Optimal. Leaf size=196 \[ \frac {a \sin (c+d x) \left (a^2 (2 A+3 C)+6 a b B+3 A b^2\right )}{3 d}+\frac {1}{2} x \left (a^3 B+3 a^2 b (A+2 C)+6 a b^2 B+2 A b^3\right )-\frac {b^2 \tan (c+d x) (3 a B+5 A b-6 b C)}{6 d}+\frac {(a B+A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}+\frac {b^2 (3 a C+b B) \tanh ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A] time = 0.60, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {4094, 4076, 4047, 8, 4045, 3770} \[ \frac {a \sin (c+d x) \left (a^2 (2 A+3 C)+6 a b B+3 A b^2\right )}{3 d}+\frac {1}{2} x \left (3 a^2 b (A+2 C)+a^3 B+6 a b^2 B+2 A b^3\right )-\frac {b^2 \tan (c+d x) (3 a B+5 A b-6 b C)}{6 d}+\frac {(a B+A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}+\frac {b^2 (3 a C+b B) \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3770
Rule 4045
Rule 4047
Rule 4076
Rule 4094
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {1}{3} \int \cos ^2(c+d x) (a+b \sec (c+d x))^2 \left (3 (A b+a B)+(2 a A+3 b B+3 a C) \sec (c+d x)-b (A-3 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {(A b+a B) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {1}{6} \int \cos (c+d x) (a+b \sec (c+d x)) \left (2 \left (3 A b^2+6 a b B+\frac {1}{2} a^2 (4 A+6 C)\right )+\left (3 a^2 B+6 b^2 B+a b (5 A+12 C)\right ) \sec (c+d x)-b (5 A b+3 a B-6 b C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {(A b+a B) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}-\frac {b^2 (5 A b+3 a B-6 b C) \tan (c+d x)}{6 d}+\frac {1}{6} \int \cos (c+d x) \left (2 a \left (3 A b^2+6 a b B+a^2 (2 A+3 C)\right )+3 \left (2 A b^3+a^3 B+6 a b^2 B+3 a^2 b (A+2 C)\right ) \sec (c+d x)+6 b^2 (b B+3 a C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {(A b+a B) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}-\frac {b^2 (5 A b+3 a B-6 b C) \tan (c+d x)}{6 d}+\frac {1}{6} \int \cos (c+d x) \left (2 a \left (3 A b^2+6 a b B+a^2 (2 A+3 C)\right )+6 b^2 (b B+3 a C) \sec ^2(c+d x)\right ) \, dx+\frac {1}{2} \left (2 A b^3+a^3 B+6 a b^2 B+3 a^2 b (A+2 C)\right ) \int 1 \, dx\\ &=\frac {1}{2} \left (2 A b^3+a^3 B+6 a b^2 B+3 a^2 b (A+2 C)\right ) x+\frac {a \left (3 A b^2+6 a b B+a^2 (2 A+3 C)\right ) \sin (c+d x)}{3 d}+\frac {(A b+a B) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}-\frac {b^2 (5 A b+3 a B-6 b C) \tan (c+d x)}{6 d}+\left (b^2 (b B+3 a C)\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{2} \left (2 A b^3+a^3 B+6 a b^2 B+3 a^2 b (A+2 C)\right ) x+\frac {b^2 (b B+3 a C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a \left (3 A b^2+6 a b B+a^2 (2 A+3 C)\right ) \sin (c+d x)}{3 d}+\frac {(A b+a B) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}-\frac {b^2 (5 A b+3 a B-6 b C) \tan (c+d x)}{6 d}\\ \end {align*}
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Mathematica [A] time = 1.32, size = 263, normalized size = 1.34 \[ \frac {a^3 A \sin (3 (c+d x))+3 a \sin (c+d x) \left (a^2 (3 A+4 C)+12 a b B+12 A b^2\right )+3 a^2 (a B+3 A b) \sin (2 (c+d x))+6 (c+d x) \left (a^3 B+3 a^2 b (A+2 C)+6 a b^2 B+2 A b^3\right )-12 b^2 (3 a C+b B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 b^2 (3 a C+b B) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+\frac {12 b^3 C \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {12 b^3 C \sin \left (\frac {1}{2} (c+d x)\right )}{\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )}}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 201, normalized size = 1.03 \[ \frac {3 \, {\left (B a^{3} + 3 \, {\left (A + 2 \, C\right )} a^{2} b + 6 \, B a b^{2} + 2 \, A b^{3}\right )} d x \cos \left (d x + c\right ) + 3 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (2 \, A a^{3} \cos \left (d x + c\right )^{3} + 6 \, C b^{3} + 3 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left ({\left (2 \, A + 3 \, C\right )} a^{3} + 9 \, B a^{2} b + 9 \, A a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.34, size = 418, normalized size = 2.13 \[ -\frac {\frac {12 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - 3 \, {\left (B a^{3} + 3 \, A a^{2} b + 6 \, C a^{2} b + 6 \, B a b^{2} + 2 \, A b^{3}\right )} {\left (d x + c\right )} - 6 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 6 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.27, size = 278, normalized size = 1.42 \[ \frac {A \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a^{3}}{3 d}+\frac {2 a^{3} A \sin \left (d x +c \right )}{3 d}+\frac {a^{3} B \sin \left (d x +c \right ) \cos \left (d x +c \right )}{2 d}+\frac {a^{3} B x}{2}+\frac {a^{3} B c}{2 d}+\frac {a^{3} C \sin \left (d x +c \right )}{d}+\frac {3 A \,a^{2} b \sin \left (d x +c \right ) \cos \left (d x +c \right )}{2 d}+\frac {3 a^{2} A b x}{2}+\frac {3 A \,a^{2} b c}{2 d}+\frac {3 a^{2} b B \sin \left (d x +c \right )}{d}+3 C x \,a^{2} b +\frac {3 C \,a^{2} b c}{d}+\frac {3 A a \,b^{2} \sin \left (d x +c \right )}{d}+3 B x a \,b^{2}+\frac {3 B a \,b^{2} c}{d}+\frac {3 C a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+A x \,b^{3}+\frac {A \,b^{3} c}{d}+\frac {b^{3} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {b^{3} C \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 216, normalized size = 1.10 \[ -\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 9 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} b - 36 \, {\left (d x + c\right )} C a^{2} b - 36 \, {\left (d x + c\right )} B a b^{2} - 12 \, {\left (d x + c\right )} A b^{3} - 18 \, C a b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 6 \, B b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, C a^{3} \sin \left (d x + c\right ) - 36 \, B a^{2} b \sin \left (d x + c\right ) - 36 \, A a b^{2} \sin \left (d x + c\right ) - 12 \, C b^{3} \tan \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.41, size = 2470, normalized size = 12.60 \[ \text {result too large to display} \]
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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