Optimal. Leaf size=212 \[ \frac {a^2 \left (a^2-b^2\right )^2 \sin (c+d x)}{b^7 d}-\frac {a \left (a^2-b^2\right )^2 \sin ^2(c+d x)}{2 b^6 d}+\frac {\left (a^2-b^2\right )^2 \sin ^3(c+d x)}{3 b^5 d}-\frac {a \left (a^2-2 b^2\right ) \sin ^4(c+d x)}{4 b^4 d}+\frac {\left (a^2-2 b^2\right ) \sin ^5(c+d x)}{5 b^3 d}-\frac {a^3 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^8 d}-\frac {a \sin ^6(c+d x)}{6 b^2 d}+\frac {\sin ^7(c+d x)}{7 b d} \]
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Rubi [A] time = 0.24, antiderivative size = 212, 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} \[ \frac {\left (a^2-2 b^2\right ) \sin ^5(c+d x)}{5 b^3 d}-\frac {a \left (a^2-2 b^2\right ) \sin ^4(c+d x)}{4 b^4 d}+\frac {\left (a^2-b^2\right )^2 \sin ^3(c+d x)}{3 b^5 d}-\frac {a \left (a^2-b^2\right )^2 \sin ^2(c+d x)}{2 b^6 d}+\frac {a^2 \left (a^2-b^2\right )^2 \sin (c+d x)}{b^7 d}-\frac {a^3 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^8 d}-\frac {a \sin ^6(c+d x)}{6 b^2 d}+\frac {\sin ^7(c+d x)}{7 b d} \]
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 ^3(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^3 \left (b^2-x^2\right )^2}{b^3 (a+x)} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^3 \left (b^2-x^2\right )^2}{a+x} \, dx,x,b \sin (c+d x)\right )}{b^8 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\left (a^3-a b^2\right )^2-a \left (a^2-b^2\right )^2 x+\left (a^2-b^2\right )^2 x^2-a \left (a^2-2 b^2\right ) x^3+\left (a^2-2 b^2\right ) x^4-a x^5+x^6-\frac {a^3 \left (a^2-b^2\right )^2}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^8 d}\\ &=-\frac {a^3 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^8 d}+\frac {a^2 \left (a^2-b^2\right )^2 \sin (c+d x)}{b^7 d}-\frac {a \left (a^2-b^2\right )^2 \sin ^2(c+d x)}{2 b^6 d}+\frac {\left (a^2-b^2\right )^2 \sin ^3(c+d x)}{3 b^5 d}-\frac {a \left (a^2-2 b^2\right ) \sin ^4(c+d x)}{4 b^4 d}+\frac {\left (a^2-2 b^2\right ) \sin ^5(c+d x)}{5 b^3 d}-\frac {a \sin ^6(c+d x)}{6 b^2 d}+\frac {\sin ^7(c+d x)}{7 b d}\\ \end {align*}
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Mathematica [A] time = 1.27, size = 180, normalized size = 0.85 \[ \frac {420 b \left (a^3-a b^2\right )^2 \sin (c+d x)-210 a b^2 \left (a^2-b^2\right )^2 \sin ^2(c+d x)+84 b^5 \left (a^2-2 b^2\right ) \sin ^5(c+d x)-105 a b^4 \left (a^2-2 b^2\right ) \sin ^4(c+d x)+140 b^3 \left (a^2-b^2\right )^2 \sin ^3(c+d x)-420 a^3 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))-70 a b^6 \sin ^6(c+d x)+60 b^7 \sin ^7(c+d x)}{420 b^8 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 199, normalized size = 0.94 \[ \frac {70 \, a b^{6} \cos \left (d x + c\right )^{6} - 105 \, a^{3} b^{4} \cos \left (d x + c\right )^{4} + 210 \, {\left (a^{5} b^{2} - a^{3} b^{4}\right )} \cos \left (d x + c\right )^{2} - 420 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 4 \, {\left (15 \, b^{7} \cos \left (d x + c\right )^{6} - 105 \, a^{6} b + 175 \, a^{4} b^{3} - 56 \, a^{2} b^{5} - 8 \, b^{7} - 3 \, {\left (7 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{4} + {\left (35 \, a^{4} b^{3} - 28 \, a^{2} b^{5} - 4 \, b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{420 \, b^{8} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 261, normalized size = 1.23 \[ \frac {\frac {60 \, b^{6} \sin \left (d x + c\right )^{7} - 70 \, a b^{5} \sin \left (d x + c\right )^{6} + 84 \, a^{2} b^{4} \sin \left (d x + c\right )^{5} - 168 \, b^{6} \sin \left (d x + c\right )^{5} - 105 \, a^{3} b^{3} \sin \left (d x + c\right )^{4} + 210 \, a b^{5} \sin \left (d x + c\right )^{4} + 140 \, a^{4} b^{2} \sin \left (d x + c\right )^{3} - 280 \, a^{2} b^{4} \sin \left (d x + c\right )^{3} + 140 \, b^{6} \sin \left (d x + c\right )^{3} - 210 \, a^{5} b \sin \left (d x + c\right )^{2} + 420 \, a^{3} b^{3} \sin \left (d x + c\right )^{2} - 210 \, a b^{5} \sin \left (d x + c\right )^{2} + 420 \, a^{6} \sin \left (d x + c\right ) - 840 \, a^{4} b^{2} \sin \left (d x + c\right ) + 420 \, a^{2} b^{4} \sin \left (d x + c\right )}{b^{7}} - \frac {420 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{8}}}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 329, normalized size = 1.55 \[ \frac {\sin ^{7}\left (d x +c \right )}{7 b d}-\frac {a \left (\sin ^{6}\left (d x +c \right )\right )}{6 b^{2} d}+\frac {\left (\sin ^{5}\left (d x +c \right )\right ) a^{2}}{5 d \,b^{3}}-\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{5 b d}-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) a^{3}}{4 d \,b^{4}}+\frac {a \left (\sin ^{4}\left (d x +c \right )\right )}{2 b^{2} d}+\frac {a^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{3 d \,b^{5}}-\frac {2 \left (\sin ^{3}\left (d x +c \right )\right ) a^{2}}{3 d \,b^{3}}+\frac {\sin ^{3}\left (d x +c \right )}{3 b d}-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) a^{5}}{2 d \,b^{6}}+\frac {\left (\sin ^{2}\left (d x +c \right )\right ) a^{3}}{d \,b^{4}}-\frac {a \left (\sin ^{2}\left (d x +c \right )\right )}{2 b^{2} d}+\frac {\sin \left (d x +c \right ) a^{6}}{d \,b^{7}}-\frac {2 a^{4} \sin \left (d x +c \right )}{d \,b^{5}}+\frac {a^{2} \sin \left (d x +c \right )}{b^{3} d}-\frac {a^{7} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,b^{8}}+\frac {2 a^{5} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,b^{6}}-\frac {a^{3} \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 205, normalized size = 0.97 \[ \frac {\frac {60 \, b^{6} \sin \left (d x + c\right )^{7} - 70 \, a b^{5} \sin \left (d x + c\right )^{6} + 84 \, {\left (a^{2} b^{4} - 2 \, b^{6}\right )} \sin \left (d x + c\right )^{5} - 105 \, {\left (a^{3} b^{3} - 2 \, a b^{5}\right )} \sin \left (d x + c\right )^{4} + 140 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \sin \left (d x + c\right )^{3} - 210 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \sin \left (d x + c\right )^{2} + 420 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sin \left (d x + c\right )}{b^{7}} - \frac {420 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{8}}}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 236, normalized size = 1.11 \[ -\frac {{\sin \left (c+d\,x\right )}^5\,\left (\frac {2}{5\,b}-\frac {a^2}{5\,b^3}\right )-\frac {{\sin \left (c+d\,x\right )}^7}{7\,b}-{\sin \left (c+d\,x\right )}^3\,\left (\frac {1}{3\,b}-\frac {a^2\,\left (\frac {2}{b}-\frac {a^2}{b^3}\right )}{3\,b^2}\right )+\frac {a\,{\sin \left (c+d\,x\right )}^6}{6\,b^2}+\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}{b^8}-\frac {a\,{\sin \left (c+d\,x\right )}^4\,\left (\frac {2}{b}-\frac {a^2}{b^3}\right )}{4\,b}+\frac {a\,{\sin \left (c+d\,x\right )}^2\,\left (\frac {1}{b}-\frac {a^2\,\left (\frac {2}{b}-\frac {a^2}{b^3}\right )}{b^2}\right )}{2\,b}-\frac {a^2\,\sin \left (c+d\,x\right )\,\left (\frac {1}{b}-\frac {a^2\,\left (\frac {2}{b}-\frac {a^2}{b^3}\right )}{b^2}\right )}{b^2}}{d} \]
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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