Optimal. Leaf size=177 \[ \frac {a^4 (10 A+7 C) \sin (c+d x)}{2 d}+\frac {(8 A+7 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{6 d}+\frac {a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {1}{2} a^4 x (12 A+7 C)+\frac {(5 A+7 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{15 d}+\frac {a C \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d} \]
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Rubi [A] time = 0.54, antiderivative size = 177, 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 = {3046, 2976, 2968, 3023, 2735, 3770} \[ \frac {a^4 (10 A+7 C) \sin (c+d x)}{2 d}+\frac {(5 A+7 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{15 d}+\frac {(8 A+7 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{6 d}+\frac {a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {1}{2} a^4 x (12 A+7 C)+\frac {a C \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2968
Rule 2976
Rule 3023
Rule 3046
Rule 3770
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx &=\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {\int (a+a \cos (c+d x))^4 (5 a A+4 a C \cos (c+d x)) \sec (c+d x) \, dx}{5 a}\\ &=\frac {a C (a+a \cos (c+d x))^3 \sin (c+d x)}{5 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {\int (a+a \cos (c+d x))^3 \left (20 a^2 A+4 a^2 (5 A+7 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{20 a}\\ &=\frac {a C (a+a \cos (c+d x))^3 \sin (c+d x)}{5 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(5 A+7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{15 d}+\frac {\int (a+a \cos (c+d x))^2 \left (60 a^3 A+20 a^3 (8 A+7 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{60 a}\\ &=\frac {a C (a+a \cos (c+d x))^3 \sin (c+d x)}{5 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(5 A+7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{15 d}+\frac {(8 A+7 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {\int (a+a \cos (c+d x)) \left (120 a^4 A+60 a^4 (10 A+7 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{120 a}\\ &=\frac {a C (a+a \cos (c+d x))^3 \sin (c+d x)}{5 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(5 A+7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{15 d}+\frac {(8 A+7 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {\int \left (120 a^5 A+\left (120 a^5 A+60 a^5 (10 A+7 C)\right ) \cos (c+d x)+60 a^5 (10 A+7 C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{120 a}\\ &=\frac {a^4 (10 A+7 C) \sin (c+d x)}{2 d}+\frac {a C (a+a \cos (c+d x))^3 \sin (c+d x)}{5 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(5 A+7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{15 d}+\frac {(8 A+7 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {\int \left (120 a^5 A+60 a^5 (12 A+7 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{120 a}\\ &=\frac {1}{2} a^4 (12 A+7 C) x+\frac {a^4 (10 A+7 C) \sin (c+d x)}{2 d}+\frac {a C (a+a \cos (c+d x))^3 \sin (c+d x)}{5 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(5 A+7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{15 d}+\frac {(8 A+7 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\left (a^4 A\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{2} a^4 (12 A+7 C) x+\frac {a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^4 (10 A+7 C) \sin (c+d x)}{2 d}+\frac {a C (a+a \cos (c+d x))^3 \sin (c+d x)}{5 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(5 A+7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{15 d}+\frac {(8 A+7 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}\\ \end {align*}
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Mathematica [A] time = 0.49, size = 147, normalized size = 0.83 \[ \frac {a^4 \left (30 (54 A+49 C) \sin (c+d x)+240 (A+2 C) \sin (2 (c+d x))+20 A \sin (3 (c+d x))-240 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+240 A \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+1440 A d x+145 C \sin (3 (c+d x))+30 C \sin (4 (c+d x))+3 C \sin (5 (c+d x))+840 C d x\right )}{240 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 138, normalized size = 0.78 \[ \frac {15 \, {\left (12 \, A + 7 \, C\right )} a^{4} d x + 15 \, A a^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, A a^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (6 \, C a^{4} \cos \left (d x + c\right )^{4} + 30 \, C a^{4} \cos \left (d x + c\right )^{3} + 2 \, {\left (5 \, A + 34 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 15 \, {\left (4 \, A + 7 \, C\right )} a^{4} \cos \left (d x + c\right ) + 2 \, {\left (100 \, A + 83 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.87, size = 248, normalized size = 1.40 \[ \frac {30 \, A a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 30 \, A a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 15 \, {\left (12 \, A a^{4} + 7 \, C a^{4}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (150 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 105 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 680 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 490 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1180 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 896 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 920 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 790 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 270 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 375 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 221, normalized size = 1.25 \[ \frac {A \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) a^{4}}{3 d}+\frac {20 A \,a^{4} \sin \left (d x +c \right )}{3 d}+\frac {83 a^{4} C \sin \left (d x +c \right )}{15 d}+\frac {a^{4} C \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{5 d}+\frac {34 a^{4} C \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{15 d}+\frac {2 A \,a^{4} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{d}+6 A \,a^{4} x +\frac {6 A \,a^{4} c}{d}+\frac {a^{4} C \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{d}+\frac {7 a^{4} C \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {7 a^{4} C x}{2}+\frac {7 a^{4} C c}{2 d}+\frac {A \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 222, normalized size = 1.25 \[ -\frac {40 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 480 \, {\left (d x + c\right )} A a^{4} - 8 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{4} + 240 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} - 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 120 \, A a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) - 720 \, A a^{4} \sin \left (d x + c\right ) - 120 \, C a^{4} \sin \left (d x + c\right )}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.52, size = 202, normalized size = 1.14 \[ \frac {12\,A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+2\,A\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+7\,C\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+A\,a^4\,\sin \left (2\,c+2\,d\,x\right )+\frac {A\,a^4\,\sin \left (3\,c+3\,d\,x\right )}{12}+2\,C\,a^4\,\sin \left (2\,c+2\,d\,x\right )+\frac {29\,C\,a^4\,\sin \left (3\,c+3\,d\,x\right )}{48}+\frac {C\,a^4\,\sin \left (4\,c+4\,d\,x\right )}{8}+\frac {C\,a^4\,\sin \left (5\,c+5\,d\,x\right )}{80}+\frac {27\,A\,a^4\,\sin \left (c+d\,x\right )}{4}+\frac {49\,C\,a^4\,\sin \left (c+d\,x\right )}{8}}{d} \]
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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