Integrand size = 23, antiderivative size = 303 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^{3/2} d}-\frac {533 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{256 \sqrt {2} a^{3/2} d}-\frac {21 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{256 a^2 d}+\frac {277 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{384 a^3 d}-\frac {81 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^3 d}-\frac {7 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{64 a^3 d}-\frac {\cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{48 a^3 d} \]
2*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/a^(3/2)/d+277/384*cot( d*x+c)^3*(a+a*sec(d*x+c))^(3/2)/a^3/d-81/128*cos(d*x+c)*cot(d*x+c)^3*sec(1 /2*d*x+1/2*c)^2*(a+a*sec(d*x+c))^(3/2)/a^3/d-7/64*cos(d*x+c)^2*cot(d*x+c)^ 3*sec(1/2*d*x+1/2*c)^4*(a+a*sec(d*x+c))^(3/2)/a^3/d-1/48*cos(d*x+c)^3*cot( d*x+c)^3*sec(1/2*d*x+1/2*c)^6*(a+a*sec(d*x+c))^(3/2)/a^3/d-533/512*arctan( 1/2*a^(1/2)*tan(d*x+c)*2^(1/2)/(a+a*sec(d*x+c))^(1/2))*2^(1/2)/a^(3/2)/d-2 1/256*cot(d*x+c)*(a+a*sec(d*x+c))^(1/2)/a^2/d
Time = 2.52 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.79 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {-\frac {1}{16} (201-472 \cos (c+d x)+516 \cos (2 (c+d x))+984 \cos (3 (c+d x))+819 \cos (4 (c+d x))) \csc ^3(c+d x) \sec (c+d x)+3072 \arctan \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\frac {1}{1+\sec (c+d x)}}}\right ) \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {\frac {\sec (c+d x)}{(1+\sec (c+d x))^2}} \sqrt {1+\sec (c+d x)}-1599 \arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )} \sec ^{\frac {3}{2}}(c+d x) \sqrt {\frac {1}{1+\sec (c+d x)}} \sqrt {1+\sec (c+d x)}}{384 d (a (1+\sec (c+d x)))^{3/2}} \]
(-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))
Time = 0.45 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.06, 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 |
| \(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {1}{2} a \left (\frac {1}{2} a \left (\frac {533 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{\sqrt {2} \sqrt {a}}-\frac {512 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{\sqrt {a}}\right )+\frac {21}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}\right )-\frac {277}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )+\frac {81 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )+\frac {7 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\) |
(-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)
3.2.93.3.1 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_)^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])
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]
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]
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]
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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(540\) vs. \(2(263)=526\).
Time = 1.87 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.79
| method | result | size |
| default | \(-\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (1599 \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\cot \left (d x +c \right )^{2}-2 \cot \left (d x +c \right ) \csc \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}\right ) \cos \left (d x +c \right )^{2}-3072 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )^{2}+3198 \sqrt {2}\, \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\cot \left (d x +c \right )^{2}-2 \cot \left (d x +c \right ) \csc \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )-6144 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )+1599 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\cot \left (d x +c \right )^{2}-2 \cot \left (d x +c \right ) \csc \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}\right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+1638 \cot \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}-3072 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+984 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{3}-1380 \cot \left (d x +c \right )^{3}-856 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )+126 \cot \left (d x +c \right ) \csc \left (d x +c \right )^{2}\right )}{1536 d \,a^{2} \left (\cos \left (d x +c \right )+1\right )^{2}}\) | \(541\) |
-1/1536/d/a^2*(a*(1+sec(d*x+c)))^(1/2)/(cos(d*x+c)+1)^2*(1599*2^(1/2)*(-co s(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln(csc(d*x+c)-cot(d*x+c)+(cot(d*x+c)^2-2*co t(d*x+c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2))*cos(d*x+c)^2-3072*(-cos(d*x+c)/ (cos(d*x+c)+1))^(1/2)*arctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos( d*x+c)+1))^(1/2))*cos(d*x+c)^2+3198*2^(1/2)*ln(csc(d*x+c)-cot(d*x+c)+(cot( d*x+c)^2-2*cot(d*x+c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2))*(-cos(d*x+c)/(cos( d*x+c)+1))^(1/2)*cos(d*x+c)-6144*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan h(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c) +1599*ln(csc(d*x+c)-cot(d*x+c)+(cot(d*x+c)^2-2*cot(d*x+c)*csc(d*x+c)+csc(d *x+c)^2-1)^(1/2))*2^(1/2)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+1638*cot(d*x+ c)^3*cos(d*x+c)^2-3072*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(sin(d*x+ c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+984*cos(d*x+c)*cot(d *x+c)^3-1380*cot(d*x+c)^3-856*cot(d*x+c)^2*csc(d*x+c)+126*cot(d*x+c)*csc(d *x+c)^2)
Time = 0.37 (sec) , antiderivative size = 712, normalized size of antiderivative = 2.35 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\left [-\frac {1599 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right ) - 1\right )} \sqrt {-a} \log \left (-\frac {2 \, \sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 3 \, a \cos \left (d x + c\right )^{2} - 2 \, a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) \sin \left (d x + c\right ) + 1536 \, {\left (\cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right ) - 1\right )} \sqrt {-a} \log \left (-\frac {8 \, a \cos \left (d x + c\right )^{3} + 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right ) - 7 \, a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right ) + 1}\right ) \sin \left (d x + c\right ) - 4 \, {\left (819 \, \cos \left (d x + c\right )^{5} + 492 \, \cos \left (d x + c\right )^{4} - 690 \, \cos \left (d x + c\right )^{3} - 428 \, \cos \left (d x + c\right )^{2} + 63 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{3072 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} + 2 \, a^{2} d \cos \left (d x + c\right )^{3} - 2 \, a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )} \sin \left (d x + c\right )}, \frac {1599 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right ) - 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) + 1536 \, {\left (\cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right ) - 1\right )} \sqrt {a} \arctan \left (\frac {2 \, \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a}\right ) \sin \left (d x + c\right ) + 2 \, {\left (819 \, \cos \left (d x + c\right )^{5} + 492 \, \cos \left (d x + c\right )^{4} - 690 \, \cos \left (d x + c\right )^{3} - 428 \, \cos \left (d x + c\right )^{2} + 63 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{1536 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} + 2 \, a^{2} d \cos \left (d x + c\right )^{3} - 2 \, a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )} \sin \left (d x + c\right )}\right ] \]
[-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))]
\[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {\cot ^{4}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int { \frac {\cot \left (d x + c\right )^{4}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Time = 1.09 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.86 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=-\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} {\left (2 \, {\left (\frac {4 \, \sqrt {2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {37 \, \sqrt {2}}{a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {417 \, \sqrt {2}}{a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {32 \, \sqrt {2} {\left (21 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{4} - 36 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} a + 19 \, a^{2}\right )}}{{\left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - a\right )}^{3} \sqrt {-a} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{1536 \, d} \]
-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
Timed out. \[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^4}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]