Optimal. Leaf size=37 \[ \frac {\sin ^2(c+d x)}{2 a d}+\frac {\cos ^3(c+d x)}{3 a d} \]
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Rubi [A] time = 0.13, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3872, 2835, 2564, 30, 2565} \[ \frac {\sin ^2(c+d x)}{2 a d}+\frac {\cos ^3(c+d x)}{3 a d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2564
Rule 2565
Rule 2835
Rule 3872
Rubi steps
\begin {align*} \int \frac {\sin ^3(c+d x)}{a+a \sec (c+d x)} \, dx &=-\int \frac {\cos (c+d x) \sin ^3(c+d x)}{-a-a \cos (c+d x)} \, dx\\ &=\frac {\int \cos (c+d x) \sin (c+d x) \, dx}{a}-\frac {\int \cos ^2(c+d x) \sin (c+d x) \, dx}{a}\\ &=\frac {\operatorname {Subst}(\int x \, dx,x,\sin (c+d x))}{a d}+\frac {\operatorname {Subst}\left (\int x^2 \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac {\cos ^3(c+d x)}{3 a d}+\frac {\sin ^2(c+d x)}{2 a d}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 32, normalized size = 0.86 \[ \frac {2 \sin ^4\left (\frac {1}{2} (c+d x)\right ) (2 \cos (c+d x)+1)}{3 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 29, normalized size = 0.78 \[ \frac {2 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2}}{6 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 32, normalized size = 0.86 \[ \frac {\frac {2 \, \cos \left (d x + c\right )^{3}}{d} - \frac {3 \, \cos \left (d x + c\right )^{2}}{d}}{6 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.43, size = 30, normalized size = 0.81 \[ -\frac {\frac {1}{2 \sec \left (d x +c \right )^{2}}-\frac {1}{3 \sec \left (d x +c \right )^{3}}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 29, normalized size = 0.78 \[ \frac {2 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2}}{6 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.88, size = 26, normalized size = 0.70 \[ \frac {{\cos \left (c+d\,x\right )}^2\,\left (2\,\cos \left (c+d\,x\right )-3\right )}{6\,a\,d} \]
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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