Optimal. Leaf size=193 \[ -\frac {\left (2-\frac {3 a^2}{b^2}\right ) \sin ^3(c+d x)}{3 b^2 d}-\frac {a^2 \left (a^2-b^2\right )^2}{b^7 d (a+b \sin (c+d x))}-\frac {2 a \left (a^2-b^2\right ) \sin ^2(c+d x)}{b^5 d}-\frac {2 a \left (3 a^4-4 a^2 b^2+b^4\right ) \log (a+b \sin (c+d x))}{b^7 d}+\frac {\left (5 a^4-6 a^2 b^2+b^4\right ) \sin (c+d x)}{b^6 d}-\frac {a \sin ^4(c+d x)}{2 b^3 d}+\frac {\sin ^5(c+d x)}{5 b^2 d} \]
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Rubi [A] time = 0.24, antiderivative size = 193, 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} \[ -\frac {\left (2-\frac {3 a^2}{b^2}\right ) \sin ^3(c+d x)}{3 b^2 d}-\frac {2 a \left (a^2-b^2\right ) \sin ^2(c+d x)}{b^5 d}+\frac {\left (-6 a^2 b^2+5 a^4+b^4\right ) \sin (c+d x)}{b^6 d}-\frac {a^2 \left (a^2-b^2\right )^2}{b^7 d (a+b \sin (c+d x))}-\frac {2 a \left (-4 a^2 b^2+3 a^4+b^4\right ) \log (a+b \sin (c+d x))}{b^7 d}-\frac {a \sin ^4(c+d x)}{2 b^3 d}+\frac {\sin ^5(c+d x)}{5 b^2 d} \]
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))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (b^2-x^2\right )^2}{b^2 (a+x)^2} \, 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)^2} \, dx,x,b \sin (c+d x)\right )}{b^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (5 a^4 \left (1+\frac {-6 a^2 b^2+b^4}{5 a^4}\right )-4 a \left (a^2-b^2\right ) x+\left (3 a^2-2 b^2\right ) x^2-2 a x^3+x^4+\frac {\left (a^3-a b^2\right )^2}{(a+x)^2}-\frac {2 a \left (3 a^4-4 a^2 b^2+b^4\right )}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^7 d}\\ &=-\frac {2 a \left (3 a^4-4 a^2 b^2+b^4\right ) \log (a+b \sin (c+d x))}{b^7 d}+\frac {\left (5 a^4-6 a^2 b^2+b^4\right ) \sin (c+d x)}{b^6 d}-\frac {2 a \left (a^2-b^2\right ) \sin ^2(c+d x)}{b^5 d}+\frac {\left (3 a^2-2 b^2\right ) \sin ^3(c+d x)}{3 b^4 d}-\frac {a \sin ^4(c+d x)}{2 b^3 d}+\frac {\sin ^5(c+d x)}{5 b^2 d}-\frac {a^2 \left (a^2-b^2\right )^2}{b^7 d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 1.49, size = 225, normalized size = 1.17 \[ \frac {\left (40 a b^5-30 a^3 b^3\right ) \sin ^3(c+d x)-30 a b \left (a^2-b^2\right ) \sin (c+d x) \left (\left (6 a^2-2 b^2\right ) \log (a+b \sin (c+d x))-5 a^2+b^2\right )-30 a^2 \left (a^2-b^2\right ) \left (\left (6 a^2-2 b^2\right ) \log (a+b \sin (c+d x))+a^2-b^2\right )+5 b^4 \left (3 a^2-4 b^2\right ) \sin ^4(c+d x)+30 b^2 \left (3 a^4-4 a^2 b^2+b^4\right ) \sin ^2(c+d x)-9 a b^5 \sin ^5(c+d x)+6 b^6 \sin ^6(c+d x)}{30 b^7 d (a+b \sin (c+d x))} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 246, normalized size = 1.27 \[ -\frac {48 \, b^{6} \cos \left (d x + c\right )^{6} + 240 \, a^{6} - 1440 \, a^{4} b^{2} + 1275 \, a^{2} b^{4} - 128 \, b^{6} - 8 \, {\left (15 \, a^{2} b^{4} - 2 \, b^{6}\right )} \cos \left (d x + c\right )^{4} + 16 \, {\left (45 \, a^{4} b^{2} - 45 \, a^{2} b^{4} + 4 \, b^{6}\right )} \cos \left (d x + c\right )^{2} + 480 \, {\left (3 \, a^{6} - 4 \, a^{4} b^{2} + a^{2} b^{4} + {\left (3 \, a^{5} b - 4 \, a^{3} b^{3} + a b^{5}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + {\left (72 \, a b^{5} \cos \left (d x + c\right )^{4} - 1200 \, a^{5} b + 1440 \, a^{3} b^{3} - 293 \, a b^{5} - 16 \, {\left (15 \, a^{3} b^{3} - 11 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{240 \, {\left (b^{8} d \sin \left (d x + c\right ) + a b^{7} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 249, normalized size = 1.29 \[ -\frac {\frac {60 \, {\left (3 \, a^{5} - 4 \, a^{3} b^{2} + a b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{7}} - \frac {30 \, {\left (6 \, a^{5} b \sin \left (d x + c\right ) - 8 \, a^{3} b^{3} \sin \left (d x + c\right ) + 2 \, a b^{5} \sin \left (d x + c\right ) + 5 \, a^{6} - 6 \, a^{4} b^{2} + a^{2} b^{4}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )} b^{7}} - \frac {6 \, b^{8} \sin \left (d x + c\right )^{5} - 15 \, a b^{7} \sin \left (d x + c\right )^{4} + 30 \, a^{2} b^{6} \sin \left (d x + c\right )^{3} - 20 \, b^{8} \sin \left (d x + c\right )^{3} - 60 \, a^{3} b^{5} \sin \left (d x + c\right )^{2} + 60 \, a b^{7} \sin \left (d x + c\right )^{2} + 150 \, a^{4} b^{4} \sin \left (d x + c\right ) - 180 \, a^{2} b^{6} \sin \left (d x + c\right ) + 30 \, b^{8} \sin \left (d x + c\right )}{b^{10}}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 285, normalized size = 1.48 \[ \frac {\sin ^{5}\left (d x +c \right )}{5 b^{2} d}-\frac {a \left (\sin ^{4}\left (d x +c \right )\right )}{2 b^{3} d}+\frac {\left (\sin ^{3}\left (d x +c \right )\right ) a^{2}}{d \,b^{4}}-\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{3 b^{2} d}-\frac {2 \left (\sin ^{2}\left (d x +c \right )\right ) a^{3}}{d \,b^{5}}+\frac {2 a \left (\sin ^{2}\left (d x +c \right )\right )}{b^{3} d}+\frac {5 a^{4} \sin \left (d x +c \right )}{d \,b^{6}}-\frac {6 \sin \left (d x +c \right ) a^{2}}{d \,b^{4}}+\frac {\sin \left (d x +c \right )}{b^{2} d}-\frac {6 a^{5} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,b^{7}}+\frac {8 a^{3} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,b^{5}}-\frac {2 a \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{3} d}-\frac {a^{6}}{d \,b^{7} \left (a +b \sin \left (d x +c \right )\right )}+\frac {2 a^{4}}{d \,b^{5} \left (a +b \sin \left (d x +c \right )\right )}-\frac {a^{2}}{d \,b^{3} \left (a +b \sin \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 184, normalized size = 0.95 \[ -\frac {\frac {30 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )}}{b^{8} \sin \left (d x + c\right ) + a b^{7}} - \frac {6 \, b^{4} \sin \left (d x + c\right )^{5} - 15 \, a b^{3} \sin \left (d x + c\right )^{4} + 10 \, {\left (3 \, a^{2} b^{2} - 2 \, b^{4}\right )} \sin \left (d x + c\right )^{3} - 60 \, {\left (a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )^{2} + 30 \, {\left (5 \, a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )}{b^{6}} + \frac {60 \, {\left (3 \, a^{5} - 4 \, a^{3} b^{2} + a b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{7}}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.46, size = 254, normalized size = 1.32 \[ \frac {{\sin \left (c+d\,x\right )}^2\,\left (\frac {a^3}{b^5}+\frac {a\,\left (\frac {2}{b^2}-\frac {3\,a^2}{b^4}\right )}{b}\right )}{d}-\frac {{\sin \left (c+d\,x\right )}^3\,\left (\frac {2}{3\,b^2}-\frac {a^2}{b^4}\right )}{d}+\frac {\sin \left (c+d\,x\right )\,\left (\frac {1}{b^2}+\frac {a^2\,\left (\frac {2}{b^2}-\frac {3\,a^2}{b^4}\right )}{b^2}-\frac {2\,a\,\left (\frac {2\,a^3}{b^5}+\frac {2\,a\,\left (\frac {2}{b^2}-\frac {3\,a^2}{b^4}\right )}{b}\right )}{b}\right )}{d}+\frac {{\sin \left (c+d\,x\right )}^5}{5\,b^2\,d}-\frac {a\,{\sin \left (c+d\,x\right )}^4}{2\,b^3\,d}-\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (6\,a^5-8\,a^3\,b^2+2\,a\,b^4\right )}{b^7\,d}-\frac {a^6-2\,a^4\,b^2+a^2\,b^4}{b\,d\,\left (\sin \left (c+d\,x\right )\,b^7+a\,b^6\right )} \]
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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