Optimal. Leaf size=68 \[ \frac {\sin ^3(c+d x)}{3 a^3 d}-\frac {3 \sin ^2(c+d x)}{2 a^3 d}+\frac {4 \sin (c+d x)}{a^3 d}-\frac {4 \log (\sin (c+d x)+1)}{a^3 d} \]
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Rubi [A] time = 0.08, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2836, 12, 77} \[ \frac {\sin ^3(c+d x)}{3 a^3 d}-\frac {3 \sin ^2(c+d x)}{2 a^3 d}+\frac {4 \sin (c+d x)}{a^3 d}-\frac {4 \log (\sin (c+d x)+1)}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 77
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2 x}{a (a+x)} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2 x}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^6 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (4 a^2-3 a x+x^2-\frac {4 a^3}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^6 d}\\ &=-\frac {4 \log (1+\sin (c+d x))}{a^3 d}+\frac {4 \sin (c+d x)}{a^3 d}-\frac {3 \sin ^2(c+d x)}{2 a^3 d}+\frac {\sin ^3(c+d x)}{3 a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 51, normalized size = 0.75 \[ \frac {32 \sin ^3(c+d x)-144 \sin ^2(c+d x)+384 \sin (c+d x)-384 \log (\sin (c+d x)+1)+15}{96 a^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 48, normalized size = 0.71 \[ \frac {9 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (\cos \left (d x + c\right )^{2} - 13\right )} \sin \left (d x + c\right ) - 24 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{6 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 141, normalized size = 2.07 \[ \frac {2 \, {\left (\frac {6 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}{a^{3}} - \frac {12 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac {11 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 42 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 28 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 42 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 11}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{3}}\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 65, normalized size = 0.96 \[ -\frac {4 \ln \left (1+\sin \left (d x +c \right )\right )}{a^{3} d}+\frac {4 \sin \left (d x +c \right )}{a^{3} d}-\frac {3 \left (\sin ^{2}\left (d x +c \right )\right )}{2 a^{3} d}+\frac {\sin ^{3}\left (d x +c \right )}{3 a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 53, normalized size = 0.78 \[ \frac {\frac {2 \, \sin \left (d x + c\right )^{3} - 9 \, \sin \left (d x + c\right )^{2} + 24 \, \sin \left (d x + c\right )}{a^{3}} - \frac {24 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 57, normalized size = 0.84 \[ -\frac {\frac {4\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{a^3}-\frac {4\,\sin \left (c+d\,x\right )}{a^3}+\frac {3\,{\sin \left (c+d\,x\right )}^2}{2\,a^3}-\frac {{\sin \left (c+d\,x\right )}^3}{3\,a^3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 86.48, size = 1102, normalized size = 16.21 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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