Optimal. Leaf size=33 \[ \frac {a \tan ^3(c+d x)}{3 d}+\frac {a \sec ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.08, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2838, 2606, 30, 2607} \[ \frac {a \tan ^3(c+d x)}{3 d}+\frac {a \sec ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2606
Rule 2607
Rule 2838
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx &=a \int \sec ^3(c+d x) \tan (c+d x) \, dx+a \int \sec ^2(c+d x) \tan ^2(c+d x) \, dx\\ &=\frac {a \operatorname {Subst}\left (\int x^2 \, dx,x,\sec (c+d x)\right )}{d}+\frac {a \operatorname {Subst}\left (\int x^2 \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {a \sec ^3(c+d x)}{3 d}+\frac {a \tan ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 1.00 \[ \frac {a \tan ^3(c+d x)}{3 d}+\frac {a \sec ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 50, normalized size = 1.52 \[ \frac {a \cos \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) - 2 \, a}{3 \, {\left (d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - d \cos \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 53, normalized size = 1.61 \[ \frac {\frac {3 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1} - \frac {3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 36, normalized size = 1.09 \[ \frac {\frac {a \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}+\frac {a}{3 \cos \left (d x +c \right )^{3}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 26, normalized size = 0.79 \[ \frac {a \tan \left (d x + c\right )^{3} + \frac {a}{\cos \left (d x + c\right )^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.03, size = 50, normalized size = 1.52 \[ -\frac {2\,a\,\left (\cos \left (c+d\,x\right )+1\right )\,\left (\cos \left (c+d\,x\right )+\sin \left (c+d\,x\right )-2\right )}{3\,d\,\left (2\,\cos \left (c+d\,x\right )-\sin \left (2\,c+2\,d\,x\right )\right )} \]
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 \[ a \left (\int \sin {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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