Optimal. Leaf size=124 \[ \frac {a \left (a^2 B+6 a b C+6 b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a^2 (a C+2 b B) \tan (c+d x)}{d}-\frac {b^2 (a B-2 b C) \sin (c+d x)}{2 d}+b^2 x (3 a C+b B)+\frac {a B \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d} \]
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Rubi [A] time = 0.41, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3029, 2989, 3031, 3023, 2735, 3770} \[ \frac {a \left (a^2 B+6 a b C+6 b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a^2 (a C+2 b B) \tan (c+d x)}{d}-\frac {b^2 (a B-2 b C) \sin (c+d x)}{2 d}+b^2 x (3 a C+b B)+\frac {a B \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2989
Rule 3023
Rule 3029
Rule 3031
Rule 3770
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx &=\int (a+b \cos (c+d x))^3 (B+C \cos (c+d x)) \sec ^3(c+d x) \, dx\\ &=\frac {a B (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int (a+b \cos (c+d x)) \left (2 a (2 b B+a C)+\left (a^2 B+2 b^2 B+4 a b C\right ) \cos (c+d x)-b (a B-2 b C) \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac {a^2 (2 b B+a C) \tan (c+d x)}{d}+\frac {a B (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {1}{2} \int \left (-a \left (a^2 B+6 b^2 B+6 a b C\right )-2 b^2 (b B+3 a C) \cos (c+d x)+b^2 (a B-2 b C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac {b^2 (a B-2 b C) \sin (c+d x)}{2 d}+\frac {a^2 (2 b B+a C) \tan (c+d x)}{d}+\frac {a B (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {1}{2} \int \left (-a \left (a^2 B+6 b^2 B+6 a b C\right )-2 b^2 (b B+3 a C) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=b^2 (b B+3 a C) x-\frac {b^2 (a B-2 b C) \sin (c+d x)}{2 d}+\frac {a^2 (2 b B+a C) \tan (c+d x)}{d}+\frac {a B (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \left (a \left (a^2 B+6 b^2 B+6 a b C\right )\right ) \int \sec (c+d x) \, dx\\ &=b^2 (b B+3 a C) x+\frac {a \left (a^2 B+6 b^2 B+6 a b C\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {b^2 (a B-2 b C) \sin (c+d x)}{2 d}+\frac {a^2 (2 b B+a C) \tan (c+d x)}{d}+\frac {a B (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}\\ \end {align*}
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Mathematica [B] time = 2.18, size = 277, normalized size = 2.23 \[ \frac {\frac {a^3 B}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {a^3 B}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}-2 a \left (a^2 B+6 a b C+6 b^2 B\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 a \left (a^2 B+6 a b C+6 b^2 B\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+\frac {4 a^2 (a C+3 b B) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {4 a^2 (a C+3 b B) \sin \left (\frac {1}{2} (c+d x)\right )}{\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )}+4 b^2 (c+d x) (3 a C+b B)+4 b^3 C \sin (c+d x)}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 167, normalized size = 1.35 \[ \frac {4 \, {\left (3 \, C a b^{2} + B b^{3}\right )} d x \cos \left (d x + c\right )^{2} + {\left (B a^{3} + 6 \, C a^{2} b + 6 \, B a b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (B a^{3} + 6 \, C a^{2} b + 6 \, B a b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, C b^{3} \cos \left (d x + c\right )^{2} + B a^{3} + 2 \, {\left (C a^{3} + 3 \, B a^{2} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 239, normalized size = 1.93 \[ \frac {\frac {4 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + 2 \, {\left (3 \, C a b^{2} + B b^{3}\right )} {\left (d x + c\right )} + {\left (B a^{3} + 6 \, C a^{2} b + 6 \, B a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (B a^{3} + 6 \, C a^{2} b + 6 \, B a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 172, normalized size = 1.39 \[ \frac {C \,a^{3} \tan \left (d x +c \right )}{d}+\frac {a^{3} B \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {a^{3} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {3 C \,a^{2} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {3 a^{2} b B \tan \left (d x +c \right )}{d}+3 a \,b^{2} C x +\frac {3 C a \,b^{2} c}{d}+\frac {3 B a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {b^{3} C \sin \left (d x +c \right )}{d}+b^{3} B x +\frac {b^{3} B c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 169, normalized size = 1.36 \[ \frac {12 \, {\left (d x + c\right )} C a b^{2} + 4 \, {\left (d x + c\right )} B b^{3} - B a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, C a^{2} b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, B a b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, C b^{3} \sin \left (d x + c\right ) + 4 \, C a^{3} \tan \left (d x + c\right ) + 12 \, B a^{2} b \tan \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.84, size = 249, normalized size = 2.01 \[ \frac {\frac {C\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{2}+\frac {C\,b^3\,\sin \left (3\,c+3\,d\,x\right )}{4}+\frac {B\,a^3\,\sin \left (c+d\,x\right )}{2}+\frac {C\,b^3\,\sin \left (c+d\,x\right )}{4}+\frac {3\,B\,a^2\,b\,\sin \left (2\,c+2\,d\,x\right )}{2}}{d\,\left (\frac {\cos \left (2\,c+2\,d\,x\right )}{2}+\frac {1}{2}\right )}-\frac {2\,\left (\frac {B\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,1{}\mathrm {i}}{2}-B\,b^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+B\,a\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,3{}\mathrm {i}-3\,C\,a\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+C\,a^2\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,3{}\mathrm {i}\right )}{d} \]
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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