{a d}\)

\(\Big \downarrow \) 1103

\(\displaystyle -\frac {\frac {b^2 (A b-a B) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 a \left (\frac {1}{2} (a (A-B)+b (A+B)) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (b (A-B)-a (A+B)) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 A \sqrt {\cot (c+d x)}}{a d}\)

\(\Big \downarrow \) 4117

\(\displaystyle -\frac {\frac {b^2 (A b-a B) \int \frac {1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}d(-\cot (c+d x))}{d \left (a^2+b^2\right )}-\frac {2 a \left (\frac {1}{2} (a (A-B)+b (A+B)) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (b (A-B)-a (A+B)) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 A \sqrt {\cot (c+d x)}}{a d}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {-\frac {2 b^2 (A b-a B) \int \frac {1}{a \cot ^2(c+d x)+b}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}-\frac {2 a \left (\frac {1}{2} (a (A-B)+b (A+B)) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (b (A-B)-a (A+B)) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 A \sqrt {\cot (c+d x)}}{a d}\)

\(\Big \downarrow \) 218

\(\displaystyle -\frac {\frac {2 b^{3/2} (A b-a B) \arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {b}}\right )}{\sqrt {a} d \left (a^2+b^2\right )}-\frac {2 a \left (\frac {1}{2} (a (A-B)+b (A+B)) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (b (A-B)-a (A+B)) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 A \sqrt {\cot (c+d x)}}{a d}\)

Input:

Int[(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]
 

Output:

(-2*A*Sqrt[Cot[c + d*x]])/(a*d) - ((2*b^(3/2)*(A*b - a*B)*ArcTan[(Sqrt[a]* 
Cot[c + d*x])/Sqrt[b]])/(Sqrt[a]*(a^2 + b^2)*d) - (2*a*(((a*(A - B) + b*(A 
 + B))*(-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqr 
t[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]))/2 - ((b*(A - B) - a*(A + B))*(-1/2*Log[ 
1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*S 
qrt[Cot[c + d*x]] + Cot[c + d*x]]/(2*Sqrt[2])))/2))/((a^2 + b^2)*d))/a
 

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 217

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])
 

rule 218

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]
 

rule 1082

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]
 

rule 1103

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]
 

rule 1476

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]
 

rule 1479

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]
 

rule 1482

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]
 

rule 3042

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

rule 4017

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]
 

rule 4064

Int[(cot[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_.) + (b_.)*tan[(e_.) + (f_.)*( 
x_)])^(m_.)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp 
[g^(m + n)   Int[(g*Cot[e + f*x])^(p - m - n)*(b + a*Cot[e + f*x])^m*(d + c 
*Cot[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&  !Integer 
Q[p] && IntegerQ[m] && IntegerQ[n]
 

rule 4090

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*B*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + 
 n))), x] + Simp[1/(d*(m + n))   Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Ta 
n[e + f*x])^n*Simp[a^2*A*d*(m + n) - b*B*(b*c*(m - 1) + a*d*(n + 1)) + d*(m 
 + n)*(2*a*A*b + B*(a^2 - b^2))*Tan[e + f*x] - (b*B*(b*c - a*d)*(m - 1) - b 
*(A*b + a*B)*d*(m + n))*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] && GtQ[m, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&  !(IGtQ[n, 1] 
&& ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
 

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 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] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(685\) vs. \(2(206)=412\).

Time = 0.37 (sec) , antiderivative size = 686, normalized size of antiderivative = 2.91

method	result	size
derivativedivides	\(-\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {3}{2}} \tan \left (d x +c \right ) \left (A \sqrt {\tan \left (d x +c \right )}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) \sqrt {2}\, \sqrt {a b}\, a b +2 A \sqrt {\tan \left (d x +c \right )}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {a b}\, a^{2}+2 A \sqrt {\tan \left (d x +c \right )}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {a b}\, a b +2 A \sqrt {\tan \left (d x +c \right )}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {a b}\, a^{2}+2 A \sqrt {\tan \left (d x +c \right )}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {a b}\, a b +A \sqrt {\tan \left (d x +c \right )}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right ) \sqrt {2}\, \sqrt {a b}\, a^{2}-B \sqrt {\tan \left (d x +c \right )}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) \sqrt {2}\, \sqrt {a b}\, a^{2}-2 B \sqrt {\tan \left (d x +c \right )}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {a b}\, a^{2}+2 B \sqrt {\tan \left (d x +c \right )}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {a b}\, a b -2 B \sqrt {\tan \left (d x +c \right )}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {a b}\, a^{2}+2 B \sqrt {\tan \left (d x +c \right )}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {a b}\, a b +B \sqrt {\tan \left (d x +c \right )}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right ) \sqrt {2}\, \sqrt {a b}\, a b +8 A \sqrt {\tan \left (d x +c \right )}\, \arctan \left (\frac {b \sqrt {\tan \left (d x +c \right )}}{\sqrt {a b}}\right ) b^{3}-8 B \sqrt {\tan \left (d x +c \right )}\, \arctan \left (\frac {b \sqrt {\tan \left (d x +c \right )}}{\sqrt {a b}}\right ) a \,b^{2}+8 A \sqrt {a b}\, a^{2}+8 A \sqrt {a b}\, b^{2}\right )}{4 d a \left (a^{2}+b^{2}\right ) \sqrt {a b}}\)	\(686\)
default	\(-\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {3}{2}} \tan \left (d x +c \right ) \left (A \sqrt {\tan \left (d x +c \right )}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) \sqrt {2}\, \sqrt {a b}\, a b +2 A \sqrt {\tan \left (d x +c \right )}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {a b}\, a^{2}+2 A \sqrt {\tan \left (d x +c \right )}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {a b}\, a b +2 A \sqrt {\tan \left (d x +c \right )}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {a b}\, a^{2}+2 A \sqrt {\tan \left (d x +c \right )}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {a b}\, a b +A \sqrt {\tan \left (d x +c \right )}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right ) \sqrt {2}\, \sqrt {a b}\, a^{2}-B \sqrt {\tan \left (d x +c \right )}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) \sqrt {2}\, \sqrt {a b}\, a^{2}-2 B \sqrt {\tan \left (d x +c \right )}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {a b}\, a^{2}+2 B \sqrt {\tan \left (d x +c \right )}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {a b}\, a b -2 B \sqrt {\tan \left (d x +c \right )}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {a b}\, a^{2}+2 B \sqrt {\tan \left (d x +c \right )}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {a b}\, a b +B \sqrt {\tan \left (d x +c \right )}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right ) \sqrt {2}\, \sqrt {a b}\, a b +8 A \sqrt {\tan \left (d x +c \right )}\, \arctan \left (\frac {b \sqrt {\tan \left (d x +c \right )}}{\sqrt {a b}}\right ) b^{3}-8 B \sqrt {\tan \left (d x +c \right )}\, \arctan \left (\frac {b \sqrt {\tan \left (d x +c \right )}}{\sqrt {a b}}\right ) a \,b^{2}+8 A \sqrt {a b}\, a^{2}+8 A \sqrt {a b}\, b^{2}\right )}{4 d a \left (a^{2}+b^{2}\right ) \sqrt {a b}}\)	\(686\)

Input:

int(cot(d*x+c)^(3/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c)),x,method=_RETURNVER 
BOSE)
 

Output:

-1/4/d*(1/tan(d*x+c))^(3/2)*tan(d*x+c)*(A*tan(d*x+c)^(1/2)*ln(-(tan(d*x+c) 
+2^(1/2)*tan(d*x+c)^(1/2)+1)/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*2^(1 
/2)*(a*b)^(1/2)*a*b+2*A*tan(d*x+c)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2) 
)*2^(1/2)*(a*b)^(1/2)*a^2+2*A*tan(d*x+c)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c) 
^(1/2))*2^(1/2)*(a*b)^(1/2)*a*b+2*A*tan(d*x+c)^(1/2)*arctan(-1+2^(1/2)*tan 
(d*x+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*a^2+2*A*tan(d*x+c)^(1/2)*arctan(-1+2^(1 
/2)*tan(d*x+c)^(1/2))*2^(1/2)*(a*b)^(1/2)*a*b+A*tan(d*x+c)^(1/2)*ln(-(2^(1 
/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1) 
)*2^(1/2)*(a*b)^(1/2)*a^2-B*tan(d*x+c)^(1/2)*ln(-(tan(d*x+c)+2^(1/2)*tan(d 
*x+c)^(1/2)+1)/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*2^(1/2)*(a*b)^(1/2 
)*a^2-2*B*tan(d*x+c)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*(a*b 
)^(1/2)*a^2+2*B*tan(d*x+c)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2 
)*(a*b)^(1/2)*a*b-2*B*tan(d*x+c)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2)) 
*2^(1/2)*(a*b)^(1/2)*a^2+2*B*tan(d*x+c)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c) 
^(1/2))*2^(1/2)*(a*b)^(1/2)*a*b+B*tan(d*x+c)^(1/2)*ln(-(2^(1/2)*tan(d*x+c) 
^(1/2)-tan(d*x+c)-1)/(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1))*2^(1/2)*(a*b 
)^(1/2)*a*b+8*A*tan(d*x+c)^(1/2)*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*b^ 
3-8*B*tan(d*x+c)^(1/2)*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))*a*b^2+8*A*(a 
*b)^(1/2)*a^2+8*A*(a*b)^(1/2)*b^2)/a/(a^2+b^2)/(a*b)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1271 vs. \(2 (206) = 412\).

Time = 10.72 (sec) , antiderivative size = 2572, normalized size of antiderivative = 10.90 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate(cot(d*x+c)^(3/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c)),x, algorithm= 
"fricas")
 

Output:

[-1/2*(2*sqrt(1/2)*(a^3 + a*b^2)*d*sqrt(((A^2 - 2*A*B + B^2)*a^2 + 2*(A^2 
- B^2)*a*b + (A^2 + 2*A*B + B^2)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^2))*arcta 
n(((a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(((A^2 + 2*A*B + B^2)*a^2 - 2*(A^2 - B^ 
2)*a*b + (A^2 - 2*A*B + B^2)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^2))*sqrt(((A^ 
2 - 2*A*B + B^2)*a^2 + 2*(A^2 - B^2)*a*b + (A^2 + 2*A*B + B^2)*b^2)/((a^4 
+ 2*a^2*b^2 + b^4)*d^2)) + 2*sqrt(1/2)*((A + B)*a^3 - (A - B)*a^2*b + (A + 
 B)*a*b^2 - (A - B)*b^3)*d*sqrt(((A^2 - 2*A*B + B^2)*a^2 + 2*(A^2 - B^2)*a 
*b + (A^2 + 2*A*B + B^2)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^2))*sqrt(tan(d*x 
+ c)))/(4*A*B*a*b + (A^2 - B^2)*a^2 - (A^2 - B^2)*b^2)) + 2*sqrt(1/2)*(a^3 
 + a*b^2)*d*sqrt(((A^2 - 2*A*B + B^2)*a^2 + 2*(A^2 - B^2)*a*b + (A^2 + 2*A 
*B + B^2)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^2))*arctan(-((a^4 + 2*a^2*b^2 + 
b^4)*d^2*sqrt(((A^2 + 2*A*B + B^2)*a^2 - 2*(A^2 - B^2)*a*b + (A^2 - 2*A*B 
+ B^2)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^2))*sqrt(((A^2 - 2*A*B + B^2)*a^2 + 
 2*(A^2 - B^2)*a*b + (A^2 + 2*A*B + B^2)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^2 
)) - 2*sqrt(1/2)*((A + B)*a^3 - (A - B)*a^2*b + (A + B)*a*b^2 - (A - B)*b^ 
3)*d*sqrt(((A^2 - 2*A*B + B^2)*a^2 + 2*(A^2 - B^2)*a*b + (A^2 + 2*A*B + B^ 
2)*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^2))*sqrt(tan(d*x + c)))/(4*A*B*a*b + (A 
^2 - B^2)*a^2 - (A^2 - B^2)*b^2)) - sqrt(1/2)*(a^3 + a*b^2)*d*sqrt(((A^2 + 
 2*A*B + B^2)*a^2 - 2*(A^2 - B^2)*a*b + (A^2 - 2*A*B + B^2)*b^2)/((a^4 + 2 
*a^2*b^2 + b^4)*d^2))*log(2*sqrt(1/2)*(a^2 + b^2)*d*sqrt(((A^2 + 2*A*B ...
 

Sympy [F]

\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \cot ^{\frac {3}{2}}{\left (c + d x \right )}}{a + b \tan {\left (c + d x \right )}}\, dx \] Input:

integrate(cot(d*x+c)**(3/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c)),x)
 

Output:

Integral((A + B*tan(c + d*x))*cot(c + d*x)**(3/2)/(a + b*tan(c + d*x)), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.01 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=-\frac {\frac {8 \, {\left (B a b^{2} - A b^{3}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{3} + a b^{2}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left ({\left (A - B\right )} a + {\left (A + B\right )} b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left ({\left (A - B\right )} a + {\left (A + B\right )} b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \sqrt {2} {\left ({\left (A + B\right )} a - {\left (A - B\right )} b\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \sqrt {2} {\left ({\left (A + B\right )} a - {\left (A - B\right )} b\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{2} + b^{2}} + \frac {8 \, A}{a \sqrt {\tan \left (d x + c\right )}}}{4 \, d} \] Input:

integrate(cot(d*x+c)^(3/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c)),x, algorithm= 
"maxima")
 

Output:

-1/4*(8*(B*a*b^2 - A*b^3)*arctan(a/(sqrt(a*b)*sqrt(tan(d*x + c))))/((a^3 + 
 a*b^2)*sqrt(a*b)) - (2*sqrt(2)*((A - B)*a + (A + B)*b)*arctan(1/2*sqrt(2) 
*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 2*sqrt(2)*((A - B)*a + (A + B)*b)*arc 
tan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) + sqrt(2)*((A + B)*a - 
(A - B)*b)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) - sqrt(2)* 
((A + B)*a - (A - B)*b)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 
 1))/(a^2 + b^2) + 8*A/(a*sqrt(tan(d*x + c))))/d
 

Giac [F]

\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \cot \left (d x + c\right )^{\frac {3}{2}}}{b \tan \left (d x + c\right ) + a} \,d x } \] Input:

integrate(cot(d*x+c)^(3/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c)),x, algorithm= 
"giac")
 

Output:

integrate((B*tan(d*x + c) + A)*cot(d*x + c)^(3/2)/(b*tan(d*x + c) + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )}{a+b\,\mathrm {tan}\left (c+d\,x\right )} \,d x \] Input:

int((cot(c + d*x)^(3/2)*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x)),x)
 

Output:

int((cot(c + d*x)^(3/2)*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x)), x)
 

Reduce [F]

\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {-2 \sqrt {\cot \left (d x +c \right )}-\left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )}d x \right ) d}{d} \] Input:

int(cot(d*x+c)^(3/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c)),x)
 

Output:

( - 2*sqrt(cot(c + d*x)) - int(sqrt(cot(c + d*x))/cot(c + d*x),x)*d)/d