Optimal. Leaf size=78 \[ \frac {a^3 A \cos (c+d x)}{d}-\frac {2 a^3 A \cot (c+d x)}{d}-\frac {a^3 A \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a^3 A \cot (c+d x) \csc (c+d x)}{2 d}-2 a^3 A x \]
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Rubi [A] time = 0.12, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2966, 3767, 8, 3768, 3770, 2638} \[ \frac {a^3 A \cos (c+d x)}{d}-\frac {2 a^3 A \cot (c+d x)}{d}-\frac {a^3 A \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a^3 A \cot (c+d x) \csc (c+d x)}{2 d}-2 a^3 A x \]
Antiderivative was successfully verified.
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Rule 8
Rule 2638
Rule 2966
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \csc ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx &=\int \left (-2 a^3 A+2 a^3 A \csc ^2(c+d x)+a^3 A \csc ^3(c+d x)-a^3 A \sin (c+d x)\right ) \, dx\\ &=-2 a^3 A x+\left (a^3 A\right ) \int \csc ^3(c+d x) \, dx-\left (a^3 A\right ) \int \sin (c+d x) \, dx+\left (2 a^3 A\right ) \int \csc ^2(c+d x) \, dx\\ &=-2 a^3 A x+\frac {a^3 A \cos (c+d x)}{d}-\frac {a^3 A \cot (c+d x) \csc (c+d x)}{2 d}+\frac {1}{2} \left (a^3 A\right ) \int \csc (c+d x) \, dx-\frac {\left (2 a^3 A\right ) \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}\\ &=-2 a^3 A x-\frac {a^3 A \tanh ^{-1}(\cos (c+d x))}{2 d}+\frac {a^3 A \cos (c+d x)}{d}-\frac {2 a^3 A \cot (c+d x)}{d}-\frac {a^3 A \cot (c+d x) \csc (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 142, normalized size = 1.82 \[ -\frac {a^3 A \sin (c) \sin (d x)}{d}+\frac {a^3 A \cos (c) \cos (d x)}{d}-\frac {2 a^3 A \cot (c+d x)}{d}-\frac {a^3 A \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {a^3 A \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {a^3 A \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {a^3 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-2 a^3 A x \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 152, normalized size = 1.95 \[ -\frac {8 \, A a^{3} d x \cos \left (d x + c\right )^{2} - 4 \, A a^{3} \cos \left (d x + c\right )^{3} - 8 \, A a^{3} d x - 8 \, A a^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 2 \, A a^{3} \cos \left (d x + c\right ) + {\left (A a^{3} \cos \left (d x + c\right )^{2} - A a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (A a^{3} \cos \left (d x + c\right )^{2} - A a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{4 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 137, normalized size = 1.76 \[ \frac {A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 16 \, {\left (d x + c\right )} A a^{3} + 4 \, A a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 8 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {16 \, A a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} - \frac {6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.56, size = 94, normalized size = 1.21 \[ \frac {a^{3} A \cos \left (d x +c \right )}{d}-2 a^{3} A x -\frac {2 a^{3} A c}{d}-\frac {2 a^{3} A \cot \left (d x +c \right )}{d}-\frac {a^{3} A \cot \left (d x +c \right ) \csc \left (d x +c \right )}{2 d}+\frac {a^{3} A \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 90, normalized size = 1.15 \[ -\frac {8 \, {\left (d x + c\right )} A a^{3} - A a^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 4 \, A a^{3} \cos \left (d x + c\right ) + \frac {8 \, A a^{3}}{\tan \left (d x + c\right )}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.50, size = 220, normalized size = 2.82 \[ \frac {A\,a^3\,\left (\frac {\cos \left (c+d\,x\right )}{2}-4\,\mathrm {atan}\left (\frac {\sqrt {17}\,\left (4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{17\,\cos \left (\frac {c}{2}-\mathrm {atan}\relax (4)+\frac {d\,x}{2}\right )}\right )-\frac {\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2}+\cos \left (2\,c+2\,d\,x\right )+\frac {\cos \left (3\,c+3\,d\,x\right )}{2}+2\,\sin \left (2\,c+2\,d\,x\right )+4\,\mathrm {atan}\left (\frac {\sqrt {17}\,\left (4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{17\,\cos \left (\frac {c}{2}-\mathrm {atan}\relax (4)+\frac {d\,x}{2}\right )}\right )\,\cos \left (2\,c+2\,d\,x\right )+\frac {\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (2\,c+2\,d\,x\right )}{2}-1\right )}{2\,d\,\left ({\cos \left (c+d\,x\right )}^2-1\right )} \]
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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