Integrand size = 23, antiderivative size = 176 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {11 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{16 \sqrt {2} a^{3/2} d}-\frac {3 a}{20 d (a+a \sec (c+d x))^{5/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{5/2}}+\frac {5}{24 d (a+a \sec (c+d x))^{3/2}}+\frac {21}{16 a d \sqrt {a+a \sec (c+d x)}} \]
[Out]
Time = 0.27 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3965, 105, 157, 162, 65, 213} \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {11 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{16 \sqrt {2} a^{3/2} d}-\frac {3 a}{20 d (a \sec (c+d x)+a)^{5/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{5/2}}+\frac {5}{24 d (a \sec (c+d x)+a)^{3/2}}+\frac {21}{16 a d \sqrt {a \sec (c+d x)+a}} \]
[In]
[Out]
Rule 65
Rule 105
Rule 157
Rule 162
Rule 213
Rule 3965
Rubi steps \begin{align*} \text {integral}& = \frac {a^4 \text {Subst}\left (\int \frac {1}{x (-a+a x)^2 (a+a x)^{7/2}} \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{5/2}}-\frac {a \text {Subst}\left (\int \frac {2 a^2+\frac {7 a^2 x}{2}}{x (-a+a x) (a+a x)^{7/2}} \, dx,x,\sec (c+d x)\right )}{2 d} \\ & = -\frac {3 a}{20 d (a+a \sec (c+d x))^{5/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{5/2}}+\frac {\text {Subst}\left (\int \frac {-10 a^4-\frac {15 a^4 x}{4}}{x (-a+a x) (a+a x)^{5/2}} \, dx,x,\sec (c+d x)\right )}{10 a^2 d} \\ & = -\frac {3 a}{20 d (a+a \sec (c+d x))^{5/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{5/2}}+\frac {5}{24 d (a+a \sec (c+d x))^{3/2}}-\frac {\text {Subst}\left (\int \frac {30 a^6-\frac {75 a^6 x}{8}}{x (-a+a x) (a+a x)^{3/2}} \, dx,x,\sec (c+d x)\right )}{30 a^5 d} \\ & = -\frac {3 a}{20 d (a+a \sec (c+d x))^{5/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{5/2}}+\frac {5}{24 d (a+a \sec (c+d x))^{3/2}}+\frac {21}{16 a d \sqrt {a+a \sec (c+d x)}}+\frac {\text {Subst}\left (\int \frac {-30 a^8+\frac {315 a^8 x}{16}}{x (-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{30 a^8 d} \\ & = -\frac {3 a}{20 d (a+a \sec (c+d x))^{5/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{5/2}}+\frac {5}{24 d (a+a \sec (c+d x))^{3/2}}+\frac {21}{16 a d \sqrt {a+a \sec (c+d x)}}-\frac {11 \text {Subst}\left (\int \frac {1}{(-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{32 d}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{a d} \\ & = -\frac {3 a}{20 d (a+a \sec (c+d x))^{5/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{5/2}}+\frac {5}{24 d (a+a \sec (c+d x))^{3/2}}+\frac {21}{16 a d \sqrt {a+a \sec (c+d x)}}+\frac {2 \text {Subst}\left (\int \frac {1}{-1+\frac {x^2}{a}} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{a^2 d}-\frac {11 \text {Subst}\left (\int \frac {1}{-2 a+x^2} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{16 a d} \\ & = -\frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {11 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{16 \sqrt {2} a^{3/2} d}-\frac {3 a}{20 d (a+a \sec (c+d x))^{5/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{5/2}}+\frac {5}{24 d (a+a \sec (c+d x))^{3/2}}+\frac {21}{16 a d \sqrt {a+a \sec (c+d x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.13 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.51 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {a \left (-10-11 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},\frac {1}{2} (1+\sec (c+d x))\right ) (-1+\sec (c+d x))+8 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},1+\sec (c+d x)\right ) (-1+\sec (c+d x))\right )}{20 d (-1+\sec (c+d x)) (a (1+\sec (c+d x)))^{5/2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(399\) vs. \(2(143)=286\).
Time = 1.92 (sec) , antiderivative size = 400, normalized size of antiderivative = 2.27
method | result | size |
default | \(\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (165 \cos \left (d x +c \right )^{2} \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+330 \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+960 \cos \left (d x +c \right )^{2} \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+165 \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+1920 \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \cos \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+960 \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-898 \cos \left (d x +c \right )^{2} \cot \left (d x +c \right )^{2}-702 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{2}+730 \cot \left (d x +c \right )^{2}+630 \cot \left (d x +c \right ) \csc \left (d x +c \right )\right )}{480 d \,a^{2} \left (\cos \left (d x +c \right )+1\right )^{2}}\) | \(400\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (141) = 282\).
Time = 0.38 (sec) , antiderivative size = 592, normalized size of antiderivative = 3.36 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\left [\frac {165 \, \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 ) + 3 \, a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right ) - 1}\right ) + 480 \, {\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 (-8 \, a \cos \left (d x + c\right )^{2} + 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 )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) + 4 \, {\left (449 \, \cos \left (d x + c\right )^{4} + 351 \, \cos \left (d x + c\right )^{3} - 365 \, \cos \left (d x + c\right )^{2} - 315 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{960 \, {\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 )}}, -\frac {165 \, \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 {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{a \cos \left (d x + c\right ) + a}\right ) - 480 \, {\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 )}{2 \, a \cos \left (d x + c\right ) + a}\right ) - 2 \, {\left (449 \, \cos \left (d x + c\right )^{4} + 351 \, \cos \left (d x + c\right )^{3} - 365 \, \cos \left (d x + c\right )^{2} - 315 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{480 \, {\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 )}}\right ] \]
[In]
[Out]
\[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {\cot ^{3}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int { \frac {\cot \left (d x + c\right )^{3}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
none
Time = 1.09 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.44 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=-\frac {\frac {165 \, \sqrt {2} \arctan \left (\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {960 \, \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} + \frac {15 \, \sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} - \frac {2 \, \sqrt {2} {\left (3 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{16} + 20 \, {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} a^{17} + 165 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{18}\right )}}{a^{20} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{480 \, d} \]
[In]
[Out]
Timed out. \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^3}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
[In]
[Out]