Optimal. Leaf size=207 \[ \frac {a^4 (7 A+10 C) \sin (c+d x)}{2 d}+\frac {(7 A+8 C) \sin (c+d x) \cos (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{6 d}+\frac {1}{2} a^4 x (7 A+12 C)+\frac {a^4 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {(7 A+5 C) \sin (c+d x) \cos ^2(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{15 d}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^4}{5 d}+\frac {a A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{5 d} \]
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Rubi [A] time = 0.55, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {4087, 4017, 3996, 3770} \[ \frac {a^4 (7 A+10 C) \sin (c+d x)}{2 d}+\frac {(7 A+5 C) \sin (c+d x) \cos ^2(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{15 d}+\frac {(7 A+8 C) \sin (c+d x) \cos (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{6 d}+\frac {1}{2} a^4 x (7 A+12 C)+\frac {a^4 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{5 d}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^4}{5 d} \]
Antiderivative was successfully verified.
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Rule 3770
Rule 3996
Rule 4017
Rule 4087
Rubi steps
\begin {align*} \int \cos ^5(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {\int \cos ^4(c+d x) (a+a \sec (c+d x))^4 (4 a A+5 a C \sec (c+d x)) \, dx}{5 a}\\ &=\frac {a A \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{5 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {\int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (4 a^2 (7 A+5 C)+20 a^2 C \sec (c+d x)\right ) \, dx}{20 a}\\ &=\frac {a A \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{5 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(7 A+5 C) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{15 d}+\frac {\int \cos ^2(c+d x) (a+a \sec (c+d x))^2 \left (20 a^3 (7 A+8 C)+60 a^3 C \sec (c+d x)\right ) \, dx}{60 a}\\ &=\frac {a A \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{5 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(7 A+5 C) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{15 d}+\frac {(7 A+8 C) \cos (c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {\int \cos (c+d x) (a+a \sec (c+d x)) \left (60 a^4 (7 A+10 C)+120 a^4 C \sec (c+d x)\right ) \, dx}{120 a}\\ &=\frac {a^4 (7 A+10 C) \sin (c+d x)}{2 d}+\frac {a A \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{5 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(7 A+5 C) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{15 d}+\frac {(7 A+8 C) \cos (c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{6 d}-\frac {\int \left (-60 a^5 (7 A+12 C)-120 a^5 C \sec (c+d x)\right ) \, dx}{120 a}\\ &=\frac {1}{2} a^4 (7 A+12 C) x+\frac {a^4 (7 A+10 C) \sin (c+d x)}{2 d}+\frac {a A \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{5 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(7 A+5 C) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{15 d}+\frac {(7 A+8 C) \cos (c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{6 d}+\left (a^4 C\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{2} a^4 (7 A+12 C) x+\frac {a^4 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^4 (7 A+10 C) \sin (c+d x)}{2 d}+\frac {a A \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{5 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(7 A+5 C) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{15 d}+\frac {(7 A+8 C) \cos (c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{6 d}\\ \end {align*}
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Mathematica [A] time = 0.56, size = 147, normalized size = 0.71 \[ \frac {a^4 \left (30 (49 A+54 C) \sin (c+d x)+240 (2 A+C) \sin (2 (c+d x))+145 A \sin (3 (c+d x))+30 A \sin (4 (c+d x))+3 A \sin (5 (c+d x))+840 A d x+20 C \sin (3 (c+d x))-240 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+240 C \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+1440 C d x\right )}{240 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 138, normalized size = 0.67 \[ \frac {15 \, {\left (7 \, A + 12 \, C\right )} a^{4} d x + 15 \, C a^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, C a^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (6 \, A a^{4} \cos \left (d x + c\right )^{4} + 30 \, A a^{4} \cos \left (d x + c\right )^{3} + 2 \, {\left (34 \, A + 5 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 15 \, {\left (7 \, A + 4 \, C\right )} a^{4} \cos \left (d x + c\right ) + 2 \, {\left (83 \, A + 100 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 248, normalized size = 1.20 \[ \frac {30 \, C a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 30 \, C a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 15 \, {\left (7 \, A a^{4} + 12 \, C a^{4}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (105 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 150 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 490 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 680 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 896 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1180 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 790 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 920 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 375 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 270 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.77, size = 221, normalized size = 1.07 \[ \frac {83 A \,a^{4} \sin \left (d x +c \right )}{15 d}+\frac {A \,a^{4} \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{5 d}+\frac {34 A \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) a^{4}}{15 d}+\frac {C \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) a^{4}}{3 d}+\frac {20 a^{4} C \sin \left (d x +c \right )}{3 d}+\frac {A \,a^{4} \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{d}+\frac {7 A \,a^{4} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {7 A \,a^{4} x}{2}+\frac {7 A \,a^{4} c}{2 d}+\frac {2 a^{4} C \sin \left (d x +c \right ) \cos \left (d x +c \right )}{d}+6 a^{4} C x +\frac {6 C \,a^{4} c}{d}+\frac {a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 229, normalized size = 1.11 \[ \frac {8 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{4} - 240 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 40 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 480 \, {\left (d x + c\right )} C a^{4} + 60 \, C a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 120 \, A a^{4} \sin \left (d x + c\right ) + 720 \, C a^{4} \sin \left (d x + c\right )}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.31, size = 202, normalized size = 0.98 \[ \frac {7\,A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+12\,C\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+2\,C\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+2\,A\,a^4\,\sin \left (2\,c+2\,d\,x\right )+\frac {29\,A\,a^4\,\sin \left (3\,c+3\,d\,x\right )}{48}+\frac {A\,a^4\,\sin \left (4\,c+4\,d\,x\right )}{8}+\frac {A\,a^4\,\sin \left (5\,c+5\,d\,x\right )}{80}+C\,a^4\,\sin \left (2\,c+2\,d\,x\right )+\frac {C\,a^4\,\sin \left (3\,c+3\,d\,x\right )}{12}+\frac {49\,A\,a^4\,\sin \left (c+d\,x\right )}{8}+\frac {27\,C\,a^4\,\sin \left (c+d\,x\right )}{4}}{d} \]
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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