Optimal. Leaf size=163 \[ \frac {a \left (a^2 (2 A+3 C)+3 A b^2\right ) \tan (c+d x)}{3 d}+\frac {b \left (3 a^2 (A+2 C)+2 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}+\frac {A b \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+3 a b^2 C x-\frac {b^3 (5 A-6 C) \sin (c+d x)}{6 d} \]
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Rubi [A] time = 0.54, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3048, 3047, 3031, 3023, 2735, 3770} \[ \frac {a \left (a^2 (2 A+3 C)+3 A b^2\right ) \tan (c+d x)}{3 d}+\frac {b \left (3 a^2 (A+2 C)+2 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}+\frac {A b \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+3 a b^2 C x-\frac {b^3 (5 A-6 C) \sin (c+d x)}{6 d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 3023
Rule 3031
Rule 3047
Rule 3048
Rule 3770
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx &=\frac {A (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {1}{3} \int (a+b \cos (c+d x))^2 \left (3 A b+a (2 A+3 C) \cos (c+d x)-b (A-3 C) \cos ^2(c+d x)\right ) \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 {A (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {1}{6} \int (a+b \cos (c+d x)) \left (2 \left (3 A b^2+\frac {1}{2} a^2 (4 A+6 C)\right )+a b (5 A+12 C) \cos (c+d x)-b^2 (5 A-6 C) \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac {a \left (3 A b^2+a^2 (2 A+3 C)\right ) \tan (c+d x)}{3 d}+\frac {A b (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {A (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac {1}{6} \int \left (-3 b \left (2 A b^2+3 a^2 (A+2 C)\right )-18 a b^2 C \cos (c+d x)+b^3 (5 A-6 C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac {b^3 (5 A-6 C) \sin (c+d x)}{6 d}+\frac {a \left (3 A b^2+a^2 (2 A+3 C)\right ) \tan (c+d x)}{3 d}+\frac {A b (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {A (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac {1}{6} \int \left (-3 b \left (2 A b^2+3 a^2 (A+2 C)\right )-18 a b^2 C \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=3 a b^2 C x-\frac {b^3 (5 A-6 C) \sin (c+d x)}{6 d}+\frac {a \left (3 A b^2+a^2 (2 A+3 C)\right ) \tan (c+d x)}{3 d}+\frac {A b (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {A (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {1}{2} \left (b \left (2 A b^2+3 a^2 (A+2 C)\right )\right ) \int \sec (c+d x) \, dx\\ &=3 a b^2 C x+\frac {b \left (2 A b^2+3 a^2 (A+2 C)\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {b^3 (5 A-6 C) \sin (c+d x)}{6 d}+\frac {a \left (3 A b^2+a^2 (2 A+3 C)\right ) \tan (c+d x)}{3 d}+\frac {A b (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {A (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}\\ \end {align*}
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Mathematica [B] time = 4.22, size = 377, normalized size = 2.31 \[ \frac {\frac {2 a^3 A \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {2 a^3 A \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {4 a \left (a^2 (2 A+3 C)+9 A b^2\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {4 a \left (a^2 (2 A+3 C)+9 A b^2\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )}-6 b \left (3 a^2 (A+2 C)+2 A b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+6 b \left (3 a^2 (A+2 C)+2 A b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+\frac {a^2 A (a+9 b)}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {a^2 A (a+9 b)}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}+36 a b^2 C (c+d x)+12 b^3 C \sin (c+d x)}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.16, size = 178, normalized size = 1.09 \[ \frac {36 \, C a b^{2} d x \cos \left (d x + c\right )^{3} + 3 \, {\left (3 \, {\left (A + 2 \, C\right )} a^{2} b + 2 \, A b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (3 \, {\left (A + 2 \, C\right )} a^{2} b + 2 \, A b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (6 \, C b^{3} \cos \left (d x + c\right )^{3} + 9 \, A a^{2} b \cos \left (d x + c\right ) + 2 \, A a^{3} + 2 \, {\left ({\left (2 \, A + 3 \, C\right )} a^{3} + 9 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.38, size = 322, normalized size = 1.98 \[ \frac {18 \, {\left (d x + c\right )} C a b^{2} + \frac {12 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + 3 \, {\left (3 \, A a^{2} b + 6 \, C a^{2} b + 2 \, A b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (3 \, A a^{2} b + 6 \, C a^{2} b + 2 \, A b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 195, normalized size = 1.20 \[ \frac {2 A \,a^{3} \tan \left (d x +c \right )}{3 d}+\frac {A \,a^{3} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {C \,a^{3} \tan \left (d x +c \right )}{d}+\frac {3 A \,a^{2} b \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {3 A \,a^{2} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {3 C \,a^{2} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {3 A a \,b^{2} \tan \left (d x +c \right )}{d}+3 a \,b^{2} C x +\frac {3 C a \,b^{2} c}{d}+\frac {A \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {b^{3} C \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 181, normalized size = 1.11 \[ \frac {4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} + 36 \, {\left (d x + c\right )} C a b^{2} - 9 \, A a^{2} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 18 \, C a^{2} b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, A b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, C b^{3} \sin \left (d x + c\right ) + 12 \, C a^{3} \tan \left (d x + c\right ) + 36 \, A a b^{2} \tan \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.80, size = 464, normalized size = 2.85 \[ \frac {\frac {A\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{6}+\frac {C\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{4}+\frac {C\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {C\,b^3\,\sin \left (4\,c+4\,d\,x\right )}{8}+\frac {A\,a^3\,\sin \left (c+d\,x\right )}{2}+\frac {C\,a^3\,\sin \left (c+d\,x\right )}{4}+\frac {3\,A\,a\,b^2\,\sin \left (c+d\,x\right )}{4}-\frac {A\,b^3\,\cos \left (c+d\,x\right )\,\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}}{2}+\frac {3\,A\,a^2\,b\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {3\,A\,a\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{4}-\frac {A\,b^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 )\,\cos \left (3\,c+3\,d\,x\right )\,1{}\mathrm {i}}{2}-\frac {A\,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 )\,\cos \left (3\,c+3\,d\,x\right )\,3{}\mathrm {i}}{4}+\frac {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 )\,\cos \left (3\,c+3\,d\,x\right )}{2}-\frac {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 )\,\cos \left (3\,c+3\,d\,x\right )\,3{}\mathrm {i}}{2}-\frac {A\,a^2\,b\,\cos \left (c+d\,x\right )\,\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 )\,9{}\mathrm {i}}{4}+\frac {9\,C\,a\,b^2\,\cos \left (c+d\,x\right )\,\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 )}{2}-\frac {C\,a^2\,b\,\cos \left (c+d\,x\right )\,\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 )\,9{}\mathrm {i}}{2}}{d\,\left (\frac {3\,\cos \left (c+d\,x\right )}{4}+\frac {\cos \left (3\,c+3\,d\,x\right )}{4}\right )} \]
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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