Optimal. Leaf size=86 \[ -\frac {2 a^3 A \cot ^3(c+d x)}{3 d}+\frac {5 a^3 A \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {a^3 A \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3 a^3 A \cot (c+d x) \csc (c+d x)}{8 d} \]
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Rubi [A] time = 0.15, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2966, 3770, 3767, 8, 3768} \[ -\frac {2 a^3 A \cot ^3(c+d x)}{3 d}+\frac {5 a^3 A \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {a^3 A \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3 a^3 A \cot (c+d x) \csc (c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2966
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \csc ^5(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx &=\int \left (-a^3 A \csc (c+d x)-2 a^3 A \csc ^2(c+d x)+2 a^3 A \csc ^4(c+d x)+a^3 A \csc ^5(c+d x)\right ) \, dx\\ &=-\left (\left (a^3 A\right ) \int \csc (c+d x) \, dx\right )+\left (a^3 A\right ) \int \csc ^5(c+d x) \, dx-\left (2 a^3 A\right ) \int \csc ^2(c+d x) \, dx+\left (2 a^3 A\right ) \int \csc ^4(c+d x) \, dx\\ &=\frac {a^3 A \tanh ^{-1}(\cos (c+d x))}{d}-\frac {a^3 A \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {1}{4} \left (3 a^3 A\right ) \int \csc ^3(c+d x) \, dx+\frac {\left (2 a^3 A\right ) \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}-\frac {\left (2 a^3 A\right ) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {a^3 A \tanh ^{-1}(\cos (c+d x))}{d}-\frac {2 a^3 A \cot ^3(c+d x)}{3 d}-\frac {3 a^3 A \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^3 A \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {1}{8} \left (3 a^3 A\right ) \int \csc (c+d x) \, dx\\ &=\frac {5 a^3 A \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {2 a^3 A \cot ^3(c+d x)}{3 d}-\frac {3 a^3 A \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^3 A \cot (c+d x) \csc ^3(c+d x)}{4 d}\\ \end {align*}
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Mathematica [B] time = 0.07, size = 210, normalized size = 2.44 \[ a^3 A \left (-\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{3 d}+\frac {\cot \left (\frac {1}{2} (c+d x)\right )}{3 d}-\frac {\csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {3 \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {3 \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {5 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {5 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{12 d}+\frac {\tan \left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{12 d}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 166, normalized size = 1.93 \[ -\frac {32 \, A a^{3} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 18 \, A a^{3} \cos \left (d x + c\right )^{3} + 30 \, A a^{3} \cos \left (d x + c\right ) - 15 \, {\left (A a^{3} \cos \left (d x + c\right )^{4} - 2 \, A a^{3} \cos \left (d x + c\right )^{2} + A a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 15 \, {\left (A a^{3} \cos \left (d x + c\right )^{4} - 2 \, A a^{3} \cos \left (d x + c\right )^{2} + A a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{48 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 174, normalized size = 2.02 \[ \frac {3 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 16 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 24 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 120 \, A a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 48 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {250 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 48 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 16 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, A a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.62, size = 109, normalized size = 1.27 \[ -\frac {5 a^{3} A \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8 d}+\frac {2 a^{3} A \cot \left (d x +c \right )}{3 d}-\frac {2 a^{3} A \cot \left (d x +c \right ) \left (\csc ^{2}\left (d x +c \right )\right )}{3 d}-\frac {a^{3} A \cot \left (d x +c \right ) \left (\csc ^{3}\left (d x +c \right )\right )}{4 d}-\frac {3 a^{3} A \cot \left (d x +c \right ) \csc \left (d x +c \right )}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 145, normalized size = 1.69 \[ \frac {3 \, A a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 24 \, A a^{3} {\left (\log \left (\cos \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {96 \, A a^{3}}{\tan \left (d x + c\right )} - \frac {32 \, {\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} A a^{3}}{\tan \left (d x + c\right )^{3}}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.12, size = 244, normalized size = 2.84 \[ -\frac {A\,a^3\,\left (3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-16\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+16\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+48\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-48\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+120\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}{192\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4} \]
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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