Optimal. Leaf size=75 \[ a^3 x+\frac {b \left (2 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {a b^2 \tan (c+d x)}{2 d}-\frac {b^2 \tan (c+d x) (a+b \sec (c+d x))}{2 d} \]
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Rubi [A] time = 0.09, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {4042, 3918, 3770, 3767, 8} \[ \frac {b \left (2 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^3 x-\frac {a b^2 \tan (c+d x)}{2 d}-\frac {b^2 \tan (c+d x) (a+b \sec (c+d x))}{2 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3918
Rule 4042
Rubi steps
\begin {align*} \int (a+b \sec (c+d x)) \left (a^2-b^2 \sec ^2(c+d x)\right ) \, dx &=-\int (-a+b \sec (c+d x)) (a+b \sec (c+d x))^2 \, dx\\ &=-\frac {b^2 (a+b \sec (c+d x)) \tan (c+d x)}{2 d}-\frac {1}{2} \int \left (-2 a^3-b \left (2 a^2-b^2\right ) \sec (c+d x)+a b^2 \sec ^2(c+d x)\right ) \, dx\\ &=a^3 x-\frac {b^2 (a+b \sec (c+d x)) \tan (c+d x)}{2 d}-\frac {1}{2} \left (a b^2\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{2} \left (b \left (2 a^2-b^2\right )\right ) \int \sec (c+d x) \, dx\\ &=a^3 x+\frac {b \left (2 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {b^2 (a+b \sec (c+d x)) \tan (c+d x)}{2 d}+\frac {\left (a b^2\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{2 d}\\ &=a^3 x+\frac {b \left (2 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {a b^2 \tan (c+d x)}{2 d}-\frac {b^2 (a+b \sec (c+d x)) \tan (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 75, normalized size = 1.00 \[ a^3 x+\frac {a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a b^2 \tan (c+d x)}{d}-\frac {b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {b^3 \tan (c+d x) \sec (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 116, normalized size = 1.55 \[ \frac {4 \, a^{3} d x \cos \left (d x + c\right )^{2} + {\left (2 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (2 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (2 \, a b^{2} \cos \left (d x + c\right ) + b^{3}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.95, size = 94, normalized size = 1.25 \[ a^{3} x +\frac {a^{3} c}{d}-\frac {a \,b^{2} \tan \left (d x +c \right )}{d}+\frac {a^{2} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}-\frac {b^{3} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}-\frac {b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 93, normalized size = 1.24 \[ \frac {4 \, {\left (d x + c\right )} a^{3} + b^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, a^{2} b \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) - 4 \, a b^{2} \tan \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.59, size = 137, normalized size = 1.83 \[ \frac {2\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {b^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {b^3\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {2\,a^2\,b\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {a\,b^2\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )} \]
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 \[ \int \left (a - b \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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