Optimal. Leaf size=95 \[ \frac {\sin (c+d x)}{a^4 d}-\frac {6}{d \left (a^4 \sin (c+d x)+a^4\right )}-\frac {4 \log (\sin (c+d x)+1)}{a^4 d}+\frac {2}{d \left (a^2 \sin (c+d x)+a^2\right )^2}-\frac {1}{3 a d (a \sin (c+d x)+a)^3} \]
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Rubi [A] time = 0.10, antiderivative size = 95, 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 = {2833, 12, 43} \[ \frac {\sin (c+d x)}{a^4 d}-\frac {6}{d \left (a^4 \sin (c+d x)+a^4\right )}+\frac {2}{d \left (a^2 \sin (c+d x)+a^2\right )^2}-\frac {4 \log (\sin (c+d x)+1)}{a^4 d}-\frac {1}{3 a d (a \sin (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4}{a^4 (a+x)^4} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^4}{(a+x)^4} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (1+\frac {a^4}{(a+x)^4}-\frac {4 a^3}{(a+x)^3}+\frac {6 a^2}{(a+x)^2}-\frac {4 a}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=-\frac {4 \log (1+\sin (c+d x))}{a^4 d}+\frac {\sin (c+d x)}{a^4 d}-\frac {1}{3 a d (a+a \sin (c+d x))^3}+\frac {2}{d \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {6}{d \left (a^4+a^4 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 6.56, size = 127, normalized size = 1.34 \[ -\frac {3 (2 \sin (c+d x)+1)^2}{16 a^4 d (\sin (c+d x)+1)^3}-\frac {\frac {252 \sin ^2(c+d x)+444 \sin (c+d x)+197}{(\sin (c+d x)+1)^3}-48 \sin (c+d x)+192 \log (\sin (c+d x)+1)}{48 a^4 d}-\frac {1}{24 a^4 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 132, normalized size = 1.39 \[ -\frac {3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} + 12 \, {\left (3 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) - 4\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 9 \, {\left (\cos \left (d x + c\right )^{2} + 2\right )} \sin \left (d x + c\right ) - 19}{3 \, {\left (3 \, a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d + {\left (a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 66, normalized size = 0.69 \[ -\frac {\frac {12 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4}} - \frac {3 \, \sin \left (d x + c\right )}{a^{4}} + \frac {18 \, \sin \left (d x + c\right )^{2} + 30 \, \sin \left (d x + c\right ) + 13}{a^{4} {\left (\sin \left (d x + c\right ) + 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 86, normalized size = 0.91 \[ \frac {\sin \left (d x +c \right )}{a^{4} d}-\frac {1}{3 d \,a^{4} \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {4 \ln \left (1+\sin \left (d x +c \right )\right )}{a^{4} d}-\frac {6}{d \,a^{4} \left (1+\sin \left (d x +c \right )\right )}+\frac {2}{d \,a^{4} \left (1+\sin \left (d x +c \right )\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 94, normalized size = 0.99 \[ -\frac {\frac {18 \, \sin \left (d x + c\right )^{2} + 30 \, \sin \left (d x + c\right ) + 13}{a^{4} \sin \left (d x + c\right )^{3} + 3 \, a^{4} \sin \left (d x + c\right )^{2} + 3 \, a^{4} \sin \left (d x + c\right ) + a^{4}} + \frac {12 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}} - \frac {3 \, \sin \left (d x + c\right )}{a^{4}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 69, normalized size = 0.73 \[ \frac {\sin \left (c+d\,x\right )}{a^4\,d}-\frac {4\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{a^4\,d}-\frac {6\,{\sin \left (c+d\,x\right )}^2+10\,\sin \left (c+d\,x\right )+\frac {13}{3}}{a^4\,d\,{\left (\sin \left (c+d\,x\right )+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.57, size = 527, normalized size = 5.55 \[ \begin {cases} - \frac {12 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin ^{3}{\left (c + d x \right )}}{3 a^{4} d \sin ^{3}{\left (c + d x \right )} + 9 a^{4} d \sin ^{2}{\left (c + d x \right )} + 9 a^{4} d \sin {\left (c + d x \right )} + 3 a^{4} d} - \frac {36 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin ^{2}{\left (c + d x \right )}}{3 a^{4} d \sin ^{3}{\left (c + d x \right )} + 9 a^{4} d \sin ^{2}{\left (c + d x \right )} + 9 a^{4} d \sin {\left (c + d x \right )} + 3 a^{4} d} - \frac {36 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin {\left (c + d x \right )}}{3 a^{4} d \sin ^{3}{\left (c + d x \right )} + 9 a^{4} d \sin ^{2}{\left (c + d x \right )} + 9 a^{4} d \sin {\left (c + d x \right )} + 3 a^{4} d} - \frac {12 \log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{3 a^{4} d \sin ^{3}{\left (c + d x \right )} + 9 a^{4} d \sin ^{2}{\left (c + d x \right )} + 9 a^{4} d \sin {\left (c + d x \right )} + 3 a^{4} d} + \frac {3 \sin ^{4}{\left (c + d x \right )}}{3 a^{4} d \sin ^{3}{\left (c + d x \right )} + 9 a^{4} d \sin ^{2}{\left (c + d x \right )} + 9 a^{4} d \sin {\left (c + d x \right )} + 3 a^{4} d} - \frac {36 \sin ^{2}{\left (c + d x \right )}}{3 a^{4} d \sin ^{3}{\left (c + d x \right )} + 9 a^{4} d \sin ^{2}{\left (c + d x \right )} + 9 a^{4} d \sin {\left (c + d x \right )} + 3 a^{4} d} - \frac {54 \sin {\left (c + d x \right )}}{3 a^{4} d \sin ^{3}{\left (c + d x \right )} + 9 a^{4} d \sin ^{2}{\left (c + d x \right )} + 9 a^{4} d \sin {\left (c + d x \right )} + 3 a^{4} d} - \frac {22}{3 a^{4} d \sin ^{3}{\left (c + d x \right )} + 9 a^{4} d \sin ^{2}{\left (c + d x \right )} + 9 a^{4} d \sin {\left (c + d x \right )} + 3 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{4}{\relax (c )} \cos {\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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