Integrand size = 23, antiderivative size = 183 \[ \int \frac {\sin ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {163 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {17 \cos (c+d x) \sin ^2(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}+\frac {197 \cos (c+d x)}{24 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {95 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{48 a^3 d} \]
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Time = 0.55 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2844, 3056, 3047, 3102, 2830, 2728, 212} \[ \int \frac {\sin ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {163 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{16 \sqrt {2} a^{5/2} d}-\frac {95 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{48 a^3 d}+\frac {197 \cos (c+d x)}{24 a^2 d \sqrt {a \sin (c+d x)+a}}+\frac {\sin ^3(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}+\frac {17 \sin ^2(c+d x) \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}} \]
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Rule 212
Rule 2728
Rule 2830
Rule 2844
Rule 3047
Rule 3056
Rule 3102
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (c+d x) \sin ^3(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}-\frac {\int \frac {\sin ^2(c+d x) \left (3 a-\frac {11}{2} a \sin (c+d x)\right )}{(a+a \sin (c+d x))^{3/2}} \, dx}{4 a^2} \\ & = \frac {\cos (c+d x) \sin ^3(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {17 \cos (c+d x) \sin ^2(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac {\int \frac {\sin (c+d x) \left (17 a^2-\frac {95}{4} a^2 \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{8 a^4} \\ & = \frac {\cos (c+d x) \sin ^3(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {17 \cos (c+d x) \sin ^2(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac {\int \frac {17 a^2 \sin (c+d x)-\frac {95}{4} a^2 \sin ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{8 a^4} \\ & = \frac {\cos (c+d x) \sin ^3(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {17 \cos (c+d x) \sin ^2(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac {95 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{48 a^3 d}-\frac {\int \frac {-\frac {95 a^3}{8}+\frac {197}{4} a^3 \sin (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{12 a^5} \\ & = \frac {\cos (c+d x) \sin ^3(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {17 \cos (c+d x) \sin ^2(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}+\frac {197 \cos (c+d x)}{24 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {95 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{48 a^3 d}+\frac {163 \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx}{32 a^2} \\ & = \frac {\cos (c+d x) \sin ^3(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {17 \cos (c+d x) \sin ^2(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}+\frac {197 \cos (c+d x)}{24 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {95 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{48 a^3 d}-\frac {163 \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 {163 \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) \sin ^3(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {17 \cos (c+d x) \sin ^2(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}+\frac {197 \cos (c+d x)}{24 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {95 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{48 a^3 d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.68 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.08 \[ \int \frac {\sin ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (279 \cos \left (\frac {1}{2} (c+d x)\right )+399 \cos \left (\frac {3}{2} (c+d x)\right )-88 \cos \left (\frac {5}{2} (c+d x)\right )+8 \cos \left (\frac {7}{2} (c+d x)\right )-279 \sin \left (\frac {1}{2} (c+d x)\right )+(978+978 i) (-1)^{3/4} \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (c+d x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4+399 \sin \left (\frac {3}{2} (c+d x)\right )+88 \sin \left (\frac {5}{2} (c+d x)\right )+8 \sin \left (\frac {7}{2} (c+d x)\right )\right )}{96 d (a (1+\sin (c+d x)))^{5/2}} \]
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Time = 0.76 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.60
method | result | size |
default | \(-\frac {\left (64 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {a}\, \left (\cos ^{2}\left (d x +c \right )\right )+384 \sqrt {a -a \sin \left (d x +c \right )}\, a^{\frac {3}{2}} \left (\cos ^{2}\left (d x +c \right )\right )-489 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} \left (\cos ^{2}\left (d x +c \right )\right )-128 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {a}\, \sin \left (d x +c \right )-768 \sqrt {a -a \sin \left (d x +c \right )}\, a^{\frac {3}{2}} \sin \left (d x +c \right )+978 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} \sin \left (d x +c \right )+46 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {a}-1092 \sqrt {a -a \sin \left (d x +c \right )}\, a^{\frac {3}{2}}+978 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2}\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{96 a^{\frac {9}{2}} \left (1+\sin \left (d x +c \right )\right ) \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(293\) |
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Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (156) = 312\).
Time = 0.29 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.97 \[ \int \frac {\sin ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {489 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) - 4\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 4\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )} + 3 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) - 4 \, {\left (32 \, \cos \left (d x + c\right )^{4} - 160 \, \cos \left (d x + c\right )^{3} + 279 \, \cos \left (d x + c\right )^{2} + {\left (32 \, \cos \left (d x + c\right )^{3} + 192 \, \cos \left (d x + c\right )^{2} + 471 \, \cos \left (d x + c\right ) + 12\right )} \sin \left (d x + c\right ) + 459 \, \cos \left (d x + c\right ) - 12\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{192 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d + {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
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\[ \int \frac {\sin ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int \frac {\sin ^{4}{\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {\sin ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int { \frac {\sin \left (d x + c\right )^{4}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 0.59 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.78 \[ \int \frac {\sin ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {\frac {3 \, \sqrt {2} {\left (29 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 27 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {128 \, \sqrt {2} {\left (a^{\frac {13}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, a^{\frac {13}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{9} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{96 \, d} \]
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Timed out. \[ \int \frac {\sin ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int \frac {{\sin \left (c+d\,x\right )}^4}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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