Optimal. Leaf size=130 \[ -\frac {a^2 c \cot ^5(e+f x)}{5 f}-\frac {a^2 c \cot ^3(e+f x)}{3 f}+\frac {a^2 c \tanh ^{-1}(\cos (e+f x))}{16 f}-\frac {a^2 c \cot (e+f x) \csc ^5(e+f x)}{6 f}+\frac {a^2 c \cot (e+f x) \csc ^3(e+f x)}{24 f}+\frac {a^2 c \cot (e+f x) \csc (e+f x)}{16 f} \]
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Rubi [A] time = 0.19, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2966, 3767, 3768, 3770} \[ -\frac {a^2 c \cot ^5(e+f x)}{5 f}-\frac {a^2 c \cot ^3(e+f x)}{3 f}+\frac {a^2 c \tanh ^{-1}(\cos (e+f x))}{16 f}-\frac {a^2 c \cot (e+f x) \csc ^5(e+f x)}{6 f}+\frac {a^2 c \cot (e+f x) \csc ^3(e+f x)}{24 f}+\frac {a^2 c \cot (e+f x) \csc (e+f x)}{16 f} \]
Antiderivative was successfully verified.
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Rule 2966
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \csc ^7(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx &=\int \left (-a^2 c \csc ^4(e+f x)-a^2 c \csc ^5(e+f x)+a^2 c \csc ^6(e+f x)+a^2 c \csc ^7(e+f x)\right ) \, dx\\ &=-\left (\left (a^2 c\right ) \int \csc ^4(e+f x) \, dx\right )-\left (a^2 c\right ) \int \csc ^5(e+f x) \, dx+\left (a^2 c\right ) \int \csc ^6(e+f x) \, dx+\left (a^2 c\right ) \int \csc ^7(e+f x) \, dx\\ &=\frac {a^2 c \cot (e+f x) \csc ^3(e+f x)}{4 f}-\frac {a^2 c \cot (e+f x) \csc ^5(e+f x)}{6 f}-\frac {1}{4} \left (3 a^2 c\right ) \int \csc ^3(e+f x) \, dx+\frac {1}{6} \left (5 a^2 c\right ) \int \csc ^5(e+f x) \, dx+\frac {\left (a^2 c\right ) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (e+f x)\right )}{f}-\frac {\left (a^2 c\right ) \operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (e+f x)\right )}{f}\\ &=-\frac {a^2 c \cot ^3(e+f x)}{3 f}-\frac {a^2 c \cot ^5(e+f x)}{5 f}+\frac {3 a^2 c \cot (e+f x) \csc (e+f x)}{8 f}+\frac {a^2 c \cot (e+f x) \csc ^3(e+f x)}{24 f}-\frac {a^2 c \cot (e+f x) \csc ^5(e+f x)}{6 f}-\frac {1}{8} \left (3 a^2 c\right ) \int \csc (e+f x) \, dx+\frac {1}{8} \left (5 a^2 c\right ) \int \csc ^3(e+f x) \, dx\\ &=\frac {3 a^2 c \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac {a^2 c \cot ^3(e+f x)}{3 f}-\frac {a^2 c \cot ^5(e+f x)}{5 f}+\frac {a^2 c \cot (e+f x) \csc (e+f x)}{16 f}+\frac {a^2 c \cot (e+f x) \csc ^3(e+f x)}{24 f}-\frac {a^2 c \cot (e+f x) \csc ^5(e+f x)}{6 f}+\frac {1}{16} \left (5 a^2 c\right ) \int \csc (e+f x) \, dx\\ &=\frac {a^2 c \tanh ^{-1}(\cos (e+f x))}{16 f}-\frac {a^2 c \cot ^3(e+f x)}{3 f}-\frac {a^2 c \cot ^5(e+f x)}{5 f}+\frac {a^2 c \cot (e+f x) \csc (e+f x)}{16 f}+\frac {a^2 c \cot (e+f x) \csc ^3(e+f x)}{24 f}-\frac {a^2 c \cot (e+f x) \csc ^5(e+f x)}{6 f}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 204, normalized size = 1.57 \[ \frac {2 a^2 c \cot (e+f x)}{15 f}-\frac {a^2 c \csc ^6\left (\frac {1}{2} (e+f x)\right )}{384 f}+\frac {a^2 c \csc ^2\left (\frac {1}{2} (e+f x)\right )}{64 f}+\frac {a^2 c \sec ^6\left (\frac {1}{2} (e+f x)\right )}{384 f}-\frac {a^2 c \sec ^2\left (\frac {1}{2} (e+f x)\right )}{64 f}-\frac {a^2 c \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{16 f}+\frac {a^2 c \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{16 f}-\frac {a^2 c \cot (e+f x) \csc ^4(e+f x)}{5 f}+\frac {a^2 c \cot (e+f x) \csc ^2(e+f x)}{15 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 240, normalized size = 1.85 \[ -\frac {30 \, a^{2} c \cos \left (f x + e\right )^{5} - 80 \, a^{2} c \cos \left (f x + e\right )^{3} - 30 \, a^{2} c \cos \left (f x + e\right ) - 15 \, {\left (a^{2} c \cos \left (f x + e\right )^{6} - 3 \, a^{2} c \cos \left (f x + e\right )^{4} + 3 \, a^{2} c \cos \left (f x + e\right )^{2} - a^{2} c\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) + 15 \, {\left (a^{2} c \cos \left (f x + e\right )^{6} - 3 \, a^{2} c \cos \left (f x + e\right )^{4} + 3 \, a^{2} c \cos \left (f x + e\right )^{2} - a^{2} c\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) + 32 \, {\left (2 \, a^{2} c \cos \left (f x + e\right )^{5} - 5 \, a^{2} c \cos \left (f x + e\right )^{3}\right )} \sin \left (f x + e\right )}{480 \, {\left (f \cos \left (f x + e\right )^{6} - 3 \, f \cos \left (f x + e\right )^{4} + 3 \, f \cos \left (f x + e\right )^{2} - f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.63, size = 155, normalized size = 1.19 \[ \frac {2 a^{2} c \cot \left (f x +e \right )}{15 f}+\frac {a^{2} c \cot \left (f x +e \right ) \left (\csc ^{2}\left (f x +e \right )\right )}{15 f}+\frac {a^{2} c \cot \left (f x +e \right ) \left (\csc ^{3}\left (f x +e \right )\right )}{24 f}+\frac {a^{2} c \cot \left (f x +e \right ) \csc \left (f x +e \right )}{16 f}-\frac {a^{2} c \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{16 f}-\frac {a^{2} c \cot \left (f x +e \right ) \left (\csc ^{4}\left (f x +e \right )\right )}{5 f}-\frac {a^{2} c \cot \left (f x +e \right ) \left (\csc ^{5}\left (f x +e \right )\right )}{6 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 232, normalized size = 1.78 \[ \frac {5 \, a^{2} c {\left (\frac {2 \, {\left (15 \, \cos \left (f x + e\right )^{5} - 40 \, \cos \left (f x + e\right )^{3} + 33 \, \cos \left (f x + e\right )\right )}}{\cos \left (f x + e\right )^{6} - 3 \, \cos \left (f x + e\right )^{4} + 3 \, \cos \left (f x + e\right )^{2} - 1} - 15 \, \log \left (\cos \left (f x + e\right ) + 1\right ) + 15 \, \log \left (\cos \left (f x + e\right ) - 1\right )\right )} - 30 \, a^{2} c {\left (\frac {2 \, {\left (3 \, \cos \left (f x + e\right )^{3} - 5 \, \cos \left (f x + e\right )\right )}}{\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\cos \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\cos \left (f x + e\right ) - 1\right )\right )} + \frac {160 \, {\left (3 \, \tan \left (f x + e\right )^{2} + 1\right )} a^{2} c}{\tan \left (f x + e\right )^{3}} - \frac {32 \, {\left (15 \, \tan \left (f x + e\right )^{4} + 10 \, \tan \left (f x + e\right )^{2} + 3\right )} a^{2} c}{\tan \left (f x + e\right )^{5}}}{480 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.54, size = 340, normalized size = 2.62 \[ -\frac {a^2\,c\,\left (5\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-12\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}+12\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )-15\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-20\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+15\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+120\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7-120\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5-15\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+20\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+15\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+120\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\ln \left (\frac {\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\right )}{1920\,f\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6} \]
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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