Optimal. Leaf size=73 \[ \frac {\sin ^7(c+d x)}{7 a d}-\frac {\sin ^6(c+d x)}{6 a d}-\frac {\sin ^5(c+d x)}{5 a d}+\frac {\sin ^4(c+d x)}{4 a d} \]
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Rubi [A] time = 0.11, antiderivative size = 73, 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, 75} \[ \frac {\sin ^7(c+d x)}{7 a d}-\frac {\sin ^6(c+d x)}{6 a d}-\frac {\sin ^5(c+d x)}{5 a d}+\frac {\sin ^4(c+d x)}{4 a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 75
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2 x^3 (a+x)}{a^3} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int (a-x)^2 x^3 (a+x) \, dx,x,a \sin (c+d x)\right )}{a^8 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^3 x^3-a^2 x^4-a x^5+x^6\right ) \, dx,x,a \sin (c+d x)\right )}{a^8 d}\\ &=\frac {\sin ^4(c+d x)}{4 a d}-\frac {\sin ^5(c+d x)}{5 a d}-\frac {\sin ^6(c+d x)}{6 a d}+\frac {\sin ^7(c+d x)}{7 a d}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 48, normalized size = 0.66 \[ \frac {\sin ^4(c+d x) \left (60 \sin ^3(c+d x)-70 \sin ^2(c+d x)-84 \sin (c+d x)+105\right )}{420 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 67, normalized size = 0.92 \[ \frac {70 \, \cos \left (d x + c\right )^{6} - 105 \, \cos \left (d x + c\right )^{4} - 12 \, {\left (5 \, \cos \left (d x + c\right )^{6} - 8 \, \cos \left (d x + c\right )^{4} + \cos \left (d x + c\right )^{2} + 2\right )} \sin \left (d x + c\right )}{420 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 49, normalized size = 0.67 \[ \frac {60 \, \sin \left (d x + c\right )^{7} - 70 \, \sin \left (d x + c\right )^{6} - 84 \, \sin \left (d x + c\right )^{5} + 105 \, \sin \left (d x + c\right )^{4}}{420 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 49, normalized size = 0.67 \[ \frac {\frac {\left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 49, normalized size = 0.67 \[ \frac {60 \, \sin \left (d x + c\right )^{7} - 70 \, \sin \left (d x + c\right )^{6} - 84 \, \sin \left (d x + c\right )^{5} + 105 \, \sin \left (d x + c\right )^{4}}{420 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 57, normalized size = 0.78 \[ \frac {\frac {{\sin \left (c+d\,x\right )}^4}{4\,a}-\frac {{\sin \left (c+d\,x\right )}^5}{5\,a}-\frac {{\sin \left (c+d\,x\right )}^6}{6\,a}+\frac {{\sin \left (c+d\,x\right )}^7}{7\,a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 81.79, size = 981, normalized size = 13.44 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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