Integrand size = 23, antiderivative size = 145 \[ \int \frac {\sin ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {11 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {\cos (c+d x) \sin ^2(c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}+\frac {13 \cos (c+d x)}{3 a d \sqrt {a+a \sin (c+d x)}}-\frac {7 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{6 a^2 d} \]
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Time = 0.18 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2844, 3047, 3102, 2830, 2728, 212} \[ \int \frac {\sin ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {11 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {7 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{6 a^2 d}+\frac {\sin ^2(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}+\frac {13 \cos (c+d x)}{3 a d \sqrt {a \sin (c+d x)+a}} \]
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Rule 212
Rule 2728
Rule 2830
Rule 2844
Rule 3047
Rule 3102
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (c+d x) \sin ^2(c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}-\frac {\int \frac {\sin (c+d x) \left (2 a-\frac {7}{2} a \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{2 a^2} \\ & = \frac {\cos (c+d x) \sin ^2(c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}-\frac {\int \frac {2 a \sin (c+d x)-\frac {7}{2} a \sin ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{2 a^2} \\ & = \frac {\cos (c+d x) \sin ^2(c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}-\frac {7 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{6 a^2 d}-\frac {\int \frac {-\frac {7 a^2}{4}+\frac {13}{2} a^2 \sin (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{3 a^3} \\ & = \frac {\cos (c+d x) \sin ^2(c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}+\frac {13 \cos (c+d x)}{3 a d \sqrt {a+a \sin (c+d x)}}-\frac {7 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{6 a^2 d}+\frac {11 \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx}{4 a} \\ & = \frac {\cos (c+d x) \sin ^2(c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}+\frac {13 \cos (c+d x)}{3 a d \sqrt {a+a \sin (c+d x)}}-\frac {7 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{6 a^2 d}-\frac {11 \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{2 a d} \\ & = -\frac {11 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {\cos (c+d x) \sin ^2(c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}+\frac {13 \cos (c+d x)}{3 a d \sqrt {a+a \sin (c+d x)}}-\frac {7 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{6 a^2 d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.47 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.08 \[ \int \frac {\sin ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (11 \cos \left (\frac {1}{2} (c+d x)\right )+7 \cos \left (\frac {3}{2} (c+d x)\right )+\cos \left (\frac {5}{2} (c+d x)\right )-11 \sin \left (\frac {1}{2} (c+d x)\right )+(33+33 i) (-1)^{3/4} \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (c+d x)\right )\right )\right ) (1+\sin (c+d x))+7 \sin \left (\frac {3}{2} (c+d x)\right )-\sin \left (\frac {5}{2} (c+d x)\right )\right )}{6 d (a (1+\sin (c+d x)))^{3/2}} \]
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Time = 0.66 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.33
method | result | size |
default | \(\frac {\left (8 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {a}\, \sin \left (d x +c \right )-33 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} \sin \left (d x +c \right )+8 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {a}+24 \sqrt {a -a \sin \left (d x +c \right )}\, a^{\frac {3}{2}} \sin \left (d x +c \right )-33 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2}+30 \sqrt {a -a \sin \left (d x +c \right )}\, a^{\frac {3}{2}}\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{12 a^{\frac {7}{2}} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(193\) |
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Leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (122) = 244\).
Time = 0.28 (sec) , antiderivative size = 295, normalized size of antiderivative = 2.03 \[ \int \frac {\sin ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {33 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )} + 3 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) - 4 \, {\left (4 \, \cos \left (d x + c\right )^{3} + 16 \, \cos \left (d x + c\right )^{2} - {\left (4 \, \cos \left (d x + c\right )^{2} - 12 \, \cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) + 15 \, \cos \left (d x + c\right ) + 3\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{24 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d \cos \left (d x + c\right ) - 2 \, a^{2} d - {\left (a^{2} d \cos \left (d x + c\right ) + 2 \, a^{2} d\right )} \sin \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \frac {\sin ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sin ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\int { \frac {\sin \left (d x + c\right )^{3}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Time = 0.79 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.36 \[ \int \frac {\sin ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {\frac {33 \, \sqrt {2} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{\frac {3}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {33 \, \sqrt {2} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{\frac {3}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} + \frac {6 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} a^{\frac {3}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {16 \, \sqrt {2} {\left (2 \, a^{\frac {9}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, a^{\frac {9}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{6} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{24 \, d} \]
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Timed out. \[ \int \frac {\sin ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\int \frac {{\sin \left (c+d\,x\right )}^3}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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