Optimal. Leaf size=109 \[ -\frac {\sin ^{10}(c+d x)}{10 a d}+\frac {\sin ^9(c+d x)}{9 a d}+\frac {\sin ^8(c+d x)}{4 a d}-\frac {2 \sin ^7(c+d x)}{7 a d}-\frac {\sin ^6(c+d x)}{6 a d}+\frac {\sin ^5(c+d x)}{5 a d} \]
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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} \[ -\frac {\sin ^{10}(c+d x)}{10 a d}+\frac {\sin ^9(c+d x)}{9 a d}+\frac {\sin ^8(c+d x)}{4 a d}-\frac {2 \sin ^7(c+d x)}{7 a d}-\frac {\sin ^6(c+d x)}{6 a d}+\frac {\sin ^5(c+d x)}{5 a d} \]
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 ^4(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^3 x^4 (a+x)^2}{a^4} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int (a-x)^3 x^4 (a+x)^2 \, dx,x,a \sin (c+d x)\right )}{a^{11} d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^5 x^4-a^4 x^5-2 a^3 x^6+2 a^2 x^7+a x^8-x^9\right ) \, dx,x,a \sin (c+d x)\right )}{a^{11} d}\\ &=\frac {\sin ^5(c+d x)}{5 a d}-\frac {\sin ^6(c+d x)}{6 a d}-\frac {2 \sin ^7(c+d x)}{7 a d}+\frac {\sin ^8(c+d x)}{4 a d}+\frac {\sin ^9(c+d x)}{9 a d}-\frac {\sin ^{10}(c+d x)}{10 a d}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 68, normalized size = 0.62 \[ \frac {\sin ^5(c+d x) \left (-126 \sin ^5(c+d x)+140 \sin ^4(c+d x)+315 \sin ^3(c+d x)-360 \sin ^2(c+d x)-210 \sin (c+d x)+252\right )}{1260 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 89, normalized size = 0.82 \[ \frac {126 \, \cos \left (d x + c\right )^{10} - 315 \, \cos \left (d x + c\right )^{8} + 210 \, \cos \left (d x + c\right )^{6} + 4 \, {\left (35 \, \cos \left (d x + c\right )^{8} - 50 \, \cos \left (d x + c\right )^{6} + 3 \, \cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right )}{1260 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 69, normalized size = 0.63 \[ -\frac {126 \, \sin \left (d x + c\right )^{10} - 140 \, \sin \left (d x + c\right )^{9} - 315 \, \sin \left (d x + c\right )^{8} + 360 \, \sin \left (d x + c\right )^{7} + 210 \, \sin \left (d x + c\right )^{6} - 252 \, \sin \left (d x + c\right )^{5}}{1260 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 69, normalized size = 0.63 \[ \frac {-\frac {\left (\sin ^{10}\left (d x +c \right )\right )}{10}+\frac {\left (\sin ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{4}-\frac {2 \left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 69, normalized size = 0.63 \[ -\frac {126 \, \sin \left (d x + c\right )^{10} - 140 \, \sin \left (d x + c\right )^{9} - 315 \, \sin \left (d x + c\right )^{8} + 360 \, \sin \left (d x + c\right )^{7} + 210 \, \sin \left (d x + c\right )^{6} - 252 \, \sin \left (d x + c\right )^{5}}{1260 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.94, size = 83, normalized size = 0.76 \[ \frac {\frac {{\sin \left (c+d\,x\right )}^5}{5\,a}-\frac {{\sin \left (c+d\,x\right )}^6}{6\,a}-\frac {2\,{\sin \left (c+d\,x\right )}^7}{7\,a}+\frac {{\sin \left (c+d\,x\right )}^8}{4\,a}+\frac {{\sin \left (c+d\,x\right )}^9}{9\,a}-\frac {{\sin \left (c+d\,x\right )}^{10}}{10\,a}}{d} \]
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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