Integrand size = 21, antiderivative size = 107 \[ \int \frac {\sin (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {5 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}+\frac {\cos (c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}-\frac {5 \cos (c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}} \]
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Time = 0.12 (sec) , antiderivative size = 107, 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 = {2829, 2729, 2728, 212} \[ \int \frac {\sin (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {5 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{16 \sqrt {2} a^{5/2} d}-\frac {5 \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}+\frac {\cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}} \]
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Rule 212
Rule 2728
Rule 2729
Rule 2829
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {5 \int \frac {1}{(a+a \sin (c+d x))^{3/2}} \, dx}{8 a} \\ & = \frac {\cos (c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}-\frac {5 \cos (c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}+\frac {5 \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx}{32 a^2} \\ & = \frac {\cos (c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}-\frac {5 \cos (c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac {5 \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 )}{16 a^2 d} \\ & = -\frac {5 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}+\frac {\cos (c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}-\frac {5 \cos (c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.83 \[ \int \frac {\sin (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (-8 \sin \left (\frac {1}{2} (c+d x)\right )+4 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+10 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2-5 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3+(5+5 i) (-1)^{3/4} \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (c+d x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4\right )}{16 d (a (1+\sin (c+d x)))^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(192\) vs. \(2(88)=176\).
Time = 0.69 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.80
method | result | size |
default | \(-\frac {\left (-5 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3} \left (\cos ^{2}\left (d x +c \right )\right )+12 \sqrt {a -a \sin \left (d x +c \right )}\, a^{\frac {5}{2}}-10 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} a^{\frac {3}{2}}+10 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3} \sin \left (d x +c \right )+10 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{32 a^{\frac {11}{2}} \left (1+\sin \left (d x +c \right )\right ) \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(193\) |
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Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (88) = 176\).
Time = 0.30 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.97 \[ \int \frac {\sin (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {5 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) - 4\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 4\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )} + 3 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) + 4 \, {\left (5 \, \cos \left (d x + c\right )^{2} + {\left (5 \, \cos \left (d x + c\right ) + 4\right )} \sin \left (d x + c\right ) + \cos \left (d x + c\right ) - 4\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{64 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d + {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
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\[ \int \frac {\sin (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int \frac {\sin {\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {\sin (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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none
Time = 0.33 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.76 \[ \int \frac {\sin (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {\sqrt {2} {\left (5 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{32 \, {\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{3} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \]
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Timed out. \[ \int \frac {\sin (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int \frac {\sin \left (c+d\,x\right )}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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