Optimal. Leaf size=106 \[ a^4 x+\frac {b^2 \left (a^2-2 b^2\right ) \tan (c+d x)}{3 d}+\frac {a b \left (2 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a b^3 \tan (c+d x) \sec (c+d x)}{3 d}-\frac {b^2 \tan (c+d x) (a+b \sec (c+d x))^2}{3 d} \]
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Rubi [A] time = 0.17, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4042, 3918, 4048, 3770, 3767, 8} \[ \frac {b^2 \left (a^2-2 b^2\right ) \tan (c+d x)}{3 d}+\frac {a b \left (2 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{d}+a^4 x-\frac {a b^3 \tan (c+d x) \sec (c+d x)}{3 d}-\frac {b^2 \tan (c+d x) (a+b \sec (c+d x))^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3918
Rule 4042
Rule 4048
Rubi steps
\begin {align*} \int (a+b \sec (c+d x))^2 \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))^3 \, dx\\ &=-\frac {b^2 (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}-\frac {1}{3} \int (a+b \sec (c+d x)) \left (-3 a^3-b \left (3 a^2-2 b^2\right ) \sec (c+d x)+2 a b^2 \sec ^2(c+d x)\right ) \, dx\\ &=-\frac {a b^3 \sec (c+d x) \tan (c+d x)}{3 d}-\frac {b^2 (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}-\frac {1}{6} \int \left (-6 a^4-6 a b \left (2 a^2-b^2\right ) \sec (c+d x)-2 b^2 \left (a^2-2 b^2\right ) \sec ^2(c+d x)\right ) \, dx\\ &=a^4 x-\frac {a b^3 \sec (c+d x) \tan (c+d x)}{3 d}-\frac {b^2 (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {1}{3} \left (b^2 \left (a^2-2 b^2\right )\right ) \int \sec ^2(c+d x) \, dx+\left (a b \left (2 a^2-b^2\right )\right ) \int \sec (c+d x) \, dx\\ &=a^4 x+\frac {a b \left (2 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a b^3 \sec (c+d x) \tan (c+d x)}{3 d}-\frac {b^2 (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}-\frac {\left (b^2 \left (a^2-2 b^2\right )\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=a^4 x+\frac {a b \left (2 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b^2 \left (a^2-2 b^2\right ) \tan (c+d x)}{3 d}-\frac {a b^3 \sec (c+d x) \tan (c+d x)}{3 d}-\frac {b^2 (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 86, normalized size = 0.81 \[ a^4 x+\frac {2 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a b^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a b^3 \tan (c+d x) \sec (c+d x)}{d}-\frac {b^4 \left (\frac {1}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 132, normalized size = 1.25 \[ \frac {6 \, a^{4} d x \cos \left (d x + c\right )^{3} + 3 \, {\left (2 \, a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (2 \, a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (2 \, b^{4} \cos \left (d x + c\right )^{2} + 3 \, a b^{3} \cos \left (d x + c\right ) + b^{4}\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 168, normalized size = 1.58 \[ \frac {3 \, {\left (d x + c\right )} a^{4} + 3 \, {\left (2 \, a^{3} b - a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (2 \, a^{3} b - a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (3 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.02, size = 118, normalized size = 1.11 \[ a^{4} x +\frac {a^{4} c}{d}+\frac {2 a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}-\frac {a \,b^{3} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{d}-\frac {a \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}-\frac {2 b^{4} \tan \left (d x +c \right )}{3 d}-\frac {b^{4} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 105, normalized size = 0.99 \[ \frac {6 \, {\left (d x + c\right )} a^{4} - 2 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} b^{4} + 3 \, a b^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, a^{3} b \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.62, size = 160, normalized size = 1.51 \[ \frac {2\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {2\,b^4\,\sin \left (c+d\,x\right )}{3\,d\,\cos \left (c+d\,x\right )}-\frac {b^4\,\sin \left (c+d\,x\right )}{3\,d\,{\cos \left (c+d\,x\right )}^3}-\frac {2\,a\,b^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {4\,a^3\,b\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {a\,b^3\,\sin \left (c+d\,x\right )}{d\,{\cos \left (c+d\,x\right )}^2} \]
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 \[ \int \left (a - b \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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