Optimal. Leaf size=17 \[ b x-\frac {a \tanh ^{-1}(\cos (e+f x))}{f} \]
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Rubi [A] time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2735, 3770} \[ b x-\frac {a \tanh ^{-1}(\cos (e+f x))}{f} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 3770
Rubi steps
\begin {align*} \int \csc (e+f x) (a+b \sin (e+f x)) \, dx &=b x+a \int \csc (e+f x) \, dx\\ &=b x-\frac {a \tanh ^{-1}(\cos (e+f x))}{f}\\ \end {align*}
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Mathematica [B] time = 0.02, size = 43, normalized size = 2.53 \[ \frac {a \log \left (\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f}-\frac {a \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f}+b x \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 38, normalized size = 2.24 \[ \frac {2 \, b f x - a \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) + a \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right )}{2 \, f} \]
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 = 0.11, size = 32, normalized size = 1.88 \[ b x +\frac {a \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{f}+\frac {b e}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 29, normalized size = 1.71 \[ \frac {{\left (f x + e\right )} b - a \log \left (\cot \left (f x + e\right ) + \csc \left (f x + e\right )\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.80, size = 85, normalized size = 5.00 \[ \frac {a\,\ln \left (\frac {\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )}{f}+\frac {2\,b\,\mathrm {atan}\left (\frac {b\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+a\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{a\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )-b\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 6.76, size = 51, normalized size = 3.00 \[ a \left (\begin {cases} \frac {x \cot {\relax (e )} \csc {\relax (e )}}{\cot {\relax (e )} + \csc {\relax (e )}} + \frac {x \csc ^{2}{\relax (e )}}{\cot {\relax (e )} + \csc {\relax (e )}} & \text {for}\: f = 0 \\- \frac {\log {\left (\cot {\left (e + f x \right )} + \csc {\left (e + f x \right )} \right )}}{f} & \text {otherwise} \end {cases}\right ) + b x \]
Verification of antiderivative is not currently implemented for this CAS.
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