Optimal. Leaf size=26 \[ -\frac {a \cot (e+f x)}{f}-\frac {b \tanh ^{-1}(\cos (e+f x))}{f} \]
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Rubi [A] time = 0.04, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2748, 3767, 8, 3770} \[ -\frac {a \cot (e+f x)}{f}-\frac {b \tanh ^{-1}(\cos (e+f x))}{f} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2748
Rule 3767
Rule 3770
Rubi steps
\begin {align*} \int \csc ^2(e+f x) (a+b \sin (e+f x)) \, dx &=a \int \csc ^2(e+f x) \, dx+b \int \csc (e+f x) \, dx\\ &=-\frac {b \tanh ^{-1}(\cos (e+f x))}{f}-\frac {a \operatorname {Subst}(\int 1 \, dx,x,\cot (e+f x))}{f}\\ &=-\frac {b \tanh ^{-1}(\cos (e+f x))}{f}-\frac {a \cot (e+f x)}{f}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 52, normalized size = 2.00 \[ -\frac {a \cot (e+f x)}{f}+\frac {b \log \left (\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f}-\frac {b \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 62, normalized size = 2.38 \[ -\frac {b \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) \sin \left (f x + e\right ) - b \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) \sin \left (f x + e\right ) + 2 \, a \cos \left (f x + e\right )}{2 \, f \sin \left (f x + e\right )} \]
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 = 0.17, size = 35, normalized size = 1.35 \[ -\frac {a \cot \left (f x +e \right )}{f}+\frac {b \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 40, normalized size = 1.54 \[ -\frac {b {\left (\log \left (\cos \left (f x + e\right ) + 1\right ) - \log \left (\cos \left (f x + e\right ) - 1\right )\right )} + \frac {2 \, a}{\tan \left (f x + e\right )}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.76, size = 28, normalized size = 1.08 \[ \frac {b\,\ln \left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{f}-\frac {a\,\mathrm {cot}\left (e+f\,x\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sin {\left (e + f x \right )}\right ) \csc ^{2}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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