Optimal. Leaf size=229 \[ 4 a^3 A b x-\frac {a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \tan (c+d x)}{6 d}-\frac {b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \tan (c+d x) \sec (c+d x)}{24 d}+\frac {\left (8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac {b (4 A-C) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}-\frac {a b (12 A-7 C) \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac {A \sin (c+d x) (a+b \sec (c+d x))^4}{d} \]
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Rubi [A] time = 0.49, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4095, 4056, 4048, 3770, 3767, 8} \[ -\frac {a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \tan (c+d x)}{6 d}+\frac {\left (24 a^2 b^2 (2 A+C)+8 a^4 C+b^4 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac {b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \tan (c+d x) \sec (c+d x)}{24 d}+4 a^3 A b x-\frac {b (4 A-C) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}-\frac {a b (12 A-7 C) \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac {A \sin (c+d x) (a+b \sec (c+d x))^4}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 4048
Rule 4056
Rule 4095
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}+\int (a+b \sec (c+d x))^3 \left (4 A b+a C \sec (c+d x)-b (4 A-C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}-\frac {b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{4} \int (a+b \sec (c+d x))^2 \left (16 a A b+\left (4 A b^2+4 a^2 C+3 b^2 C\right ) \sec (c+d x)-a b (12 A-7 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}-\frac {a b (12 A-7 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac {b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{12} \int (a+b \sec (c+d x)) \left (48 a^2 A b+a \left (36 A b^2+12 a^2 C+23 b^2 C\right ) \sec (c+d x)-b \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}-\frac {b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac {a b (12 A-7 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac {b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{24} \int \left (96 a^3 A b+3 \left (8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \sec (c+d x)-4 a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=4 a^3 A b x+\frac {A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}-\frac {b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac {a b (12 A-7 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac {b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac {1}{6} \left (a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right )\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{8} \left (8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \int \sec (c+d x) \, dx\\ &=4 a^3 A b x+\frac {\left (8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}-\frac {b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac {a b (12 A-7 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac {b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {\left (a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right )\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{6 d}\\ &=4 a^3 A b x+\frac {\left (8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}-\frac {a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \tan (c+d x)}{6 d}-\frac {b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac {a b (12 A-7 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac {b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}\\ \end {align*}
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Mathematica [B] time = 6.56, size = 1357, normalized size = 5.93 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 236, normalized size = 1.03 \[ \frac {192 \, A a^{3} b d x \cos \left (d x + c\right )^{4} + 3 \, {\left (8 \, C a^{4} + 24 \, {\left (2 \, A + C\right )} a^{2} b^{2} + {\left (4 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (8 \, C a^{4} + 24 \, {\left (2 \, A + C\right )} a^{2} b^{2} + {\left (4 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (24 \, A a^{4} \cos \left (d x + c\right )^{4} + 32 \, C a b^{3} \cos \left (d x + c\right ) + 6 \, C b^{4} + 32 \, {\left (3 \, C a^{3} b + {\left (3 \, A + 2 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (24 \, C a^{2} b^{2} + {\left (4 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.34, size = 590, normalized size = 2.58 \[ \frac {96 \, {\left (d x + c\right )} A a^{3} b + \frac {48 \, A a^{4} \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 (8 \, C a^{4} + 48 \, A a^{2} b^{2} + 24 \, C a^{2} b^{2} + 4 \, A b^{4} + 3 \, C b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (8 \, C a^{4} + 48 \, A a^{2} b^{2} + 24 \, C a^{2} b^{2} + 4 \, A b^{4} + 3 \, C b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (96 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 72 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 96 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 96 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 12 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 288 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 288 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 160 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 288 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 288 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 160 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 96 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 72 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, C b^{4} \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}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.56, size = 316, normalized size = 1.38 \[ \frac {A \,a^{4} \sin \left (d x +c \right )}{d}+\frac {a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+4 a^{3} A b x +\frac {4 A \,a^{3} b c}{d}+\frac {4 a^{3} b C \tan \left (d x +c \right )}{d}+\frac {6 A \,a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {3 C \,a^{2} b^{2} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{d}+\frac {3 C \,a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {4 a A \,b^{3} \tan \left (d x +c \right )}{d}+\frac {8 C a \,b^{3} \tan \left (d x +c \right )}{3 d}+\frac {4 C a \,b^{3} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {A \,b^{4} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {A \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {C \,b^{4} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{4 d}+\frac {3 C \,b^{4} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {3 C \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 306, normalized size = 1.34 \[ \frac {192 \, {\left (d x + c\right )} A a^{3} b + 64 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a b^{3} - 3 \, C b^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 72 \, C a^{2} b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, A b^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, C a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 144 \, A a^{2} b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, A a^{4} \sin \left (d x + c\right ) + 192 \, C a^{3} b \tan \left (d x + c\right ) + 192 \, A a b^{3} \tan \left (d x + c\right )}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.55, size = 1988, normalized size = 8.68 \[ \text {result too large to display} \]
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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