Integrand size = 23, antiderivative size = 96 \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\frac {2 a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}+\frac {2 a^2 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{d}-\frac {2 a \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{3 d} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3972, 331, 209} \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\frac {2 a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}+\frac {2 a^2 \cot (c+d x) \sqrt {a \sec (c+d x)+a}}{d}-\frac {2 a \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d} \]
[In]
[Out]
Rule 209
Rule 331
Rule 3972
Rubi steps \begin{align*} \text {integral}& = -\frac {(2 a) \text {Subst}\left (\int \frac {1}{x^4 \left (1+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d} \\ & = -\frac {2 a \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{3 d}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (1+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d} \\ & = \frac {2 a^2 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{d}-\frac {2 a \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{3 d}-\frac {\left (2 a^3\right ) \text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d} \\ & = \frac {2 a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}+\frac {2 a^2 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{d}-\frac {2 a \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{3 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.20 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.84 \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=-\frac {2 \left (\frac {1}{1+\cos (c+d x)}\right )^{3/2} \cot ^3(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right ) (a (1+\sec (c+d x)))^{5/2}}{3 d \sqrt {\frac {1}{1+\sec (c+d x)}}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(179\) vs. \(2(84)=168\).
Time = 40.39 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.88
method | result | size |
default | \(\frac {2 a^{2} \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (3 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )-3 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+4 \cot \left (d x +c \right ) \cos \left (d x +c \right )-3 \cot \left (d x +c \right )\right )}{3 d \left (\cos \left (d x +c \right )-1\right )}\) | \(180\) |
[In]
[Out]
none
Time = 0.34 (sec) , antiderivative size = 355, normalized size of antiderivative = 3.70 \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\left [\frac {3 \, {\left (a^{2} \cos \left (d x + c\right ) - a^{2}\right )} \sqrt {-a} \log \left (-\frac {8 \, a \cos \left (d x + c\right )^{3} - 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right ) - 7 \, a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right ) + 1}\right ) \sin \left (d x + c\right ) + 4 \, {\left (4 \, a^{2} \cos \left (d x + c\right )^{2} - 3 \, a^{2} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{6 \, {\left (d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right )}, \frac {3 \, {\left (a^{2} \cos \left (d x + c\right ) - a^{2}\right )} \sqrt {a} \arctan \left (\frac {2 \, \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a}\right ) \sin \left (d x + c\right ) + 2 \, {\left (4 \, a^{2} \cos \left (d x + c\right )^{2} - 3 \, a^{2} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{3 \, {\left (d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right )}\right ] \]
[In]
[Out]
Timed out. \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{4} \,d x } \]
[In]
[Out]
\[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{4} \,d x } \]
[In]
[Out]
Timed out. \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^4\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \]
[In]
[Out]