Optimal. Leaf size=110 \[ \frac {a^2 (A+C) \sin (c+d x)}{d}+\frac {A \sin (c+d x) \cos (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{3 d}+a^2 x (A+2 C)+\frac {a^2 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^2}{3 d} \]
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Rubi [A] time = 0.27, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {4087, 4017, 3996, 3770} \[ \frac {a^2 (A+C) \sin (c+d x)}{d}+\frac {A \sin (c+d x) \cos (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{3 d}+a^2 x (A+2 C)+\frac {a^2 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 3770
Rule 3996
Rule 4017
Rule 4087
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac {\int \cos ^2(c+d x) (a+a \sec (c+d x))^2 (2 a A+3 a C \sec (c+d x)) \, dx}{3 a}\\ &=\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac {A \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{3 d}+\frac {\int \cos (c+d x) (a+a \sec (c+d x)) \left (6 a^2 (A+C)+6 a^2 C \sec (c+d x)\right ) \, dx}{6 a}\\ &=\frac {a^2 (A+C) \sin (c+d x)}{d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac {A \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{3 d}-\frac {\int \left (-6 a^3 (A+2 C)-6 a^3 C \sec (c+d x)\right ) \, dx}{6 a}\\ &=a^2 (A+2 C) x+\frac {a^2 (A+C) \sin (c+d x)}{d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac {A \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{3 d}+\left (a^2 C\right ) \int \sec (c+d x) \, dx\\ &=a^2 (A+2 C) x+\frac {a^2 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^2 (A+C) \sin (c+d x)}{d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac {A \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 109, normalized size = 0.99 \[ \frac {a^2 \left (3 (7 A+4 C) \sin (c+d x)+6 A \sin (2 (c+d x))+A \sin (3 (c+d x))+12 A d x-12 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 C \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+24 C d x\right )}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 95, normalized size = 0.86 \[ \frac {6 \, {\left (A + 2 \, C\right )} a^{2} d x + 3 \, C a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, C a^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (A a^{2} \cos \left (d x + c\right )^{2} + 3 \, A a^{2} \cos \left (d x + c\right ) + {\left (5 \, A + 3 \, C\right )} a^{2}\right )} \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.03, size = 179, normalized size = 1.63 \[ \frac {3 \, C a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, C a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 3 \, {\left (A a^{2} + 2 \, C a^{2}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (3 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 8 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.09, size = 128, normalized size = 1.16 \[ \frac {a^{2} A \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {5 a^{2} A \sin \left (d x +c \right )}{3 d}+\frac {a^{2} C \sin \left (d x +c \right )}{d}+\frac {a^{2} A \cos \left (d x +c \right ) \sin \left (d x +c \right )}{d}+a^{2} A x +\frac {A \,a^{2} c}{d}+2 a^{2} C x +\frac {2 C \,a^{2} c}{d}+\frac {a^{2} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 114, normalized size = 1.04 \[ -\frac {2 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} - 12 \, {\left (d x + c\right )} C a^{2} - 3 \, C a^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 6 \, A a^{2} \sin \left (d x + c\right ) - 6 \, C a^{2} \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.68, size = 159, normalized size = 1.45 \[ \frac {7\,A\,a^2\,\sin \left (c+d\,x\right )}{4\,d}+\frac {C\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {2\,A\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {4\,C\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {A\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {A\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{12\,d} \]
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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