Optimal. Leaf size=167 \[ \frac {a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a \left (C \left (a^2+4 b^2\right )+6 A b^2\right ) \sin (c+d x)}{2 d}+\frac {b \left (2 a^2 C+b^2 (4 A+3 C)\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} b x \left (12 a^2 (2 A+C)+b^2 (4 A+3 C)\right )+\frac {a C \sin (c+d x) (a+b \cos (c+d x))^2}{4 d}+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d} \]
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Rubi [A] time = 0.54, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3050, 3049, 3033, 3023, 2735, 3770} \[ \frac {a \left (C \left (a^2+4 b^2\right )+6 A b^2\right ) \sin (c+d x)}{2 d}+\frac {b \left (2 a^2 C+b^2 (4 A+3 C)\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} b x \left (12 a^2 (2 A+C)+b^2 (4 A+3 C)\right )+\frac {a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a C \sin (c+d x) (a+b \cos (c+d x))^2}{4 d}+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 3023
Rule 3033
Rule 3049
Rule 3050
Rule 3770
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx &=\frac {C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {1}{4} \int (a+b \cos (c+d x))^2 \left (4 a A+b (4 A+3 C) \cos (c+d x)+3 a C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {a C (a+b \cos (c+d x))^2 \sin (c+d x)}{4 d}+\frac {C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {1}{12} \int (a+b \cos (c+d x)) \left (12 a^2 A+3 a b (8 A+5 C) \cos (c+d x)+3 \left (2 a^2 C+b^2 (4 A+3 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {b \left (2 a^2 C+b^2 (4 A+3 C)\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a C (a+b \cos (c+d x))^2 \sin (c+d x)}{4 d}+\frac {C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {1}{24} \int \left (24 a^3 A+3 b \left (12 a^2 (2 A+C)+b^2 (4 A+3 C)\right ) \cos (c+d x)+12 a \left (6 A b^2+\left (a^2+4 b^2\right ) C\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {a \left (6 A b^2+\left (a^2+4 b^2\right ) C\right ) \sin (c+d x)}{2 d}+\frac {b \left (2 a^2 C+b^2 (4 A+3 C)\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a C (a+b \cos (c+d x))^2 \sin (c+d x)}{4 d}+\frac {C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {1}{24} \int \left (24 a^3 A+3 b \left (12 a^2 (2 A+C)+b^2 (4 A+3 C)\right ) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {1}{8} b \left (12 a^2 (2 A+C)+b^2 (4 A+3 C)\right ) x+\frac {a \left (6 A b^2+\left (a^2+4 b^2\right ) C\right ) \sin (c+d x)}{2 d}+\frac {b \left (2 a^2 C+b^2 (4 A+3 C)\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a C (a+b \cos (c+d x))^2 \sin (c+d x)}{4 d}+\frac {C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\left (a^3 A\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{8} b \left (12 a^2 (2 A+C)+b^2 (4 A+3 C)\right ) x+\frac {a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a \left (6 A b^2+\left (a^2+4 b^2\right ) C\right ) \sin (c+d x)}{2 d}+\frac {b \left (2 a^2 C+b^2 (4 A+3 C)\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a C (a+b \cos (c+d x))^2 \sin (c+d x)}{4 d}+\frac {C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.60, size = 180, normalized size = 1.08 \[ \frac {-32 a^3 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+32 a^3 A \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+4 b (c+d x) \left (12 a^2 (2 A+C)+b^2 (4 A+3 C)\right )+8 a \left (4 a^2 C+12 A b^2+9 b^2 C\right ) \sin (c+d x)+8 b \left (C \left (3 a^2+b^2\right )+A b^2\right ) \sin (2 (c+d x))+8 a b^2 C \sin (3 (c+d x))+b^3 C \sin (4 (c+d x))}{32 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 146, normalized size = 0.87 \[ \frac {4 \, A a^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 4 \, A a^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (12 \, {\left (2 \, A + C\right )} a^{2} b + {\left (4 \, A + 3 \, C\right )} b^{3}\right )} d x + {\left (2 \, C b^{3} \cos \left (d x + c\right )^{3} + 8 \, C a b^{2} \cos \left (d x + c\right )^{2} + 8 \, C a^{3} + 8 \, {\left (3 \, A + 2 \, C\right )} a b^{2} + {\left (12 \, C a^{2} b + {\left (4 \, A + 3 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.89, size = 503, normalized size = 3.01 \[ \frac {8 \, A a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 8 \, A a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + {\left (24 \, A a^{2} b + 12 \, C a^{2} b + 4 \, A b^{3} + 3 \, C b^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (8 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 12 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 4 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 5 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 24 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, 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 )}^{4}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 252, normalized size = 1.51 \[ \frac {A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a^{3} C \sin \left (d x +c \right )}{d}+3 A x \,a^{2} b +\frac {3 A \,a^{2} b c}{d}+\frac {3 C \,a^{2} b \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {3 C \,a^{2} b x}{2}+\frac {3 C \,a^{2} b c}{2 d}+\frac {3 A a \,b^{2} \sin \left (d x +c \right )}{d}+\frac {C \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) a \,b^{2}}{d}+\frac {2 C a \,b^{2} \sin \left (d x +c \right )}{d}+\frac {A \,b^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {A x \,b^{3}}{2}+\frac {A \,b^{3} c}{2 d}+\frac {b^{3} C \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4 d}+\frac {3 b^{3} C \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8 d}+\frac {3 b^{3} C x}{8}+\frac {3 b^{3} C c}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 167, normalized size = 1.00 \[ \frac {96 \, {\left (d x + c\right )} A a^{2} b + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} b - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a b^{2} + 8 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{3} + {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{3} + 32 \, A a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 32 \, C a^{3} \sin \left (d x + c\right ) + 96 \, A a b^{2} \sin \left (d x + c\right )}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.02, size = 2008, normalized size = 12.02 \[ \text {result too large to display} \]
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 \[ \int \left (A + C \cos ^{2}{\left (c + d x \right )}\right ) \left (a + b \cos {\left (c + d x \right )}\right )^{3} \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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