Integrand size = 21, antiderivative size = 98 \[ \int \csc (c+d x) (a+a \sin (c+d x))^{5/2} \, dx=-\frac {2 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d}-\frac {14 a^3 \cos (c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a^2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d} \]
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Time = 0.14 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2842, 3060, 2852, 212} \[ \int \csc (c+d x) (a+a \sin (c+d x))^{5/2} \, dx=-\frac {2 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {14 a^3 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a^2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d} \]
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Rule 212
Rule 2842
Rule 2852
Rule 3060
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}+\frac {2}{3} \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \left (\frac {3 a^2}{2}+\frac {7}{2} a^2 \sin (c+d x)\right ) \, dx \\ & = -\frac {14 a^3 \cos (c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a^2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}+a^2 \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {14 a^3 \cos (c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a^2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}-\frac {\left (2 a^3\right ) \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d} \\ & = -\frac {2 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d}-\frac {14 a^3 \cos (c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a^2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d} \\ \end{align*}
Time = 5.71 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.46 \[ \int \csc (c+d x) (a+a \sin (c+d x))^{5/2} \, dx=-\frac {(a (1+\sin (c+d x)))^{5/2} \left (15 \cos \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {3}{2} (c+d x)\right )+3 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-3 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-15 \sin \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {3}{2} (c+d x)\right )\right )}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^5} \]
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Time = 1.09 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.05
method | result | size |
default | \(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a \left (3 a^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}}{\sqrt {a}}\right )-\left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}}+9 a \sqrt {a -a \sin \left (d x +c \right )}\right )}{3 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(103\) |
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Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (84) = 168\).
Time = 0.31 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.85 \[ \int \csc (c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\frac {3 \, {\left (a^{2} \cos \left (d x + c\right ) + a^{2} \sin \left (d x + c\right ) + a^{2}\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) - 4 \, {\left (a^{2} \cos \left (d x + c\right )^{2} + 8 \, a^{2} \cos \left (d x + c\right ) + 7 \, a^{2} + {\left (a^{2} \cos \left (d x + c\right ) - 7 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{6 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
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Timed out. \[ \int \csc (c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int \csc (c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \csc \left (d x + c\right ) \,d x } \]
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none
Time = 0.34 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.44 \[ \int \csc (c+d x) (a+a \sin (c+d x))^{5/2} \, dx=-\frac {\sqrt {2} {\left (8 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, \sqrt {2} a^{2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 36 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{6 \, d} \]
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Timed out. \[ \int \csc (c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}}{\sin \left (c+d\,x\right )} \,d x \]
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