Optimal. Leaf size=47 \[ \frac {(a-a \sin (c+d x))^5}{5 a^7 d}-\frac {(a-a \sin (c+d x))^4}{2 a^6 d} \]
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Rubi [A] time = 0.05, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2667, 43} \[ \frac {(a-a \sin (c+d x))^5}{5 a^7 d}-\frac {(a-a \sin (c+d x))^4}{2 a^6 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \frac {\cos ^7(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^3 (a+x) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (2 a (a-x)^3-(a-x)^4\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=-\frac {(a-a \sin (c+d x))^4}{2 a^6 d}+\frac {(a-a \sin (c+d x))^5}{5 a^7 d}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 46, normalized size = 0.98 \[ -\frac {\sin (c+d x) \left (2 \sin ^4(c+d x)-5 \sin ^3(c+d x)+10 \sin (c+d x)-10\right )}{10 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 47, normalized size = 1.00 \[ \frac {5 \, \cos \left (d x + c\right )^{4} - 2 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right )}{10 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 47, normalized size = 1.00 \[ -\frac {2 \, \sin \left (d x + c\right )^{5} - 5 \, \sin \left (d x + c\right )^{4} + 10 \, \sin \left (d x + c\right )^{2} - 10 \, \sin \left (d x + c\right )}{10 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 45, normalized size = 0.96 \[ \frac {-\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{2}-\left (\sin ^{2}\left (d x +c \right )\right )+\sin \left (d x +c \right )}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 47, normalized size = 1.00 \[ -\frac {2 \, \sin \left (d x + c\right )^{5} - 5 \, \sin \left (d x + c\right )^{4} + 10 \, \sin \left (d x + c\right )^{2} - 10 \, \sin \left (d x + c\right )}{10 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.66, size = 54, normalized size = 1.15 \[ \frac {\frac {\sin \left (c+d\,x\right )}{a^2}-\frac {{\sin \left (c+d\,x\right )}^2}{a^2}+\frac {{\sin \left (c+d\,x\right )}^4}{2\,a^2}-\frac {{\sin \left (c+d\,x\right )}^5}{5\,a^2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 92.86, size = 1037, normalized size = 22.06 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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