Optimal. Leaf size=109 \[ -\frac {\sin ^{12}(c+d x)}{12 a d}+\frac {\sin ^{11}(c+d x)}{11 a d}+\frac {\sin ^{10}(c+d x)}{5 a d}-\frac {2 \sin ^9(c+d x)}{9 a d}-\frac {\sin ^8(c+d x)}{8 a d}+\frac {\sin ^7(c+d x)}{7 a d} \]
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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} \[ -\frac {\sin ^{12}(c+d x)}{12 a d}+\frac {\sin ^{11}(c+d x)}{11 a d}+\frac {\sin ^{10}(c+d x)}{5 a d}-\frac {2 \sin ^9(c+d x)}{9 a d}-\frac {\sin ^8(c+d x)}{8 a d}+\frac {\sin ^7(c+d x)}{7 a d} \]
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 ^6(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^3 x^6 (a+x)^2}{a^6} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int (a-x)^3 x^6 (a+x)^2 \, dx,x,a \sin (c+d x)\right )}{a^{13} d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^5 x^6-a^4 x^7-2 a^3 x^8+2 a^2 x^9+a x^{10}-x^{11}\right ) \, dx,x,a \sin (c+d x)\right )}{a^{13} d}\\ &=\frac {\sin ^7(c+d x)}{7 a d}-\frac {\sin ^8(c+d x)}{8 a d}-\frac {2 \sin ^9(c+d x)}{9 a d}+\frac {\sin ^{10}(c+d x)}{5 a d}+\frac {\sin ^{11}(c+d x)}{11 a d}-\frac {\sin ^{12}(c+d x)}{12 a d}\\ \end {align*}
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Mathematica [A] time = 0.60, size = 68, normalized size = 0.62 \[ \frac {\sin ^7(c+d x) \left (-2310 \sin ^5(c+d x)+2520 \sin ^4(c+d x)+5544 \sin ^3(c+d x)-6160 \sin ^2(c+d x)-3465 \sin (c+d x)+3960\right )}{27720 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 109, normalized size = 1.00 \[ -\frac {2310 \, \cos \left (d x + c\right )^{12} - 8316 \, \cos \left (d x + c\right )^{10} + 10395 \, \cos \left (d x + c\right )^{8} - 4620 \, \cos \left (d x + c\right )^{6} + 40 \, {\left (63 \, \cos \left (d x + c\right )^{10} - 161 \, \cos \left (d x + c\right )^{8} + 113 \, \cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right )}{27720 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 69, normalized size = 0.63 \[ -\frac {2310 \, \sin \left (d x + c\right )^{12} - 2520 \, \sin \left (d x + c\right )^{11} - 5544 \, \sin \left (d x + c\right )^{10} + 6160 \, \sin \left (d x + c\right )^{9} + 3465 \, \sin \left (d x + c\right )^{8} - 3960 \, \sin \left (d x + c\right )^{7}}{27720 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 69, normalized size = 0.63 \[ \frac {-\frac {\left (\sin ^{12}\left (d x +c \right )\right )}{12}+\frac {\left (\sin ^{11}\left (d x +c \right )\right )}{11}+\frac {\left (\sin ^{10}\left (d x +c \right )\right )}{5}-\frac {2 \left (\sin ^{9}\left (d x +c \right )\right )}{9}-\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{8}+\frac {\left (\sin ^{7}\left (d x +c \right )\right )}{7}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 69, normalized size = 0.63 \[ -\frac {2310 \, \sin \left (d x + c\right )^{12} - 2520 \, \sin \left (d x + c\right )^{11} - 5544 \, \sin \left (d x + c\right )^{10} + 6160 \, \sin \left (d x + c\right )^{9} + 3465 \, \sin \left (d x + c\right )^{8} - 3960 \, \sin \left (d x + c\right )^{7}}{27720 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 83, normalized size = 0.76 \[ \frac {\frac {{\sin \left (c+d\,x\right )}^7}{7\,a}-\frac {{\sin \left (c+d\,x\right )}^8}{8\,a}-\frac {2\,{\sin \left (c+d\,x\right )}^9}{9\,a}+\frac {{\sin \left (c+d\,x\right )}^{10}}{5\,a}+\frac {{\sin \left (c+d\,x\right )}^{11}}{11\,a}-\frac {{\sin \left (c+d\,x\right )}^{12}}{12\,a}}{d} \]
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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