Integrand size = 19, antiderivative size = 74 \[ \int \csc (e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {1}{2} b \left (6 a^2+b^2\right ) x-\frac {a^3 \text {arctanh}(\cos (e+f x))}{f}-\frac {5 a b^2 \cos (e+f x)}{2 f}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x))}{2 f} \]
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Time = 0.08 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2872, 3102, 2814, 3855} \[ \int \csc (e+f x) (a+b \sin (e+f x))^3 \, dx=-\frac {a^3 \text {arctanh}(\cos (e+f x))}{f}+\frac {1}{2} b x \left (6 a^2+b^2\right )-\frac {5 a b^2 \cos (e+f x)}{2 f}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x))}{2 f} \]
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Rule 2814
Rule 2872
Rule 3102
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {b^2 \cos (e+f x) (a+b \sin (e+f x))}{2 f}+\frac {1}{2} \int \csc (e+f x) \left (2 a^3+b \left (6 a^2+b^2\right ) \sin (e+f x)+5 a b^2 \sin ^2(e+f x)\right ) \, dx \\ & = -\frac {5 a b^2 \cos (e+f x)}{2 f}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x))}{2 f}+\frac {1}{2} \int \csc (e+f x) \left (2 a^3+b \left (6 a^2+b^2\right ) \sin (e+f x)\right ) \, dx \\ & = \frac {1}{2} b \left (6 a^2+b^2\right ) x-\frac {5 a b^2 \cos (e+f x)}{2 f}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x))}{2 f}+a^3 \int \csc (e+f x) \, dx \\ & = \frac {1}{2} b \left (6 a^2+b^2\right ) x-\frac {a^3 \text {arctanh}(\cos (e+f x))}{f}-\frac {5 a b^2 \cos (e+f x)}{2 f}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x))}{2 f} \\ \end{align*}
Time = 1.15 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.09 \[ \int \csc (e+f x) (a+b \sin (e+f x))^3 \, dx=-\frac {-2 b \left (6 a^2+b^2\right ) (e+f x)+12 a b^2 \cos (e+f x)+4 a^3 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )-4 a^3 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )+b^3 \sin (2 (e+f x))}{4 f} \]
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Time = 0.69 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.93
method | result | size |
parallelrisch | \(\frac {12 a^{2} b f x +2 b^{3} f x +4 a^{3} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-12 \cos \left (f x +e \right ) a \,b^{2}-\sin \left (2 f x +2 e \right ) b^{3}-12 a \,b^{2}}{4 f}\) | \(69\) |
derivativedivides | \(\frac {a^{3} \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+3 a^{2} b \left (f x +e \right )-3 \cos \left (f x +e \right ) a \,b^{2}+b^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}\) | \(75\) |
default | \(\frac {a^{3} \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+3 a^{2} b \left (f x +e \right )-3 \cos \left (f x +e \right ) a \,b^{2}+b^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}\) | \(75\) |
risch | \(3 a^{2} b x +\frac {b^{3} x}{2}-\frac {3 a \,b^{2} {\mathrm e}^{i \left (f x +e \right )}}{2 f}-\frac {3 a \,b^{2} {\mathrm e}^{-i \left (f x +e \right )}}{2 f}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{f}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{f}-\frac {b^{3} \sin \left (2 f x +2 e \right )}{4 f}\) | \(107\) |
norman | \(\frac {\left (3 a^{2} b +\frac {1}{2} b^{3}\right ) x +\frac {b^{3} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\left (3 a^{2} b +\frac {1}{2} b^{3}\right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (9 a^{2} b +\frac {3}{2} b^{3}\right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (9 a^{2} b +\frac {3}{2} b^{3}\right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {6 a \,b^{2}}{f}-\frac {b^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {6 a \,b^{2} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {12 a \,b^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}+\frac {a^{3} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}\) | \(209\) |
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Time = 0.35 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.07 \[ \int \csc (e+f x) (a+b \sin (e+f x))^3 \, dx=-\frac {b^{3} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 6 \, a b^{2} \cos \left (f x + e\right ) + a^{3} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) - a^{3} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) - {\left (6 \, a^{2} b + b^{3}\right )} f x}{2 \, f} \]
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\[ \int \csc (e+f x) (a+b \sin (e+f x))^3 \, dx=\int \left (a + b \sin {\left (e + f x \right )}\right )^{3} \csc {\left (e + f x \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.96 \[ \int \csc (e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {12 \, {\left (f x + e\right )} a^{2} b + {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} b^{3} - 12 \, a b^{2} \cos \left (f x + e\right ) - 4 \, a^{3} \log \left (\cot \left (f x + e\right ) + \csc \left (f x + e\right )\right )}{4 \, f} \]
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Time = 0.32 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.46 \[ \int \csc (e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {2 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right ) + {\left (6 \, a^{2} b + b^{3}\right )} {\left (f x + e\right )} + \frac {2 \, {\left (b^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 6 \, a b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - b^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 6 \, a b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{2}}}{2 \, f} \]
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Time = 6.47 (sec) , antiderivative size = 259, normalized size of antiderivative = 3.50 \[ \int \csc (e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {a^3\,\ln \left (\frac {\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )}{f}-\frac {b^3\,\mathrm {atan}\left (\frac {2\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,a^3+6\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,a^2\,b+\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,b^3}{-2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,a^3+6\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,a^2\,b+\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,b^3}\right )}{f}-\frac {b^3\,\sin \left (2\,e+2\,f\,x\right )}{4\,f}-\frac {6\,a^2\,b\,\mathrm {atan}\left (\frac {2\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,a^3+6\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,a^2\,b+\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,b^3}{-2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,a^3+6\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,a^2\,b+\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,b^3}\right )}{f}-\frac {3\,a\,b^2\,\cos \left (e+f\,x\right )}{f} \]
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