Optimal. Leaf size=130 \[ -\frac {2 a^3 A \cot ^5(c+d x)}{5 d}-\frac {2 a^3 A \cot ^3(c+d x)}{3 d}+\frac {3 a^3 A \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {a^3 A \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {5 a^3 A \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac {3 a^3 A \cot (c+d x) \csc (c+d x)}{16 d} \]
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Rubi [A] time = 0.20, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2966, 3768, 3770, 3767} \[ -\frac {2 a^3 A \cot ^5(c+d x)}{5 d}-\frac {2 a^3 A \cot ^3(c+d x)}{3 d}+\frac {3 a^3 A \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {a^3 A \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {5 a^3 A \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac {3 a^3 A \cot (c+d x) \csc (c+d x)}{16 d} \]
Antiderivative was successfully verified.
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Rule 2966
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \csc ^7(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx &=\int \left (-a^3 A \csc ^3(c+d x)-2 a^3 A \csc ^4(c+d x)+2 a^3 A \csc ^6(c+d x)+a^3 A \csc ^7(c+d x)\right ) \, dx\\ &=-\left (\left (a^3 A\right ) \int \csc ^3(c+d x) \, dx\right )+\left (a^3 A\right ) \int \csc ^7(c+d x) \, dx-\left (2 a^3 A\right ) \int \csc ^4(c+d x) \, dx+\left (2 a^3 A\right ) \int \csc ^6(c+d x) \, dx\\ &=\frac {a^3 A \cot (c+d x) \csc (c+d x)}{2 d}-\frac {a^3 A \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {1}{2} \left (a^3 A\right ) \int \csc (c+d x) \, dx+\frac {1}{6} \left (5 a^3 A\right ) \int \csc ^5(c+d x) \, dx+\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 {\left (2 a^3 A\right ) \operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {a^3 A \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {2 a^3 A \cot ^3(c+d x)}{3 d}-\frac {2 a^3 A \cot ^5(c+d x)}{5 d}+\frac {a^3 A \cot (c+d x) \csc (c+d x)}{2 d}-\frac {5 a^3 A \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^3 A \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {1}{8} \left (5 a^3 A\right ) \int \csc ^3(c+d x) \, dx\\ &=\frac {a^3 A \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {2 a^3 A \cot ^3(c+d x)}{3 d}-\frac {2 a^3 A \cot ^5(c+d x)}{5 d}+\frac {3 a^3 A \cot (c+d x) \csc (c+d x)}{16 d}-\frac {5 a^3 A \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^3 A \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {1}{16} \left (5 a^3 A\right ) \int \csc (c+d x) \, dx\\ &=\frac {3 a^3 A \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {2 a^3 A \cot ^3(c+d x)}{3 d}-\frac {2 a^3 A \cot ^5(c+d x)}{5 d}+\frac {3 a^3 A \cot (c+d x) \csc (c+d x)}{16 d}-\frac {5 a^3 A \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^3 A \cot (c+d x) \csc ^5(c+d x)}{6 d}\\ \end {align*}
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Mathematica [B] time = 0.08, size = 306, normalized size = 2.35 \[ a^3 A \left (-\frac {2 \tan \left (\frac {1}{2} (c+d x)\right )}{15 d}+\frac {2 \cot \left (\frac {1}{2} (c+d x)\right )}{15 d}-\frac {\csc ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}-\frac {\csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {3 \csc ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {\sec ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}+\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {3 \sec ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}+\frac {3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^4\left (\frac {1}{2} (c+d x)\right )}{80 d}+\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{240 d}+\frac {\tan \left (\frac {1}{2} (c+d x)\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right )}{80 d}-\frac {\tan \left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{240 d}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 240, normalized size = 1.85 \[ -\frac {90 \, A a^{3} \cos \left (d x + c\right )^{5} - 80 \, A a^{3} \cos \left (d x + c\right )^{3} - 90 \, A a^{3} \cos \left (d x + c\right ) - 45 \, {\left (A a^{3} \cos \left (d x + c\right )^{6} - 3 \, A a^{3} \cos \left (d x + c\right )^{4} + 3 \, A a^{3} \cos \left (d x + c\right )^{2} - A a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 45 \, {\left (A a^{3} \cos \left (d x + c\right )^{6} - 3 \, A a^{3} \cos \left (d x + c\right )^{4} + 3 \, A a^{3} \cos \left (d x + c\right )^{2} - A a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 64 \, {\left (2 \, A a^{3} \cos \left (d x + c\right )^{5} - 5 \, A a^{3} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 242, normalized size = 1.86 \[ \frac {5 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 24 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 45 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 40 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 360 \, A a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 240 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {882 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 240 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 40 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 45 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, A a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.73, size = 155, normalized size = 1.19 \[ \frac {3 a^{3} A \cot \left (d x +c \right ) \csc \left (d x +c \right )}{16 d}-\frac {3 a^{3} A \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16 d}+\frac {4 a^{3} A \cot \left (d x +c \right )}{15 d}+\frac {2 a^{3} A \cot \left (d x +c \right ) \left (\csc ^{2}\left (d x +c \right )\right )}{15 d}-\frac {2 a^{3} A \cot \left (d x +c \right ) \left (\csc ^{4}\left (d x +c \right )\right )}{5 d}-\frac {a^{3} A \cot \left (d x +c \right ) \left (\csc ^{5}\left (d x +c \right )\right )}{6 d}-\frac {5 a^{3} A \cot \left (d x +c \right ) \left (\csc ^{3}\left (d x +c \right )\right )}{24 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 207, normalized size = 1.59 \[ \frac {5 \, A a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 33 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 120 \, A a^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {320 \, {\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} A a^{3}}{\tan \left (d x + c\right )^{3}} - \frac {64 \, {\left (15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} + 3\right )} A a^{3}}{\tan \left (d x + c\right )^{5}}}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.37, size = 340, normalized size = 2.62 \[ -\frac {A\,a^3\,\left (5\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-24\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-40\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+40\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+360\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\right )}{1920\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6} \]
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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