Optimal. Leaf size=148 \[ -\frac {a \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^6 d}+\frac {\left (a^2-b^2\right )^2 \sin (c+d x)}{b^5 d}-\frac {a \left (a^2-2 b^2\right ) \sin ^2(c+d x)}{2 b^4 d}+\frac {\left (a^2-2 b^2\right ) \sin ^3(c+d x)}{3 b^3 d}-\frac {a \sin ^4(c+d x)}{4 b^2 d}+\frac {\sin ^5(c+d x)}{5 b d} \]
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Rubi [A] time = 0.14, antiderivative size = 148, 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 = {2837, 12, 772} \[ \frac {\left (a^2-2 b^2\right ) \sin ^3(c+d x)}{3 b^3 d}-\frac {a \left (a^2-2 b^2\right ) \sin ^2(c+d x)}{2 b^4 d}+\frac {\left (a^2-b^2\right )^2 \sin (c+d x)}{b^5 d}-\frac {a \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^6 d}-\frac {a \sin ^4(c+d x)}{4 b^2 d}+\frac {\sin ^5(c+d x)}{5 b d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 772
Rule 2837
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x \left (b^2-x^2\right )^2}{b (a+x)} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x \left (b^2-x^2\right )^2}{a+x} \, dx,x,b \sin (c+d x)\right )}{b^6 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\left (a^2-b^2\right )^2-a \left (a^2-2 b^2\right ) x+\left (a^2-2 b^2\right ) x^2-a x^3+x^4-\frac {a \left (a^2-b^2\right )^2}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^6 d}\\ &=-\frac {a \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^6 d}+\frac {\left (a^2-b^2\right )^2 \sin (c+d x)}{b^5 d}-\frac {a \left (a^2-2 b^2\right ) \sin ^2(c+d x)}{2 b^4 d}+\frac {\left (a^2-2 b^2\right ) \sin ^3(c+d x)}{3 b^3 d}-\frac {a \sin ^4(c+d x)}{4 b^2 d}+\frac {\sin ^5(c+d x)}{5 b d}\\ \end {align*}
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Mathematica [A] time = 0.64, size = 128, normalized size = 0.86 \[ \frac {-30 a b^2 \left (a^2-2 b^2\right ) \sin ^2(c+d x)+60 b \left (a^2-b^2\right )^2 \sin (c+d x)-60 a \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))+20 b^3 \left (a^2-2 b^2\right ) \sin ^3(c+d x)-15 a b^4 \sin ^4(c+d x)+12 b^5 \sin ^5(c+d x)}{60 b^6 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 142, normalized size = 0.96 \[ -\frac {15 \, a b^{4} \cos \left (d x + c\right )^{4} - 30 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )^{2} + 60 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 4 \, {\left (3 \, b^{5} \cos \left (d x + c\right )^{4} + 15 \, a^{4} b - 25 \, a^{2} b^{3} + 8 \, b^{5} - {\left (5 \, a^{2} b^{3} - 4 \, b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{60 \, b^{6} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 165, normalized size = 1.11 \[ \frac {\frac {12 \, b^{4} \sin \left (d x + c\right )^{5} - 15 \, a b^{3} \sin \left (d x + c\right )^{4} + 20 \, a^{2} b^{2} \sin \left (d x + c\right )^{3} - 40 \, b^{4} \sin \left (d x + c\right )^{3} - 30 \, a^{3} b \sin \left (d x + c\right )^{2} + 60 \, a b^{3} \sin \left (d x + c\right )^{2} + 60 \, a^{4} \sin \left (d x + c\right ) - 120 \, a^{2} b^{2} \sin \left (d x + c\right ) + 60 \, b^{4} \sin \left (d x + c\right )}{b^{5}} - \frac {60 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{6}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 215, normalized size = 1.45 \[ \frac {\sin ^{5}\left (d x +c \right )}{5 b d}-\frac {a \left (\sin ^{4}\left (d x +c \right )\right )}{4 b^{2} d}+\frac {\left (\sin ^{3}\left (d x +c \right )\right ) a^{2}}{3 d \,b^{3}}-\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{3 b d}-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) a^{3}}{2 d \,b^{4}}+\frac {a \left (\sin ^{2}\left (d x +c \right )\right )}{b^{2} d}+\frac {a^{4} \sin \left (d x +c \right )}{d \,b^{5}}-\frac {2 a^{2} \sin \left (d x +c \right )}{b^{3} d}+\frac {\sin \left (d x +c \right )}{b d}-\frac {a^{5} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,b^{6}}+\frac {2 a^{3} \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{4} d}-\frac {a \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 139, normalized size = 0.94 \[ \frac {\frac {12 \, b^{4} \sin \left (d x + c\right )^{5} - 15 \, a b^{3} \sin \left (d x + c\right )^{4} + 20 \, {\left (a^{2} b^{2} - 2 \, b^{4}\right )} \sin \left (d x + c\right )^{3} - 30 \, {\left (a^{3} b - 2 \, a b^{3}\right )} \sin \left (d x + c\right )^{2} + 60 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )}{b^{5}} - \frac {60 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{6}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 150, normalized size = 1.01 \[ \frac {\sin \left (c+d\,x\right )\,\left (\frac {1}{b}-\frac {a^2\,\left (\frac {2}{b}-\frac {a^2}{b^3}\right )}{b^2}\right )-{\sin \left (c+d\,x\right )}^3\,\left (\frac {2}{3\,b}-\frac {a^2}{3\,b^3}\right )+\frac {{\sin \left (c+d\,x\right )}^5}{5\,b}-\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (a^5-2\,a^3\,b^2+a\,b^4\right )}{b^6}-\frac {a\,{\sin \left (c+d\,x\right )}^4}{4\,b^2}+\frac {a\,{\sin \left (c+d\,x\right )}^2\,\left (\frac {2}{b}-\frac {a^2}{b^3}\right )}{2\,b}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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