Optimal. Leaf size=103 \[ \frac {A \cos (c+d x)}{a^3 d}+\frac {104 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)}-\frac {31 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)^2}+\frac {2 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^3}+\frac {4 A x}{a^3} \]
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Rubi [A] time = 0.19, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2966, 2638, 2650, 2648} \[ \frac {A \cos (c+d x)}{a^3 d}+\frac {104 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)}-\frac {31 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)^2}+\frac {2 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^3}+\frac {4 A x}{a^3} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 2648
Rule 2650
Rule 2966
Rubi steps
\begin {align*} \int \frac {\sin ^3(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx &=\int \left (\frac {4 A}{a^3}-\frac {A \sin (c+d x)}{a^3}-\frac {2 A}{a^3 (1+\sin (c+d x))^3}+\frac {7 A}{a^3 (1+\sin (c+d x))^2}-\frac {9 A}{a^3 (1+\sin (c+d x))}\right ) \, dx\\ &=\frac {4 A x}{a^3}-\frac {A \int \sin (c+d x) \, dx}{a^3}-\frac {(2 A) \int \frac {1}{(1+\sin (c+d x))^3} \, dx}{a^3}+\frac {(7 A) \int \frac {1}{(1+\sin (c+d x))^2} \, dx}{a^3}-\frac {(9 A) \int \frac {1}{1+\sin (c+d x)} \, dx}{a^3}\\ &=\frac {4 A x}{a^3}+\frac {A \cos (c+d x)}{a^3 d}+\frac {2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}-\frac {7 A \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))^2}+\frac {9 A \cos (c+d x)}{a^3 d (1+\sin (c+d x))}-\frac {(4 A) \int \frac {1}{(1+\sin (c+d x))^2} \, dx}{5 a^3}+\frac {(7 A) \int \frac {1}{1+\sin (c+d x)} \, dx}{3 a^3}\\ &=\frac {4 A x}{a^3}+\frac {A \cos (c+d x)}{a^3 d}+\frac {2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}-\frac {31 A \cos (c+d x)}{15 a^3 d (1+\sin (c+d x))^2}+\frac {20 A \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))}-\frac {(4 A) \int \frac {1}{1+\sin (c+d x)} \, dx}{15 a^3}\\ &=\frac {4 A x}{a^3}+\frac {A \cos (c+d x)}{a^3 d}+\frac {2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}-\frac {31 A \cos (c+d x)}{15 a^3 d (1+\sin (c+d x))^2}+\frac {104 A \cos (c+d x)}{15 a^3 d (1+\sin (c+d x))}\\ \end {align*}
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Mathematica [B] time = 0.80, size = 228, normalized size = 2.21 \[ -\frac {A \left (-1200 d x \sin \left (c+\frac {d x}{2}\right )-600 d x \sin \left (c+\frac {3 d x}{2}\right )+405 \sin \left (2 c+\frac {3 d x}{2}\right )-491 \sin \left (2 c+\frac {5 d x}{2}\right )+120 d x \sin \left (3 c+\frac {5 d x}{2}\right )+15 \sin \left (4 c+\frac {7 d x}{2}\right )+1665 \cos \left (c+\frac {d x}{2}\right )-1675 \cos \left (c+\frac {3 d x}{2}\right )+600 d x \cos \left (2 c+\frac {3 d x}{2}\right )+120 d x \cos \left (2 c+\frac {5 d x}{2}\right )+75 \cos \left (3 c+\frac {5 d x}{2}\right )+15 \cos \left (3 c+\frac {7 d x}{2}\right )+2495 \sin \left (\frac {d x}{2}\right )-1200 d x \cos \left (\frac {d x}{2}\right )\right )}{120 a^3 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 225, normalized size = 2.18 \[ \frac {15 \, A \cos \left (d x + c\right )^{4} + {\left (60 \, A d x + 149 \, A\right )} \cos \left (d x + c\right )^{3} - 240 \, A d x + {\left (180 \, A d x - 103 \, A\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (40 \, A d x + 81 \, A\right )} \cos \left (d x + c\right ) + {\left (15 \, A \cos \left (d x + c\right )^{3} - 240 \, A d x + 2 \, {\left (30 \, A d x - 67 \, A\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (40 \, A d x + 79 \, A\right )} \cos \left (d x + c\right ) + 6 \, A\right )} \sin \left (d x + c\right ) - 6 \, A}{15 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d + {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 113, normalized size = 1.10 \[ \frac {2 \, {\left (\frac {30 \, {\left (d x + c\right )} A}{a^{3}} + \frac {15 \, A}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{3}} + \frac {60 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 285 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 505 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 335 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 79 \, A}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}\right )}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.43, size = 155, normalized size = 1.50 \[ \frac {2 A}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {8 A \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}+\frac {16 A}{5 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {8 A}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {4 A}{3 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {6 A}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {8 A}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.56, size = 543, normalized size = 5.27 \[ \frac {2 \, {\left (3 \, A {\left (\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {189 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {200 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {160 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {75 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 24}{a^{3} + \frac {5 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {11 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {11 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}} + \frac {15 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} + A {\left (\frac {\frac {95 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {145 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {75 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 22}{a^{3} + \frac {5 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}} + \frac {15 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}\right )}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 16.93, size = 261, normalized size = 2.53 \[ \frac {4\,A\,x}{a^3}-\frac {\left (20\,A\,\left (c+d\,x\right )-\frac {4\,A\,\left (75\,c+75\,d\,x+30\right )}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (44\,A\,\left (c+d\,x\right )-\frac {4\,A\,\left (165\,c+165\,d\,x+150\right )}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (60\,A\,\left (c+d\,x\right )-\frac {4\,A\,\left (225\,c+225\,d\,x+320\right )}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (60\,A\,\left (c+d\,x\right )-\frac {4\,A\,\left (225\,c+225\,d\,x+385\right )}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (44\,A\,\left (c+d\,x\right )-\frac {4\,A\,\left (165\,c+165\,d\,x+367\right )}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\left (20\,A\,\left (c+d\,x\right )-\frac {4\,A\,\left (75\,c+75\,d\,x+205\right )}{15}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+4\,A\,\left (c+d\,x\right )-\frac {4\,A\,\left (15\,c+15\,d\,x+47\right )}{15}}{a^3\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^5\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 45.41, size = 2290, normalized size = 22.23 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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