Optimal. Leaf size=87 \[ \frac {\sin ^3(c+d x)}{3 a^2 d}-\frac {\sin ^2(c+d x)}{a^2 d}+\frac {3 \sin (c+d x)}{a^2 d}-\frac {1}{d \left (a^2 \sin (c+d x)+a^2\right )}-\frac {4 \log (\sin (c+d x)+1)}{a^2 d} \]
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Rubi [A] time = 0.09, antiderivative size = 87, 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} \[ \frac {\sin ^3(c+d x)}{3 a^2 d}-\frac {\sin ^2(c+d x)}{a^2 d}+\frac {3 \sin (c+d x)}{a^2 d}-\frac {1}{d \left (a^2 \sin (c+d x)+a^2\right )}-\frac {4 \log (\sin (c+d x)+1)}{a^2 d} \]
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))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4}{a^4 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^4}{(a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (3 a^2-2 a x+x^2+\frac {a^4}{(a+x)^2}-\frac {4 a^3}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=-\frac {4 \log (1+\sin (c+d x))}{a^2 d}+\frac {3 \sin (c+d x)}{a^2 d}-\frac {\sin ^2(c+d x)}{a^2 d}+\frac {\sin ^3(c+d x)}{3 a^2 d}-\frac {1}{d \left (a^2+a^2 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.58, size = 73, normalized size = 0.84 \[ \frac {\sin ^3(c+d x)-3 \sin ^2(c+d x)+9 \sin (c+d x)-12 \log (\sin (c+d x)+1)-\frac {3}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}}{3 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 81, normalized size = 0.93 \[ \frac {2 \, \cos \left (d x + c\right )^{4} - 16 \, \cos \left (d x + c\right )^{2} - 24 \, {\left (\sin \left (d x + c\right ) + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (4 \, \cos \left (d x + c\right )^{2} + 17\right )} \sin \left (d x + c\right ) + 11}{6 \, {\left (a^{2} d \sin \left (d x + c\right ) + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 107, normalized size = 1.23 \[ -\frac {\frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{3} {\left (\frac {6 \, a}{a \sin \left (d x + c\right ) + a} - \frac {18 \, a^{2}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}} - 1\right )}}{a^{5}} - \frac {12 \, \log \left (\frac {{\left | a \sin \left (d x + c\right ) + a \right |}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2} {\left | a \right |}}\right )}{a^{2}} + \frac {3}{{\left (a \sin \left (d x + c\right ) + a\right )} a}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 83, normalized size = 0.95 \[ \frac {\sin ^{3}\left (d x +c \right )}{3 a^{2} d}-\frac {\sin ^{2}\left (d x +c \right )}{a^{2} d}+\frac {3 \sin \left (d x +c \right )}{a^{2} d}-\frac {4 \ln \left (1+\sin \left (d x +c \right )\right )}{a^{2} d}-\frac {1}{d \,a^{2} \left (1+\sin \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 70, normalized size = 0.80 \[ -\frac {\frac {3}{a^{2} \sin \left (d x + c\right ) + a^{2}} - \frac {\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )^{2} + 9 \, \sin \left (d x + c\right )}{a^{2}} + \frac {12 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 72, normalized size = 0.83 \[ -\frac {\frac {1}{a^2\,\sin \left (c+d\,x\right )+a^2}+\frac {4\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{a^2}-\frac {3\,\sin \left (c+d\,x\right )}{a^2}+\frac {{\sin \left (c+d\,x\right )}^2}{a^2}-\frac {{\sin \left (c+d\,x\right )}^3}{3\,a^2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.34, size = 201, normalized size = 2.31 \[ \begin {cases} - \frac {12 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin {\left (c + d x \right )}}{3 a^{2} d \sin {\left (c + d x \right )} + 3 a^{2} d} - \frac {12 \log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{3 a^{2} d \sin {\left (c + d x \right )} + 3 a^{2} d} + \frac {\sin ^{4}{\left (c + d x \right )}}{3 a^{2} d \sin {\left (c + d x \right )} + 3 a^{2} d} - \frac {2 \sin ^{3}{\left (c + d x \right )}}{3 a^{2} d \sin {\left (c + d x \right )} + 3 a^{2} d} + \frac {6 \sin ^{2}{\left (c + d x \right )}}{3 a^{2} d \sin {\left (c + d x \right )} + 3 a^{2} d} - \frac {12}{3 a^{2} d \sin {\left (c + d x \right )} + 3 a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{4}{\relax (c )} \cos {\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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