\(\int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx\) [193]

Optimal result

Integrand size = 23, antiderivative size = 314 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^{3/2} d}-\frac {533 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{256 \sqrt {2} a^{3/2} d}-\frac {21 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{256 a^2 d}+\frac {277 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{384 a^3 d}-\frac {\cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{6 a^3 d \left (2+\frac {\tan ^2(c+d x)}{1+\sec (c+d x)}\right )^3}-\frac {7 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{16 a^3 d \left (2+\frac {\tan ^2(c+d x)}{1+\sec (c+d x)}\right )^2}-\frac {81 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{64 a^3 d \left (2+\frac {\tan ^2(c+d x)}{1+\sec (c+d x)}\right )} \] Output:

2*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/a^(3/2)/d-533/512*2^(1 
/2)*arctan(1/2*a^(1/2)*tan(d*x+c)*2^(1/2)/(a+a*sec(d*x+c))^(1/2))/a^(3/2)/ 
d-21/256*cot(d*x+c)*(a+a*sec(d*x+c))^(1/2)/a^2/d+277/384*cot(d*x+c)^3*(a+a 
*sec(d*x+c))^(3/2)/a^3/d-1/6*cot(d*x+c)^3*(a+a*sec(d*x+c))^(3/2)/a^3/d/(2+ 
tan(d*x+c)^2/(1+sec(d*x+c)))^3-7/16*cot(d*x+c)^3*(a+a*sec(d*x+c))^(3/2)/a^ 
3/d/(2+tan(d*x+c)^2/(1+sec(d*x+c)))^2-81/64*cot(d*x+c)^3*(a+a*sec(d*x+c))^ 
(3/2)/a^3/d/(2+tan(d*x+c)^2/(1+sec(d*x+c)))
 

Mathematica [A] (warning: unable to verify)

Time = 2.02 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.76 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {-\frac {1}{16} (201-472 \cos (c+d x)+516 \cos (2 (c+d x))+984 \cos (3 (c+d x))+819 \cos (4 (c+d x))) \csc ^3(c+d x) \sec (c+d x)+3072 \arctan \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\frac {1}{1+\sec (c+d x)}}}\right ) \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {\frac {\sec (c+d x)}{(1+\sec (c+d x))^2}} \sqrt {1+\sec (c+d x)}-1599 \arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )} \sec ^{\frac {3}{2}}(c+d x) \sqrt {\frac {1}{1+\sec (c+d x)}} \sqrt {1+\sec (c+d x)}}{384 d (a (1+\sec (c+d x)))^{3/2}} \] Input:

Integrate[Cot[c + d*x]^4/(a + a*Sec[c + d*x])^(3/2),x]
 

Output:

(-1/16*((201 - 472*Cos[c + d*x] + 516*Cos[2*(c + d*x)] + 984*Cos[3*(c + d* 
x)] + 819*Cos[4*(c + d*x)])*Csc[c + d*x]^3*Sec[c + d*x]) + 3072*ArcTan[Tan 
[(c + d*x)/2]/Sqrt[(1 + Sec[c + d*x])^(-1)]]*Cos[(c + d*x)/2]^4*Sec[c + d* 
x]^(3/2)*Sqrt[Sec[c + d*x]/(1 + Sec[c + d*x])^2]*Sqrt[1 + Sec[c + d*x]] - 
1599*ArcSin[Tan[(c + d*x)/2]]*Cos[(c + d*x)/2]^4*Sqrt[Sec[(c + d*x)/2]^2]* 
Sec[c + d*x]^(3/2)*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[1 + Sec[c + d*x]])/( 
384*d*(a*(1 + Sec[c + d*x]))^(3/2))
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.02, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.696, Rules used = {3042, 4375, 374, 27, 441, 25, 27, 441, 27, 445, 27, 445, 25, 27, 397, 216}

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.

\(\displaystyle \int \frac {\cot ^4(c+d x)}{(a \sec (c+d x)+a)^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\cot \left (c+d x+\frac {\pi }{2}\right )^4 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^{3/2}}dx\)

\(\Big \downarrow \) 4375

\(\displaystyle -\frac {2 \int \frac {\cot ^4(c+d x) (\sec (c+d x) a+a)^2}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^4}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{a^3 d}\)

\(\Big \downarrow \) 374

\(\displaystyle -\frac {2 \left (\frac {\int \frac {3 a \cot ^4(c+d x) (\sec (c+d x) a+a)^2 \left (1-\frac {3 a \tan ^2(c+d x)}{\sec (c+d x) a+a}\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^3}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{12 a}+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{4} \int \frac {\cot ^4(c+d x) (\sec (c+d x) a+a)^2 \left (1-\frac {3 a \tan ^2(c+d x)}{\sec (c+d x) a+a}\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^3}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 441

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {\int -\frac {a \cot ^4(c+d x) (\sec (c+d x) a+a)^2 \left (\frac {49 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+17\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^2}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{8 a}+\frac {7 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {7 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {\int \frac {a \cot ^4(c+d x) (\sec (c+d x) a+a)^2 \left (\frac {49 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+17\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^2}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{8 a}\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {7 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {1}{8} \int \frac {\cot ^4(c+d x) (\sec (c+d x) a+a)^2 \left (\frac {49 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+17\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^2}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 441

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{8} \left (\frac {81 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}-\frac {\int \frac {a \cot ^4(c+d x) (\sec (c+d x) a+a)^2 \left (\frac {405 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+277\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{4 a}\right )+\frac {7 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{8} \left (\frac {81 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}-\frac {1}{4} \int \frac {\cot ^4(c+d x) (\sec (c+d x) a+a)^2 \left (\frac {405 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+277\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )+\frac {7 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 445

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {1}{6} \int \frac {3 a \cot ^2(c+d x) (\sec (c+d x) a+a) \left (\frac {277 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+21\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )-\frac {277}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )+\frac {81 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )+\frac {7 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {1}{2} a \int \frac {\cot ^2(c+d x) (\sec (c+d x) a+a) \left (\frac {277 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+21\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )-\frac {277}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )+\frac {81 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )+\frac {7 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 445

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {1}{2} a \left (\frac {21}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} \int -\frac {a \left (491-\frac {21 a \tan ^2(c+d x)}{\sec (c+d x) a+a}\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )-\frac {277}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )+\frac {81 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )+\frac {7 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {1}{2} a \left (\frac {1}{2} \int \frac {a \left (491-\frac {21 a \tan ^2(c+d x)}{\sec (c+d x) a+a}\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )+\frac {21}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}\right )-\frac {277}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )+\frac {81 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )+\frac {7 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {1}{2} a \left (\frac {1}{2} a \int \frac {491-\frac {21 a \tan ^2(c+d x)}{\sec (c+d x) a+a}}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )+\frac {21}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}\right )-\frac {277}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )+\frac {81 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )+\frac {7 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 397

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {1}{2} a \left (\frac {1}{2} a \left (512 \int \frac {1}{\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )-533 \int \frac {1}{\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )+\frac {21}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}\right )-\frac {277}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )+\frac {81 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )+\frac {7 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {1}{2} a \left (\frac {1}{2} a \left (\frac {533 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{\sqrt {2} \sqrt {a}}-\frac {512 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{\sqrt {a}}\right )+\frac {21}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}\right )-\frac {277}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )+\frac {81 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )+\frac {7 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

Input:

Int[Cot[c + d*x]^4/(a + a*Sec[c + d*x])^(3/2),x]
 

Output:

(-2*((Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(12*(2 + (a*Tan[c + d*x]^ 
2)/(a + a*Sec[c + d*x]))^3) + ((7*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2 
))/(8*(2 + (a*Tan[c + d*x]^2)/(a + a*Sec[c + d*x]))^2) + (((-277*Cot[c + d 
*x]^3*(a + a*Sec[c + d*x])^(3/2))/6 + (a*((a*((-512*ArcTan[(Sqrt[a]*Tan[c 
+ d*x])/Sqrt[a + a*Sec[c + d*x]]])/Sqrt[a] + (533*ArcTan[(Sqrt[a]*Tan[c + 
d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[2]*Sqrt[a])))/2 + (21*Cot 
[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/2))/2)/4 + (81*Cot[c + d*x]^3*(a + a*S 
ec[c + d*x])^(3/2))/(4*(2 + (a*Tan[c + d*x]^2)/(a + a*Sec[c + d*x]))))/8)/ 
4))/(a^3*d)
 

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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 374

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q 
 + 1)/(a*e*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(e*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[b*c*(m + 1) + 2*(b*c - 
a*d)*(p + 1) + d*b*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, 
c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IntBinomialQ[a, b, 
 c, d, e, m, 2, p, q, x]
 

rule 397

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ 
Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^2), x], x] - Simp[ 
(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e 
, f}, x]
 

rule 441

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a 
+ b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*g*2*(b*c - a*d)*(p + 1))), x] + Si 
mp[1/(a*2*(b*c - a*d)*(p + 1))   Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2 
)^q*Simp[c*(b*e - a*f)*(m + 1) + e*2*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m 
 + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, 
 x] && LtQ[p, -1]
 

rule 445

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ 
.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p 
+ 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) 
 Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c 
+ a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
 

rule 3042

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

rule 4375

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n 
_.), x_Symbol] :> Simp[-2*(a^(m/2 + n + 1/2)/d)   Subst[Int[x^m*((2 + a*x^2 
)^(m/2 + n - 1/2)/(1 + a*x^2)), x], x, Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x] 
]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m/2] && I 
ntegerQ[n - 1/2]
 

Maple [A] (warning: unable to verify)

Time = 1.20 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.50

method	result	size
default	\(\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (\left (-30720 \cos \left (d x +c \right )^{4}-122880 \cos \left (d x +c \right )^{3}-184320 \cos \left (d x +c \right )^{2}-122880 \cos \left (d x +c \right )-30720\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}{\sqrt {\cot \left (d x +c \right )^{2}-2 \csc \left (d x +c \right ) \cot \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}}\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+\left (-31980 \cos \left (d x +c \right )^{4}-127920 \cos \left (d x +c \right )^{3}-191880 \cos \left (d x +c \right )^{2}-127920 \cos \left (d x +c \right )-31980\right ) \ln \left (\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+\sqrt {2}\, \left (161 \cos \left (d x +c \right )^{6}+39642 \cos \left (d x +c \right )^{5}+85527 \cos \left (d x +c \right )^{4}+34892 \cos \left (d x +c \right )^{3}-50193 \cos \left (d x +c \right )^{2}-54054 \cos \left (d x +c \right )-15015\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cot \left (d x +c \right ) \csc \left (d x +c \right )^{2}+\left (-32438 \cos \left (d x +c \right )^{6}-6238 \cos \left (d x +c \right )^{5}+47572 \cos \left (d x +c \right )^{4}+30332 \cos \left (d x +c \right )^{3}-18758 \cos \left (d x +c \right )^{2}-17950 \cos \left (d x +c \right )-2520\right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )^{2}\right )}{30720 d \,a^{2} \left (1+\cos \left (d x +c \right )\right ) \left (\cos \left (d x +c \right )^{3}+3 \cos \left (d x +c \right )^{2}+3 \cos \left (d x +c \right )+1\right )}\)	\(470\)

Input:

int(cot(d*x+c)^4/(a+a*sec(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/30720/d/a^2*(a*(1+sec(d*x+c)))^(1/2)/(1+cos(d*x+c))/(cos(d*x+c)^3+3*cos( 
d*x+c)^2+3*cos(d*x+c)+1)*((-30720*cos(d*x+c)^4-122880*cos(d*x+c)^3-184320* 
cos(d*x+c)^2-122880*cos(d*x+c)-30720)*2^(1/2)*arctanh(2^(1/2)*(-csc(d*x+c) 
+cot(d*x+c))/(cot(d*x+c)^2-2*csc(d*x+c)*cot(d*x+c)+csc(d*x+c)^2-1)^(1/2))* 
(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+(-31980*cos(d*x+c)^4-127920*cos(d*x+c 
)^3-191880*cos(d*x+c)^2-127920*cos(d*x+c)-31980)*ln((-2*cos(d*x+c)/(1+cos( 
d*x+c)))^(1/2)-cot(d*x+c)+csc(d*x+c))*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2) 
+2^(1/2)*(161*cos(d*x+c)^6+39642*cos(d*x+c)^5+85527*cos(d*x+c)^4+34892*cos 
(d*x+c)^3-50193*cos(d*x+c)^2-54054*cos(d*x+c)-15015)*(-cos(d*x+c)/(1+cos(d 
*x+c)))^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cot(d*x+c)*csc(d*x+c)^2 
+(-32438*cos(d*x+c)^6-6238*cos(d*x+c)^5+47572*cos(d*x+c)^4+30332*cos(d*x+c 
)^3-18758*cos(d*x+c)^2-17950*cos(d*x+c)-2520)*cot(d*x+c)*csc(d*x+c)^2)
                                                                                    
                                                                                    
 

Fricas [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 712, normalized size of antiderivative = 2.27 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx =\text {Too large to display} \] Input:

integrate(cot(d*x+c)^4/(a+a*sec(d*x+c))^(3/2),x, algorithm="fricas")
 

Output:

[-1/3072*(1599*sqrt(2)*(cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c) 
 - 1)*sqrt(-a)*log(-(2*sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x 
+ c))*cos(d*x + c)*sin(d*x + c) - 3*a*cos(d*x + c)^2 - 2*a*cos(d*x + c) + 
a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1))*sin(d*x + c) + 1536*(cos(d*x + c 
)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c) - 1)*sqrt(-a)*log(-(8*a*cos(d*x + 
c)^3 + 4*(2*cos(d*x + c)^2 - cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) + 
 a)/cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x + c) + 1)) 
*sin(d*x + c) - 4*(819*cos(d*x + c)^5 + 492*cos(d*x + c)^4 - 690*cos(d*x + 
 c)^3 - 428*cos(d*x + c)^2 + 63*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/co 
s(d*x + c)))/((a^2*d*cos(d*x + c)^4 + 2*a^2*d*cos(d*x + c)^3 - 2*a^2*d*cos 
(d*x + c) - a^2*d)*sin(d*x + c)), 1/1536*(1599*sqrt(2)*(cos(d*x + c)^4 + 2 
*cos(d*x + c)^3 - 2*cos(d*x + c) - 1)*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(d 
*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x + 
c) + 1536*(cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c) - 1)*sqrt(a) 
*arctan(2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin 
(d*x + c)/(2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a))*sin(d*x + c) + 2*(819 
*cos(d*x + c)^5 + 492*cos(d*x + c)^4 - 690*cos(d*x + c)^3 - 428*cos(d*x + 
c)^2 + 63*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((a^2*d*c 
os(d*x + c)^4 + 2*a^2*d*cos(d*x + c)^3 - 2*a^2*d*cos(d*x + c) - a^2*d)*sin 
(d*x + c))]
 

Sympy [F]

\[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {\cot ^{4}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(cot(d*x+c)**4/(a+a*sec(d*x+c))**(3/2),x)
 

Output:

Integral(cot(c + d*x)**4/(a*(sec(c + d*x) + 1))**(3/2), x)
 

Maxima [F]

\[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int { \frac {\cot \left (d x + c\right )^{4}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(cot(d*x+c)^4/(a+a*sec(d*x+c))^(3/2),x, algorithm="maxima")
 

Output:

integrate(cot(d*x + c)^4/(a*sec(d*x + c) + a)^(3/2), x)
 

Giac [A] (verification not implemented)

Time = 0.69 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.83 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=-\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} {\left (2 \, {\left (\frac {4 \, \sqrt {2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {37 \, \sqrt {2}}{a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {417 \, \sqrt {2}}{a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {32 \, \sqrt {2} {\left (21 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{4} - 36 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} a + 19 \, a^{2}\right )}}{{\left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - a\right )}^{3} \sqrt {-a} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{1536 \, d} \] Input:

integrate(cot(d*x+c)^4/(a+a*sec(d*x+c))^(3/2),x, algorithm="giac")
 

Output:

-1/1536*(sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*(2*(4*sqrt(2)*tan(1/2*d*x + 1 
/2*c)^2/(a^2*sgn(cos(d*x + c))) - 37*sqrt(2)/(a^2*sgn(cos(d*x + c))))*tan( 
1/2*d*x + 1/2*c)^2 + 417*sqrt(2)/(a^2*sgn(cos(d*x + c))))*tan(1/2*d*x + 1/ 
2*c) - 32*sqrt(2)*(21*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x 
 + 1/2*c)^2 + a))^4 - 36*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2* 
d*x + 1/2*c)^2 + a))^2*a + 19*a^2)/(((sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt 
(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2 - a)^3*sqrt(-a)*sgn(cos(d*x + c))))/d
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^4}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \] Input:

int(cot(c + d*x)^4/(a + a/cos(c + d*x))^(3/2),x)
 

Output:

int(cot(c + d*x)^4/(a + a/cos(c + d*x))^(3/2), x)
 

Reduce [F]

\[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sec \left (d x +c \right )+1}\, \cot \left (d x +c \right )^{4}}{\sec \left (d x +c \right )^{2}+2 \sec \left (d x +c \right )+1}d x \right )}{a^{2}} \] Input:

int(cot(d*x+c)^4/(a+a*sec(d*x+c))^(3/2),x)
 

Output:

(sqrt(a)*int((sqrt(sec(c + d*x) + 1)*cot(c + d*x)**4)/(sec(c + d*x)**2 + 2 
*sec(c + d*x) + 1),x))/a**2