Optimal. Leaf size=235 \[ -\frac {4 a \left (a^2-b^2\right ) \sin ^3(c+d x)}{3 b^5 d}+\frac {\left (3 a^2-2 b^2\right ) \sin ^4(c+d x)}{4 b^4 d}+\frac {a^2 \left (7 a^4-10 a^2 b^2+3 b^4\right ) \log (a+b \sin (c+d x))}{b^8 d}-\frac {2 a \left (3 a^4-4 a^2 b^2+b^4\right ) \sin (c+d x)}{b^7 d}+\frac {\left (5 a^4-6 a^2 b^2+b^4\right ) \sin ^2(c+d x)}{2 b^6 d}+\frac {a^3 \left (a^2-b^2\right )^2}{b^8 d (a+b \sin (c+d x))}-\frac {2 a \sin ^5(c+d x)}{5 b^3 d}+\frac {\sin ^6(c+d x)}{6 b^2 d} \]
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Rubi [A] time = 0.28, antiderivative size = 235, 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} \[ \frac {\left (3 a^2-2 b^2\right ) \sin ^4(c+d x)}{4 b^4 d}-\frac {4 a \left (a^2-b^2\right ) \sin ^3(c+d x)}{3 b^5 d}+\frac {\left (-6 a^2 b^2+5 a^4+b^4\right ) \sin ^2(c+d x)}{2 b^6 d}-\frac {2 a \left (-4 a^2 b^2+3 a^4+b^4\right ) \sin (c+d x)}{b^7 d}+\frac {a^3 \left (a^2-b^2\right )^2}{b^8 d (a+b \sin (c+d x))}+\frac {a^2 \left (-10 a^2 b^2+7 a^4+3 b^4\right ) \log (a+b \sin (c+d x))}{b^8 d}-\frac {2 a \sin ^5(c+d x)}{5 b^3 d}+\frac {\sin ^6(c+d x)}{6 b^2 d} \]
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 ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^3 \left (b^2-x^2\right )^2}{b^3 (a+x)^2} \, 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)^2} \, dx,x,b \sin (c+d x)\right )}{b^8 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-2 a \left (3 a^4-4 a^2 b^2+b^4\right )+\left (5 a^4-6 a^2 b^2+b^4\right ) x-4 a \left (a^2-b^2\right ) x^2+\left (3 a^2-2 b^2\right ) x^3-2 a x^4+x^5-\frac {a^3 \left (a^2-b^2\right )^2}{(a+x)^2}+\frac {7 a^6-10 a^4 b^2+3 a^2 b^4}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^8 d}\\ &=\frac {a^2 \left (7 a^4-10 a^2 b^2+3 b^4\right ) \log (a+b \sin (c+d x))}{b^8 d}-\frac {2 a \left (3 a^4-4 a^2 b^2+b^4\right ) \sin (c+d x)}{b^7 d}+\frac {\left (5 a^4-6 a^2 b^2+b^4\right ) \sin ^2(c+d x)}{2 b^6 d}-\frac {4 a \left (a^2-b^2\right ) \sin ^3(c+d x)}{3 b^5 d}+\frac {\left (3 a^2-2 b^2\right ) \sin ^4(c+d x)}{4 b^4 d}-\frac {2 a \sin ^5(c+d x)}{5 b^3 d}+\frac {\sin ^6(c+d x)}{6 b^2 d}+\frac {a^3 \left (a^2-b^2\right )^2}{b^8 d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 2.19, size = 264, normalized size = 1.12 \[ \frac {\left (50 a b^6-35 a^3 b^4\right ) \sin ^4(c+d x)+60 a^2 b \left (a^2-b^2\right ) \sin (c+d x) \left (\left (7 a^2-3 b^2\right ) \log (a+b \sin (c+d x))-6 a^2+2 b^2\right )+3 b^5 \left (7 a^2-10 b^2\right ) \sin ^5(c+d x)-30 a b^2 \left (7 a^4-10 a^2 b^2+3 b^4\right ) \sin ^2(c+d x)+10 b^3 \left (7 a^4-10 a^2 b^2+3 b^4\right ) \sin ^3(c+d x)+60 a^3 \left (a^2-b^2\right ) \left (\left (7 a^2-3 b^2\right ) \log (a+b \sin (c+d x))+a^2-b^2\right )-14 a b^6 \sin ^6(c+d x)+10 b^7 \sin ^7(c+d x)}{60 b^8 d (a+b \sin (c+d x))} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 279, normalized size = 1.19 \[ \frac {112 \, a b^{6} \cos \left (d x + c\right )^{6} + 480 \, a^{7} - 3240 \, a^{5} b^{2} + 3185 \, a^{3} b^{4} - 487 \, a b^{6} - 8 \, {\left (35 \, a^{3} b^{4} - 8 \, a b^{6}\right )} \cos \left (d x + c\right )^{4} + 16 \, {\left (105 \, a^{5} b^{2} - 115 \, a^{3} b^{4} + 16 \, a b^{6}\right )} \cos \left (d x + c\right )^{2} + 480 \, {\left (7 \, a^{7} - 10 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + {\left (7 \, a^{6} b - 10 \, a^{4} b^{3} + 3 \, a^{2} b^{5}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - {\left (80 \, b^{7} \cos \left (d x + c\right )^{6} - 168 \, a^{2} b^{5} \cos \left (d x + c\right )^{4} + 2880 \, a^{6} b - 3800 \, a^{4} b^{3} + 1007 \, a^{2} b^{5} - 25 \, b^{7} + 16 \, {\left (35 \, a^{4} b^{3} - 29 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{480 \, {\left (b^{9} d \sin \left (d x + c\right ) + a b^{8} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 300, normalized size = 1.28 \[ \frac {\frac {60 \, {\left (7 \, a^{6} - 10 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{8}} - \frac {60 \, {\left (7 \, a^{6} b \sin \left (d x + c\right ) - 10 \, a^{4} b^{3} \sin \left (d x + c\right ) + 3 \, a^{2} b^{5} \sin \left (d x + c\right ) + 6 \, a^{7} - 8 \, a^{5} b^{2} + 2 \, a^{3} b^{4}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )} b^{8}} + \frac {10 \, b^{10} \sin \left (d x + c\right )^{6} - 24 \, a b^{9} \sin \left (d x + c\right )^{5} + 45 \, a^{2} b^{8} \sin \left (d x + c\right )^{4} - 30 \, b^{10} \sin \left (d x + c\right )^{4} - 80 \, a^{3} b^{7} \sin \left (d x + c\right )^{3} + 80 \, a b^{9} \sin \left (d x + c\right )^{3} + 150 \, a^{4} b^{6} \sin \left (d x + c\right )^{2} - 180 \, a^{2} b^{8} \sin \left (d x + c\right )^{2} + 30 \, b^{10} \sin \left (d x + c\right )^{2} - 360 \, a^{5} b^{5} \sin \left (d x + c\right ) + 480 \, a^{3} b^{7} \sin \left (d x + c\right ) - 120 \, a b^{9} \sin \left (d x + c\right )}{b^{12}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.55, size = 342, normalized size = 1.46 \[ \frac {\sin ^{6}\left (d x +c \right )}{6 b^{2} d}-\frac {2 a \left (\sin ^{5}\left (d x +c \right )\right )}{5 b^{3} d}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right ) a^{2}}{4 d \,b^{4}}-\frac {\sin ^{4}\left (d x +c \right )}{2 b^{2} d}-\frac {4 \left (\sin ^{3}\left (d x +c \right )\right ) a^{3}}{3 d \,b^{5}}+\frac {4 a \left (\sin ^{3}\left (d x +c \right )\right )}{3 b^{3} d}+\frac {5 \left (\sin ^{2}\left (d x +c \right )\right ) a^{4}}{2 d \,b^{6}}-\frac {3 \left (\sin ^{2}\left (d x +c \right )\right ) a^{2}}{d \,b^{4}}+\frac {\sin ^{2}\left (d x +c \right )}{2 b^{2} d}-\frac {6 a^{5} \sin \left (d x +c \right )}{d \,b^{7}}+\frac {8 a^{3} \sin \left (d x +c \right )}{d \,b^{5}}-\frac {2 a \sin \left (d x +c \right )}{b^{3} d}+\frac {7 a^{6} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,b^{8}}-\frac {10 a^{4} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,b^{6}}+\frac {3 a^{2} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,b^{4}}+\frac {a^{7}}{d \,b^{8} \left (a +b \sin \left (d x +c \right )\right )}-\frac {2 a^{5}}{d \,b^{6} \left (a +b \sin \left (d x +c \right )\right )}+\frac {a^{3}}{d \,b^{4} \left (a +b \sin \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 218, normalized size = 0.93 \[ \frac {\frac {60 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )}}{b^{9} \sin \left (d x + c\right ) + a b^{8}} + \frac {10 \, b^{5} \sin \left (d x + c\right )^{6} - 24 \, a b^{4} \sin \left (d x + c\right )^{5} + 15 \, {\left (3 \, a^{2} b^{3} - 2 \, b^{5}\right )} \sin \left (d x + c\right )^{4} - 80 \, {\left (a^{3} b^{2} - a b^{4}\right )} \sin \left (d x + c\right )^{3} + 30 \, {\left (5 \, a^{4} b - 6 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )^{2} - 120 \, {\left (3 \, a^{5} - 4 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (d x + c\right )}{b^{7}} + \frac {60 \, {\left (7 \, a^{6} - 10 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{8}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 375, normalized size = 1.60 \[ \frac {{\sin \left (c+d\,x\right )}^3\,\left (\frac {2\,a^3}{3\,b^5}+\frac {2\,a\,\left (\frac {2}{b^2}-\frac {3\,a^2}{b^4}\right )}{3\,b}\right )}{d}-\frac {{\sin \left (c+d\,x\right )}^4\,\left (\frac {1}{2\,b^2}-\frac {3\,a^2}{4\,b^4}\right )}{d}+\frac {{\sin \left (c+d\,x\right )}^2\,\left (\frac {1}{2\,b^2}+\frac {a^2\,\left (\frac {2}{b^2}-\frac {3\,a^2}{b^4}\right )}{2\,b^2}-\frac {a\,\left (\frac {2\,a^3}{b^5}+\frac {2\,a\,\left (\frac {2}{b^2}-\frac {3\,a^2}{b^4}\right )}{b}\right )}{b}\right )}{d}-\frac {\sin \left (c+d\,x\right )\,\left (\frac {a^2\,\left (\frac {2\,a^3}{b^5}+\frac {2\,a\,\left (\frac {2}{b^2}-\frac {3\,a^2}{b^4}\right )}{b}\right )}{b^2}+\frac {2\,a\,\left (\frac {1}{b^2}+\frac {a^2\,\left (\frac {2}{b^2}-\frac {3\,a^2}{b^4}\right )}{b^2}-\frac {2\,a\,\left (\frac {2\,a^3}{b^5}+\frac {2\,a\,\left (\frac {2}{b^2}-\frac {3\,a^2}{b^4}\right )}{b}\right )}{b}\right )}{b}\right )}{d}+\frac {{\sin \left (c+d\,x\right )}^6}{6\,b^2\,d}-\frac {2\,a\,{\sin \left (c+d\,x\right )}^5}{5\,b^3\,d}+\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (7\,a^6-10\,a^4\,b^2+3\,a^2\,b^4\right )}{b^8\,d}+\frac {a^7-2\,a^5\,b^2+a^3\,b^4}{b\,d\,\left (\sin \left (c+d\,x\right )\,b^8+a\,b^7\right )} \]
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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