Optimal. Leaf size=180 \[ \frac {a^2 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^7 d}-\frac {a \left (a^2-b^2\right )^2 \sin (c+d x)}{b^6 d}+\frac {\left (a^2-b^2\right )^2 \sin ^2(c+d x)}{2 b^5 d}-\frac {a \left (a^2-2 b^2\right ) \sin ^3(c+d x)}{3 b^4 d}+\frac {\left (a^2-2 b^2\right ) \sin ^4(c+d x)}{4 b^3 d}-\frac {a \sin ^5(c+d x)}{5 b^2 d}+\frac {\sin ^6(c+d x)}{6 b d} \]
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Rubi [A] time = 0.21, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2837, 12, 948} \[ \frac {\left (a^2-2 b^2\right ) \sin ^4(c+d x)}{4 b^3 d}-\frac {a \left (a^2-2 b^2\right ) \sin ^3(c+d x)}{3 b^4 d}+\frac {\left (a^2-b^2\right )^2 \sin ^2(c+d x)}{2 b^5 d}-\frac {a \left (a^2-b^2\right )^2 \sin (c+d x)}{b^6 d}+\frac {a^2 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^7 d}-\frac {a \sin ^5(c+d x)}{5 b^2 d}+\frac {\sin ^6(c+d x)}{6 b d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 948
Rule 2837
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (b^2-x^2\right )^2}{b^2 (a+x)} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (b^2-x^2\right )^2}{a+x} \, dx,x,b \sin (c+d x)\right )}{b^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-a \left (a^2-b^2\right )^2+\left (a^2-b^2\right )^2 x-a \left (a^2-2 b^2\right ) x^2+\left (a^2-2 b^2\right ) x^3-a x^4+x^5+\frac {\left (a^3-a b^2\right )^2}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^7 d}\\ &=\frac {a^2 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^7 d}-\frac {a \left (a^2-b^2\right )^2 \sin (c+d x)}{b^6 d}+\frac {\left (a^2-b^2\right )^2 \sin ^2(c+d x)}{2 b^5 d}-\frac {a \left (a^2-2 b^2\right ) \sin ^3(c+d x)}{3 b^4 d}+\frac {\left (a^2-2 b^2\right ) \sin ^4(c+d x)}{4 b^3 d}-\frac {a \sin ^5(c+d x)}{5 b^2 d}+\frac {\sin ^6(c+d x)}{6 b d}\\ \end {align*}
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Mathematica [A] time = 0.72, size = 153, normalized size = 0.85 \[ \frac {60 \left (a^3-a b^2\right )^2 \log (a+b \sin (c+d x))+30 b^2 \left (a^2-b^2\right )^2 \sin ^2(c+d x)-60 a b \left (a^2-b^2\right )^2 \sin (c+d x)+15 b^4 \left (a^2-2 b^2\right ) \sin ^4(c+d x)-20 a b^3 \left (a^2-2 b^2\right ) \sin ^3(c+d x)-12 a b^5 \sin ^5(c+d x)+10 b^6 \sin ^6(c+d x)}{60 b^7 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 164, normalized size = 0.91 \[ -\frac {10 \, b^{6} \cos \left (d x + c\right )^{6} - 15 \, a^{2} b^{4} \cos \left (d x + c\right )^{4} + 30 \, {\left (a^{4} b^{2} - a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} - 60 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + 4 \, {\left (3 \, a b^{5} \cos \left (d x + c\right )^{4} + 15 \, a^{5} b - 25 \, a^{3} b^{3} + 8 \, a b^{5} - {\left (5 \, a^{3} b^{3} - 4 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{60 \, b^{7} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 213, normalized size = 1.18 \[ \frac {\frac {10 \, b^{5} \sin \left (d x + c\right )^{6} - 12 \, a b^{4} \sin \left (d x + c\right )^{5} + 15 \, a^{2} b^{3} \sin \left (d x + c\right )^{4} - 30 \, b^{5} \sin \left (d x + c\right )^{4} - 20 \, a^{3} b^{2} \sin \left (d x + c\right )^{3} + 40 \, a b^{4} \sin \left (d x + c\right )^{3} + 30 \, a^{4} b \sin \left (d x + c\right )^{2} - 60 \, a^{2} b^{3} \sin \left (d x + c\right )^{2} + 30 \, b^{5} \sin \left (d x + c\right )^{2} - 60 \, a^{5} \sin \left (d x + c\right ) + 120 \, a^{3} b^{2} \sin \left (d x + c\right ) - 60 \, a b^{4} \sin \left (d x + c\right )}{b^{6}} + \frac {60 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{7}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 273, normalized size = 1.52 \[ \frac {\sin ^{6}\left (d x +c \right )}{6 b d}-\frac {a \left (\sin ^{5}\left (d x +c \right )\right )}{5 b^{2} d}+\frac {\left (\sin ^{4}\left (d x +c \right )\right ) a^{2}}{4 d \,b^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{2 b d}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) a^{3}}{3 d \,b^{4}}+\frac {2 a \left (\sin ^{3}\left (d x +c \right )\right )}{3 b^{2} d}+\frac {\left (\sin ^{2}\left (d x +c \right )\right ) a^{4}}{2 d \,b^{5}}-\frac {a^{2} \left (\sin ^{2}\left (d x +c \right )\right )}{d \,b^{3}}+\frac {\sin ^{2}\left (d x +c \right )}{2 b d}-\frac {\sin \left (d x +c \right ) a^{5}}{d \,b^{6}}+\frac {2 \sin \left (d x +c \right ) a^{3}}{d \,b^{4}}-\frac {a \sin \left (d x +c \right )}{b^{2} d}+\frac {a^{6} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,b^{7}}-\frac {2 a^{4} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,b^{5}}+\frac {\ln \left (a +b \sin \left (d x +c \right )\right ) a^{2}}{d \,b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 172, normalized size = 0.96 \[ \frac {\frac {10 \, b^{5} \sin \left (d x + c\right )^{6} - 12 \, a b^{4} \sin \left (d x + c\right )^{5} + 15 \, {\left (a^{2} b^{3} - 2 \, b^{5}\right )} \sin \left (d x + c\right )^{4} - 20 \, {\left (a^{3} b^{2} - 2 \, a b^{4}\right )} \sin \left (d x + c\right )^{3} + 30 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )^{2} - 60 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (d x + c\right )}{b^{6}} + \frac {60 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{7}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.78, size = 191, normalized size = 1.06 \[ \frac {{\sin \left (c+d\,x\right )}^2\,\left (\frac {1}{2\,b}-\frac {a^2\,\left (\frac {1}{b}-\frac {a^2}{2\,b^3}\right )}{b^2}\right )-{\sin \left (c+d\,x\right )}^4\,\left (\frac {1}{2\,b}-\frac {a^2}{4\,b^3}\right )+\frac {{\sin \left (c+d\,x\right )}^6}{6\,b}-\frac {a\,{\sin \left (c+d\,x\right )}^5}{5\,b^2}+\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (a^6-2\,a^4\,b^2+a^2\,b^4\right )}{b^7}-\frac {a\,\sin \left (c+d\,x\right )\,\left (\frac {1}{b}-\frac {a^2\,\left (\frac {2}{b}-\frac {a^2}{b^3}\right )}{b^2}\right )}{b}+\frac {a\,{\sin \left (c+d\,x\right )}^3\,\left (\frac {2}{b}-\frac {a^2}{b^3}\right )}{3\,b}}{d} \]
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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