Optimal. Leaf size=99 \[ \frac {2 b \left (4 a^2+b^2\right ) \tan (c+d x)}{3 d}+\frac {a \left (2 a^2+3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {5 a b^2 \tan (c+d x) \sec (c+d x)}{6 d}+\frac {b \tan (c+d x) (a+b \sec (c+d x))^2}{3 d} \]
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Rubi [A] time = 0.13, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3830, 3997, 3787, 3770, 3767, 8} \[ \frac {2 b \left (4 a^2+b^2\right ) \tan (c+d x)}{3 d}+\frac {a \left (2 a^2+3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {5 a b^2 \tan (c+d x) \sec (c+d x)}{6 d}+\frac {b \tan (c+d x) (a+b \sec (c+d x))^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3787
Rule 3830
Rule 3997
Rubi steps
\begin {align*} \int \sec (c+d x) (a+b \sec (c+d x))^3 \, dx &=\frac {b (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {1}{3} \int \sec (c+d x) (a+b \sec (c+d x)) \left (3 a^2+2 b^2+5 a b \sec (c+d x)\right ) \, dx\\ &=\frac {5 a b^2 \sec (c+d x) \tan (c+d x)}{6 d}+\frac {b (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {1}{6} \int \sec (c+d x) \left (3 a \left (2 a^2+3 b^2\right )+4 b \left (4 a^2+b^2\right ) \sec (c+d x)\right ) \, dx\\ &=\frac {5 a b^2 \sec (c+d x) \tan (c+d x)}{6 d}+\frac {b (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {1}{3} \left (2 b \left (4 a^2+b^2\right )\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{2} \left (a \left (2 a^2+3 b^2\right )\right ) \int \sec (c+d x) \, dx\\ &=\frac {a \left (2 a^2+3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {5 a b^2 \sec (c+d x) \tan (c+d x)}{6 d}+\frac {b (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}-\frac {\left (2 b \left (4 a^2+b^2\right )\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=\frac {a \left (2 a^2+3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {2 b \left (4 a^2+b^2\right ) \tan (c+d x)}{3 d}+\frac {5 a b^2 \sec (c+d x) \tan (c+d x)}{6 d}+\frac {b (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 = 70, normalized size = 0.71 \[ \frac {\left (6 a^3+9 a b^2\right ) \tanh ^{-1}(\sin (c+d x))+b \tan (c+d x) \left (18 a^2+9 a b \sec (c+d x)+2 b^2 \tan ^2(c+d x)+6 b^2\right )}{6 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 126, normalized size = 1.27 \[ \frac {3 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (9 \, a b^{2} \cos \left (d x + c\right ) + 2 \, b^{3} + 2 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 205, normalized size = 2.07 \[ \frac {3 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (18 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 36 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.91, size = 118, normalized size = 1.19 \[ \frac {a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {3 a^{2} b \tan \left (d x +c \right )}{d}+\frac {3 a \,b^{2} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {3 b^{2} a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {2 b^{3} \tan \left (d x +c \right )}{3 d}+\frac {b^{3} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 106, normalized size = 1.07 \[ \frac {4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} b^{3} - 9 \, a b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 36 \, a^{2} b \tan \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.15, size = 157, normalized size = 1.59 \[ \frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (2\,a^3+3\,a\,b^2\right )}{d}-\frac {\left (6\,a^2\,b-3\,a\,b^2+2\,b^3\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-12\,a^2\,b-\frac {4\,b^3}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (6\,a^2\,b+3\,a\,b^2+2\,b^3\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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 \[ \int \left (a + b \sec {\left (c + d x \right )}\right )^{3} \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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