Optimal. Leaf size=111 \[ \frac {5 a^3 (A+B) \tan (c+d x)}{2 d}+\frac {a^3 (7 A+5 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {(3 A+5 B) \tan (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{6 d}+a^3 A x+\frac {a B \tan (c+d x) (a \sec (c+d x)+a)^2}{3 d} \]
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Rubi [A] time = 0.14, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3917, 3914, 3767, 8, 3770} \[ \frac {5 a^3 (A+B) \tan (c+d x)}{2 d}+\frac {a^3 (7 A+5 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {(3 A+5 B) \tan (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{6 d}+a^3 A x+\frac {a B \tan (c+d x) (a \sec (c+d x)+a)^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3914
Rule 3917
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx &=\frac {a B (a+a \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {1}{3} \int (a+a \sec (c+d x))^2 (3 a A+a (3 A+5 B) \sec (c+d x)) \, dx\\ &=\frac {a B (a+a \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {(3 A+5 B) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{6 d}+\frac {1}{6} \int (a+a \sec (c+d x)) \left (6 a^2 A+15 a^2 (A+B) \sec (c+d x)\right ) \, dx\\ &=a^3 A x+\frac {a B (a+a \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {(3 A+5 B) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{6 d}+\frac {1}{2} \left (5 a^3 (A+B)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{2} \left (a^3 (7 A+5 B)\right ) \int \sec (c+d x) \, dx\\ &=a^3 A x+\frac {a^3 (7 A+5 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a B (a+a \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {(3 A+5 B) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{6 d}-\frac {\left (5 a^3 (A+B)\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{2 d}\\ &=a^3 A x+\frac {a^3 (7 A+5 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {5 a^3 (A+B) \tan (c+d x)}{2 d}+\frac {a B (a+a \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {(3 A+5 B) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{6 d}\\ \end {align*}
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Mathematica [B] time = 6.41, size = 1056, normalized size = 9.51 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 141, normalized size = 1.27 \[ \frac {12 \, A a^{3} d x \cos \left (d x + c\right )^{3} + 3 \, {\left (7 \, A + 5 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (7 \, A + 5 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, {\left (9 \, A + 11 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 3 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right ) + 2 \, B a^{3}\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.77, size = 189, normalized size = 1.70 \[ \frac {6 \, {\left (d x + c\right )} A a^{3} + 3 \, {\left (7 \, A a^{3} + 5 \, B a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (7 \, A a^{3} + 5 \, B a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (15 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 36 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 33 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.16, size = 158, normalized size = 1.42 \[ a^{3} A x +\frac {A \,a^{3} c}{d}+\frac {5 a^{3} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {7 A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {11 a^{3} B \tan \left (d x +c \right )}{3 d}+\frac {3 A \,a^{3} \tan \left (d x +c \right )}{d}+\frac {3 a^{3} B \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {A \,a^{3} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {a^{3} B \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 198, normalized size = 1.78 \[ \frac {12 \, {\left (d x + c\right )} A a^{3} + 4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{3} - 3 \, A a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 9 \, B a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, A a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 12 \, B a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 36 \, A a^{3} \tan \left (d x + c\right ) + 36 \, B a^{3} \tan \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.08, size = 209, normalized size = 1.88 \[ \frac {2\,A\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {7\,A\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {5\,B\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {3\,A\,a^3\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {A\,a^3\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {11\,B\,a^3\,\sin \left (c+d\,x\right )}{3\,d\,\cos \left (c+d\,x\right )}+\frac {3\,B\,a^3\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {B\,a^3\,\sin \left (c+d\,x\right )}{3\,d\,{\cos \left (c+d\,x\right )}^3} \]
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 \[ a^{3} \left (\int A\, dx + \int 3 A \sec {\left (c + d x \right )}\, dx + \int 3 A \sec ^{2}{\left (c + d x \right )}\, dx + \int A \sec ^{3}{\left (c + d x \right )}\, dx + \int B \sec {\left (c + d x \right )}\, dx + \int 3 B \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 B \sec ^{3}{\left (c + d x \right )}\, dx + \int B \sec ^{4}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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