Optimal. Leaf size=45 \[ \frac {a \tan ^3(c+d x)}{3 d}+\frac {a \sec ^3(c+d x)}{3 d}-\frac {a \sec (c+d x)}{d} \]
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Rubi [A] time = 0.10, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2838, 2607, 30, 2606} \[ \frac {a \tan ^3(c+d x)}{3 d}+\frac {a \sec ^3(c+d x)}{3 d}-\frac {a \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2606
Rule 2607
Rule 2838
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx &=a \int \sec ^2(c+d x) \tan ^2(c+d x) \, dx+a \int \sec (c+d x) \tan ^3(c+d x) \, dx\\ &=\frac {a \operatorname {Subst}\left (\int x^2 \, dx,x,\tan (c+d x)\right )}{d}+\frac {a \operatorname {Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac {a \sec (c+d x)}{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 = 0.04, size = 45, normalized size = 1.00 \[ \frac {a \tan ^3(c+d x)}{3 d}+\frac {a \sec ^3(c+d x)}{3 d}-\frac {a \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 49, normalized size = 1.09 \[ \frac {a \cos \left (d x + c\right )^{2} - 2 \, a \sin \left (d x + c\right ) + a}{3 \, {\left (d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - d \cos \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 67, normalized size = 1.49 \[ -\frac {\frac {3 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1} - \frac {3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 82, normalized size = 1.82 \[ \frac {a \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )+\frac {a \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 39, normalized size = 0.87 \[ \frac {a \tan \left (d x + c\right )^{3} - \frac {{\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} a}{\cos \left (d x + c\right )^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.03, size = 74, normalized size = 1.64 \[ -\frac {4\,a\,\left ({\sin \left (c+d\,x\right )}^2+2\,\sin \left (c+d\,x\right )+4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\sin \left (2\,c+2\,d\,x\right )-4\right )}{3\,d\,\left (8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,\sin \left (2\,c+2\,d\,x\right )-4\right )} \]
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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