Optimal. Leaf size=109 \[ -\frac {\sin ^{11}(c+d x)}{11 a d}+\frac {\sin ^{10}(c+d x)}{10 a d}+\frac {2 \sin ^9(c+d x)}{9 a d}-\frac {\sin ^8(c+d x)}{4 a d}-\frac {\sin ^7(c+d x)}{7 a d}+\frac {\sin ^6(c+d x)}{6 a d} \]
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Rubi [A] time = 0.13, antiderivative size = 109, 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 = {2836, 12, 88} \[ -\frac {\sin ^{11}(c+d x)}{11 a d}+\frac {\sin ^{10}(c+d x)}{10 a d}+\frac {2 \sin ^9(c+d x)}{9 a d}-\frac {\sin ^8(c+d x)}{4 a d}-\frac {\sin ^7(c+d x)}{7 a d}+\frac {\sin ^6(c+d x)}{6 a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos ^7(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^3 x^5 (a+x)^2}{a^5} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int (a-x)^3 x^5 (a+x)^2 \, dx,x,a \sin (c+d x)\right )}{a^{12} d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^5 x^5-a^4 x^6-2 a^3 x^7+2 a^2 x^8+a x^9-x^{10}\right ) \, dx,x,a \sin (c+d x)\right )}{a^{12} d}\\ &=\frac {\sin ^6(c+d x)}{6 a d}-\frac {\sin ^7(c+d x)}{7 a d}-\frac {\sin ^8(c+d x)}{4 a d}+\frac {2 \sin ^9(c+d x)}{9 a d}+\frac {\sin ^{10}(c+d x)}{10 a d}-\frac {\sin ^{11}(c+d x)}{11 a d}\\ \end {align*}
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Mathematica [A] time = 0.92, size = 68, normalized size = 0.62 \[ \frac {\sin ^6(c+d x) \left (-1260 \sin ^5(c+d x)+1386 \sin ^4(c+d x)+3080 \sin ^3(c+d x)-3465 \sin ^2(c+d x)-1980 \sin (c+d x)+2310\right )}{13860 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 99, normalized size = 0.91 \[ -\frac {1386 \, \cos \left (d x + c\right )^{10} - 3465 \, \cos \left (d x + c\right )^{8} + 2310 \, \cos \left (d x + c\right )^{6} - 20 \, {\left (63 \, \cos \left (d x + c\right )^{10} - 161 \, \cos \left (d x + c\right )^{8} + 113 \, \cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right )}{13860 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 69, normalized size = 0.63 \[ -\frac {1260 \, \sin \left (d x + c\right )^{11} - 1386 \, \sin \left (d x + c\right )^{10} - 3080 \, \sin \left (d x + c\right )^{9} + 3465 \, \sin \left (d x + c\right )^{8} + 1980 \, \sin \left (d x + c\right )^{7} - 2310 \, \sin \left (d x + c\right )^{6}}{13860 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 69, normalized size = 0.63 \[ \frac {-\frac {\left (\sin ^{11}\left (d x +c \right )\right )}{11}+\frac {\left (\sin ^{10}\left (d x +c \right )\right )}{10}+\frac {2 \left (\sin ^{9}\left (d x +c \right )\right )}{9}-\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 69, normalized size = 0.63 \[ -\frac {1260 \, \sin \left (d x + c\right )^{11} - 1386 \, \sin \left (d x + c\right )^{10} - 3080 \, \sin \left (d x + c\right )^{9} + 3465 \, \sin \left (d x + c\right )^{8} + 1980 \, \sin \left (d x + c\right )^{7} - 2310 \, \sin \left (d x + c\right )^{6}}{13860 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.99, size = 83, normalized size = 0.76 \[ \frac {\frac {{\sin \left (c+d\,x\right )}^6}{6\,a}-\frac {{\sin \left (c+d\,x\right )}^7}{7\,a}-\frac {{\sin \left (c+d\,x\right )}^8}{4\,a}+\frac {2\,{\sin \left (c+d\,x\right )}^9}{9\,a}+\frac {{\sin \left (c+d\,x\right )}^{10}}{10\,a}-\frac {{\sin \left (c+d\,x\right )}^{11}}{11\,a}}{d} \]
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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