Optimal. Leaf size=40 \[ \frac {2 a^4}{d (a-a \sin (c+d x))}+\frac {a^3 \log (1-\sin (c+d x))}{d} \]
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Rubi [A] time = 0.05, antiderivative size = 40, 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} \[ \frac {2 a^4}{d (a-a \sin (c+d x))}+\frac {a^3 \log (1-\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {a^3 \operatorname {Subst}\left (\int \frac {a+x}{(a-x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^3 \operatorname {Subst}\left (\int \left (\frac {2 a}{(a-x)^2}+\frac {1}{-a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^3 \log (1-\sin (c+d x))}{d}+\frac {2 a^4}{d (a-a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 59, normalized size = 1.48 \[ \frac {a^3 (1-\sin (c+d x)) (\sin (c+d x)+1) \sec ^2(c+d x) \left (\frac {2}{1-\sin (c+d x)}+\log (1-\sin (c+d x))\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 51, normalized size = 1.28 \[ -\frac {2 \, a^{3} - {\left (a^{3} \sin \left (d x + c\right ) - a^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{d \sin \left (d x + c\right ) - d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.50, size = 92, normalized size = 2.30 \[ -\frac {a^{3} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 2 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 10 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{3}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 128, normalized size = 3.20 \[ \frac {a^{3} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a^{3} \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {3 a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {3 a^{3} \sin \left (d x +c \right )}{2 d}-\frac {a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {3 a^{3}}{2 d \cos \left (d x +c \right )^{2}}+\frac {a^{3} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 33, normalized size = 0.82 \[ \frac {a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, a^{3}}{\sin \left (d x + c\right ) - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.52, size = 35, normalized size = 0.88 \[ \frac {a^3\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{d}-\frac {2\,a^3}{d\,\left (\sin \left (c+d\,x\right )-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{3} \left (\int 3 \sin {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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