Optimal. Leaf size=52 \[ -\frac {a^3 \sin ^2(c+d x)}{2 d}-\frac {3 a^3 \sin (c+d x)}{d}-\frac {4 a^3 \log (1-\sin (c+d x))}{d} \]
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Rubi [A] time = 0.05, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2667, 43} \[ -\frac {a^3 \sin ^2(c+d x)}{2 d}-\frac {3 a^3 \sin (c+d x)}{d}-\frac {4 a^3 \log (1-\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \sec (c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {a \operatorname {Subst}\left (\int \frac {(a+x)^2}{a-x} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a \operatorname {Subst}\left (\int \left (-3 a+\frac {4 a^2}{a-x}-x\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {4 a^3 \log (1-\sin (c+d x))}{d}-\frac {3 a^3 \sin (c+d x)}{d}-\frac {a^3 \sin ^2(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 41, normalized size = 0.79 \[ \frac {a^3 \left (-\frac {1}{2} \sin ^2(c+d x)-3 \sin (c+d x)-4 \log (1-\sin (c+d x))\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 45, normalized size = 0.87 \[ \frac {a^{3} \cos \left (d x + c\right )^{2} - 8 \, a^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 6 \, a^{3} \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.90, size = 128, normalized size = 2.46 \[ \frac {2 \, {\left (2 \, a^{3} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 4 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{3}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 69, normalized size = 1.33 \[ -\frac {a^{3} \left (\sin ^{2}\left (d x +c \right )\right )}{2 d}-\frac {4 a^{3} \ln \left (\cos \left (d x +c \right )\right )}{d}-\frac {3 a^{3} \sin \left (d x +c \right )}{d}+\frac {4 a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 43, normalized size = 0.83 \[ -\frac {a^{3} \sin \left (d x + c\right )^{2} + 8 \, a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) + 6 \, a^{3} \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 36, normalized size = 0.69 \[ -\frac {a^3\,\left (8\,\ln \left (\sin \left (c+d\,x\right )-1\right )+6\,\sin \left (c+d\,x\right )+{\sin \left (c+d\,x\right )}^2\right )}{2\,d} \]
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 \[ a^{3} \left (\int 3 \sin {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \sec {\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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