Optimal. Leaf size=182 \[ \frac {b \left (a^2 (4 A+6 C)+A b^2\right ) \sin (c+d x)}{2 d}+\frac {a \left (a^2 (3 A+4 C)+2 A b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} a x \left (a^2 (3 A+4 C)+12 b^2 (A+2 C)\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}+\frac {A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{4 d}+\frac {b^3 C \tanh ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A] time = 0.56, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4095, 4094, 4074, 4047, 8, 4045, 3770} \[ \frac {b \left (a^2 (4 A+6 C)+A b^2\right ) \sin (c+d x)}{2 d}+\frac {a \left (a^2 (3 A+4 C)+2 A b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} a x \left (a^2 (3 A+4 C)+12 b^2 (A+2 C)\right )+\frac {A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{4 d}+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}+\frac {b^3 C \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3770
Rule 4045
Rule 4047
Rule 4074
Rule 4094
Rule 4095
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d}+\frac {1}{4} \int \cos ^3(c+d x) (a+b \sec (c+d x))^2 \left (3 A b+a (3 A+4 C) \sec (c+d x)+4 b C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {A b \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d}+\frac {1}{12} \int \cos ^2(c+d x) (a+b \sec (c+d x)) \left (3 \left (2 A b^2+a^2 (3 A+4 C)\right )+3 a b (5 A+8 C) \sec (c+d x)+12 b^2 C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a \left (2 A b^2+a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {A b \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d}-\frac {1}{24} \int \cos (c+d x) \left (-12 b \left (A b^2+a^2 (4 A+6 C)\right )-3 a \left (12 b^2 (A+2 C)+a^2 (3 A+4 C)\right ) \sec (c+d x)-24 b^3 C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a \left (2 A b^2+a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {A b \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d}-\frac {1}{24} \int \cos (c+d x) \left (-12 b \left (A b^2+a^2 (4 A+6 C)\right )-24 b^3 C \sec ^2(c+d x)\right ) \, dx+\frac {1}{8} \left (a \left (12 b^2 (A+2 C)+a^2 (3 A+4 C)\right )\right ) \int 1 \, dx\\ &=\frac {1}{8} a \left (12 b^2 (A+2 C)+a^2 (3 A+4 C)\right ) x+\frac {b \left (A b^2+a^2 (4 A+6 C)\right ) \sin (c+d x)}{2 d}+\frac {a \left (2 A b^2+a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {A b \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d}+\left (b^3 C\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{8} a \left (12 b^2 (A+2 C)+a^2 (3 A+4 C)\right ) x+\frac {b^3 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b \left (A b^2+a^2 (4 A+6 C)\right ) \sin (c+d x)}{2 d}+\frac {a \left (2 A b^2+a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {A b \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.64, size = 177, normalized size = 0.97 \[ \frac {a^3 A \sin (4 (c+d x))+4 a (c+d x) \left (a^2 (3 A+4 C)+12 b^2 (A+2 C)\right )+8 a \left (a^2 (A+C)+3 A b^2\right ) \sin (2 (c+d x))+8 b \left (3 a^2 (3 A+4 C)+4 A b^2\right ) \sin (c+d x)+8 a^2 A b \sin (3 (c+d x))-32 b^3 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+32 b^3 C \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{32 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 146, normalized size = 0.80 \[ \frac {4 \, C b^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 4 \, C b^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left ({\left (3 \, A + 4 \, C\right )} a^{3} + 12 \, {\left (A + 2 \, C\right )} a b^{2}\right )} d x + {\left (2 \, A a^{3} \cos \left (d x + c\right )^{3} + 8 \, A a^{2} b \cos \left (d x + c\right )^{2} + 8 \, {\left (2 \, A + 3 \, C\right )} a^{2} b + 8 \, A b^{3} + {\left ({\left (3 \, A + 4 \, C\right )} a^{3} + 12 \, A a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 503, normalized size = 2.76 \[ \frac {8 \, C b^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 8 \, C b^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + {\left (3 \, A a^{3} + 4 \, C a^{3} + 12 \, A a b^{2} + 24 \, C a b^{2}\right )} {\left (d x + c\right )} - \frac {2 \, {\left (5 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 4 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 8 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 72 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 24 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, A 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 = 1.12, size = 252, normalized size = 1.38 \[ \frac {A \,a^{3} \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4 d}+\frac {3 A \,a^{3} \sin \left (d x +c \right ) \cos \left (d x +c \right )}{8 d}+\frac {3 a^{3} A x}{8}+\frac {3 A \,a^{3} c}{8 d}+\frac {C \,a^{3} \sin \left (d x +c \right ) \cos \left (d x +c \right )}{2 d}+\frac {a^{3} C x}{2}+\frac {C \,a^{3} c}{2 d}+\frac {A \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) a^{2} b}{d}+\frac {2 A \,a^{2} b \sin \left (d x +c \right )}{d}+\frac {3 C \,a^{2} b \sin \left (d x +c \right )}{d}+\frac {3 A a \,b^{2} \sin \left (d x +c \right ) \cos \left (d x +c \right )}{2 d}+\frac {3 A x a \,b^{2}}{2}+\frac {3 A a \,b^{2} c}{2 d}+3 a \,b^{2} C x +\frac {3 C a \,b^{2} c}{d}+\frac {A \,b^{3} \sin \left (d x +c \right )}{d}+\frac {b^{3} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 174, normalized size = 0.96 \[ \frac {{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} + 8 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} b + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{2} + 96 \, {\left (d x + c\right )} C a b^{2} + 16 \, C b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 96 \, C a^{2} b \sin \left (d x + c\right ) + 32 \, A b^{3} \sin \left (d x + c\right )}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.79, size = 2008, normalized size = 11.03 \[ \text {result too large to display} \]
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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