Integrand size = 23, antiderivative size = 109 \[ \int \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {5 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} d}+\frac {a \sqrt {a+a \sec (c+d x)}}{2 d (1-\sec (c+d x))} \]
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Time = 0.14 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3965, 105, 162, 65, 213} \[ \int \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{d}+\frac {5 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} d}+\frac {a \sqrt {a \sec (c+d x)+a}}{2 d (1-\sec (c+d x))} \]
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Rule 65
Rule 105
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 \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {a \sqrt {a+a \sec (c+d x)}}{2 d (1-\sec (c+d x))}-\frac {a \text {Subst}\left (\int \frac {2 a^2+\frac {a^2 x}{2}}{x (-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{2 d} \\ & = \frac {a \sqrt {a+a \sec (c+d x)}}{2 d (1-\sec (c+d x))}+\frac {a^2 \text {Subst}\left (\int \frac {1}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{d}-\frac {\left (5 a^3\right ) \text {Subst}\left (\int \frac {1}{(-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{4 d} \\ & = \frac {a \sqrt {a+a \sec (c+d x)}}{2 d (1-\sec (c+d x))}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{-1+\frac {x^2}{a}} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{d}-\frac {\left (5 a^2\right ) \text {Subst}\left (\int \frac {1}{-2 a+x^2} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{2 d} \\ & = -\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {5 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} d}+\frac {a \sqrt {a+a \sec (c+d x)}}{2 d (1-\sec (c+d x))} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.92 \[ \int \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\frac {(a (1+\sec (c+d x)))^{3/2} \left (-2 \text {arctanh}\left (\sqrt {1+\sec (c+d x)}\right )+\frac {5 \text {arctanh}\left (\frac {\sqrt {1+\sec (c+d x)}}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {\sqrt {1+\sec (c+d x)}}{2 (-1+\sec (c+d x))}\right )}{d (1+\sec (c+d x))^{3/2}} \]
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Time = 1.94 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.27
method | result | size |
default | \(\frac {a \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (5 \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+8 \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-2 \cot \left (d x +c \right )^{2}-2 \cot \left (d x +c \right ) \csc \left (d x +c \right )\right )}{4 d}\) | \(138\) |
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Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (86) = 172\).
Time = 0.33 (sec) , antiderivative size = 378, normalized size of antiderivative = 3.47 \[ \int \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\left [\frac {4 \, a \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) + 8 \, {\left (a \cos \left (d x + c\right ) - a\right )} \sqrt {a} \log \left (-2 \, a \cos \left (d x + c\right ) + 2 \, \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) - a\right ) + 5 \, {\left (\sqrt {2} a \cos \left (d x + c\right ) - \sqrt {2} a\right )} \sqrt {a} \log \left (\frac {2 \, \sqrt {2} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) + 3 \, a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right ) - 1}\right )}{8 \, {\left (d \cos \left (d x + c\right ) - d\right )}}, -\frac {5 \, {\left (\sqrt {2} a \cos \left (d x + c\right ) - \sqrt {2} a\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{a \cos \left (d x + c\right ) + a}\right ) - 8 \, {\left (a \cos \left (d x + c\right ) - a\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{a \cos \left (d x + c\right ) + a}\right ) - 2 \, a \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{4 \, {\left (d \cos \left (d x + c\right ) - d\right )}}\right ] \]
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\[ \int \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}} \cot ^{3}{\left (c + d x \right )}\, dx \]
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\[ \int \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cot \left (d x + c\right )^{3} \,d x } \]
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\[ \int \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cot \left (d x + c\right )^{3} \,d x } \]
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Timed out. \[ \int \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^3\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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