Optimal. Leaf size=167 \[ -\frac {b \left (a^2 (6 A-8 C)-b^2 (3 A+2 C)\right ) \sin (c+d x)}{3 d}+\frac {1}{2} a x \left (2 a^2 C+6 A b^2+3 b^2 C\right )+\frac {3 a^2 A b \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a b^2 (6 A-5 C) \sin (c+d x) \cos (c+d x)}{6 d}-\frac {b (3 A-C) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^3}{d} \]
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Rubi [A] time = 0.50, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3048, 3049, 3033, 3023, 2735, 3770} \[ -\frac {b \left (a^2 (6 A-8 C)-b^2 (3 A+2 C)\right ) \sin (c+d x)}{3 d}+\frac {1}{2} a x \left (2 a^2 C+6 A b^2+3 b^2 C\right )+\frac {3 a^2 A b \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a b^2 (6 A-5 C) \sin (c+d x) \cos (c+d x)}{6 d}-\frac {b (3 A-C) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^3}{d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 3023
Rule 3033
Rule 3048
Rule 3049
Rule 3770
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx &=\frac {A (a+b \cos (c+d x))^3 \tan (c+d x)}{d}+\int (a+b \cos (c+d x))^2 \left (3 A b+a C \cos (c+d x)-b (3 A-C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac {b (3 A-C) (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {1}{3} \int (a+b \cos (c+d x)) \left (9 a A b+\left (3 A b^2+3 a^2 C+2 b^2 C\right ) \cos (c+d x)-a b (6 A-5 C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac {a b^2 (6 A-5 C) \cos (c+d x) \sin (c+d x)}{6 d}-\frac {b (3 A-C) (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {1}{6} \int \left (18 a^2 A b+3 a \left (6 A b^2+2 a^2 C+3 b^2 C\right ) \cos (c+d x)-2 b \left (a^2 (6 A-8 C)-b^2 (3 A+2 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac {b \left (a^2 (6 A-8 C)-b^2 (3 A+2 C)\right ) \sin (c+d x)}{3 d}-\frac {a b^2 (6 A-5 C) \cos (c+d x) \sin (c+d x)}{6 d}-\frac {b (3 A-C) (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {1}{6} \int \left (18 a^2 A b+3 a \left (6 A b^2+2 a^2 C+3 b^2 C\right ) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {1}{2} a \left (6 A b^2+2 a^2 C+3 b^2 C\right ) x-\frac {b \left (a^2 (6 A-8 C)-b^2 (3 A+2 C)\right ) \sin (c+d x)}{3 d}-\frac {a b^2 (6 A-5 C) \cos (c+d x) \sin (c+d x)}{6 d}-\frac {b (3 A-C) (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^3 \tan (c+d x)}{d}+\left (3 a^2 A b\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{2} a \left (6 A b^2+2 a^2 C+3 b^2 C\right ) x+\frac {3 a^2 A b \tanh ^{-1}(\sin (c+d x))}{d}-\frac {b \left (a^2 (6 A-8 C)-b^2 (3 A+2 C)\right ) \sin (c+d x)}{3 d}-\frac {a b^2 (6 A-5 C) \cos (c+d x) \sin (c+d x)}{6 d}-\frac {b (3 A-C) (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^3 \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.89, size = 185, normalized size = 1.11 \[ \frac {12 a^3 A \tan (c+d x)+12 a^3 c C+12 a^3 C d x+3 b \left (3 C \left (4 a^2+b^2\right )+4 A b^2\right ) \sin (c+d x)-36 a^2 A b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+36 a^2 A b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+36 a A b^2 c+36 a A b^2 d x+9 a b^2 C \sin (2 (c+d x))+18 a b^2 c C+18 a b^2 C d x+b^3 C \sin (3 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 158, normalized size = 0.95 \[ \frac {9 \, A a^{2} b \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 9 \, A a^{2} b \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (2 \, C a^{3} + 3 \, {\left (2 \, A + C\right )} a b^{2}\right )} d x \cos \left (d x + c\right ) + {\left (2 \, C b^{3} \cos \left (d x + c\right )^{3} + 9 \, C a b^{2} \cos \left (d x + c\right )^{2} + 6 \, A a^{3} + 2 \, {\left (9 \, C a^{2} b + {\left (3 \, A + 2 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.14, size = 306, normalized size = 1.83 \[ \frac {18 \, A a^{2} b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 18 \, A a^{2} b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {12 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + 3 \, {\left (2 \, C a^{3} + 6 \, A a b^{2} + 3 \, C a b^{2}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (18 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 183, normalized size = 1.10 \[ \frac {A \,a^{3} \tan \left (d x +c \right )}{d}+a^{3} C x +\frac {C \,a^{3} c}{d}+\frac {3 A \,a^{2} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {3 C \,a^{2} b \sin \left (d x +c \right )}{d}+3 A x a \,b^{2}+\frac {3 A a \,b^{2} c}{d}+\frac {3 C a \,b^{2} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {3 a \,b^{2} C x}{2}+\frac {3 C a \,b^{2} c}{2 d}+\frac {A \,b^{3} \sin \left (d x +c \right )}{d}+\frac {C \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) b^{3}}{3 d}+\frac {2 b^{3} C \sin \left (d x +c \right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 141, normalized size = 0.84 \[ \frac {12 \, {\left (d x + c\right )} C a^{3} + 36 \, {\left (d x + c\right )} A a b^{2} + 9 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a b^{2} - 4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C b^{3} + 18 \, A a^{2} b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, C a^{2} b \sin \left (d x + c\right ) + 12 \, A b^{3} \sin \left (d x + c\right ) + 12 \, A a^{3} \tan \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.28, size = 238, normalized size = 1.43 \[ \frac {2\,C\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+6\,A\,a\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+3\,C\,a\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )-A\,a^2\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,6{}\mathrm {i}}{d}+\frac {\frac {A\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{2}+\frac {5\,C\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{12}+\frac {C\,b^3\,\sin \left (4\,c+4\,d\,x\right )}{24}+A\,a^3\,\sin \left (c+d\,x\right )+\frac {3\,C\,a\,b^2\,\sin \left (c+d\,x\right )}{8}+\frac {3\,C\,a^2\,b\,\sin \left (2\,c+2\,d\,x\right )}{2}+\frac {3\,C\,a\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{8}}{d\,\cos \left (c+d\,x\right )} \]
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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