Integrand size = 23, antiderivative size = 144 \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\frac {2 a^{3/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}-\frac {a^{3/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{2 \sqrt {2} d}+\frac {3 a \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{2 d}-\frac {\cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{3 d} \]
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Time = 0.18 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3972, 491, 597, 536, 209} \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\frac {2 a^{3/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}-\frac {a^{3/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{2 \sqrt {2} d}-\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}+\frac {3 a \cot (c+d x) \sqrt {a \sec (c+d x)+a}}{2 d} \]
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Rule 209
Rule 491
Rule 536
Rule 597
Rule 3972
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \text {Subst}\left (\int \frac {1}{x^4 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d} \\ & = -\frac {\cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{3 d}-\frac {\text {Subst}\left (\int \frac {-9 a-3 a^2 x^2}{x^2 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{3 d} \\ & = \frac {3 a \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{2 d}-\frac {\cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{3 d}+\frac {\text {Subst}\left (\int \frac {-21 a^2-9 a^3 x^2}{\left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{6 d} \\ & = \frac {3 a \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{2 d}-\frac {\cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{3 d}+\frac {a^2 \text {Subst}\left (\int \frac {1}{2+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{2 d}-\frac {\left (2 a^2\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^{3/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}-\frac {a^{3/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{2 \sqrt {2} d}+\frac {3 a \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{2 d}-\frac {\cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{3 d} \\ \end{align*}
Time = 6.69 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.57 \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=-\frac {\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )-4 \sqrt {2} \arctan \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}}}\right )\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sec ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {1+\sec (c+d x)} (a (1+\sec (c+d x)))^{3/2}}{8 d \sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )} \sec ^{\frac {3}{2}}(c+d x)}+\frac {\cos (c+d x) \sec ^3\left (\frac {1}{2} (c+d x)\right ) (a (1+\sec (c+d x)))^{3/2} \left (\frac {13}{24} \csc \left (\frac {1}{2} (c+d x)\right )-\frac {1}{24} \csc ^3\left (\frac {1}{2} (c+d x)\right )-\frac {11}{12} \sin \left (\frac {1}{2} (c+d x)\right )\right )}{d} \]
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Time = 1.74 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.37
method | result | size |
default | \(-\frac {a \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\cot \left (d x +c \right )^{2}-2 \cot \left (d x +c \right ) \csc \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}\right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-24 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+22 \cot \left (d x +c \right )^{3}+4 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )-18 \cot \left (d x +c \right ) \csc \left (d x +c \right )^{2}\right )}{12 d}\) | \(197\) |
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Time = 0.40 (sec) , antiderivative size = 531, normalized size of antiderivative = 3.69 \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\left [\frac {3 \, {\left (\sqrt {2} a \cos \left (d x + c\right ) - \sqrt {2} a\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 ) + 12 \, {\left (a \cos \left (d x + c\right ) - a\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 (11 \, a \cos \left (d x + c\right )^{2} - 9 \, a \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{24 \, {\left (d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right )}, \frac {12 \, {\left (a \cos \left (d x + c\right ) - a\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 ) + 3 \, {\left (\sqrt {2} a \cos \left (d x + c\right ) - \sqrt {2} a\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 ) + 2 \, {\left (11 \, a \cos \left (d x + c\right )^{2} - 9 \, a \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{12 \, {\left (d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right )}\right ] \]
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Timed out. \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\text {Timed out} \]
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\[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cot \left (d x + c\right )^{4} \,d x } \]
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Timed out. \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^4\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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