Optimal. Leaf size=16 \[ \frac {\cot (e+f x) \csc (e+f x)}{f} \]
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Rubi [A]
time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {3090}
\begin {gather*} \frac {\cot (e+f x) \csc (e+f x)}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 3090
Rubi steps
\begin {align*} \int \csc ^3(e+f x) \left (-2+\sin ^2(e+f x)\right ) \, dx &=\frac {\cot (e+f x) \csc (e+f x)}{f}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(107\) vs. \(2(16)=32\).
time = 0.02, size = 107, normalized size = 6.69 \begin {gather*} \frac {\csc ^2\left (\frac {1}{2} (e+f x)\right )}{4 f}-\frac {\log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f}+\frac {\log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{f}+\frac {\log \left (\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f}-\frac {\log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{f}-\frac {\sec ^2\left (\frac {1}{2} (e+f x)\right )}{4 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 17, normalized size = 1.06
method | result | size |
derivativedivides | \(\frac {\cot \left (f x +e \right ) \csc \left (f x +e \right )}{f}\) | \(17\) |
default | \(\frac {\cot \left (f x +e \right ) \csc \left (f x +e \right )}{f}\) | \(17\) |
risch | \(-\frac {2 \left ({\mathrm e}^{3 i \left (f x +e \right )}+{\mathrm e}^{i \left (f x +e \right )}\right )}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{2}}\) | \(38\) |
norman | \(\frac {\frac {\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {1}{4 f}-\frac {\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 25, normalized size = 1.56 \begin {gather*} -\frac {\cos \left (f x + e\right )}{{\left (\cos \left (f x + e\right )^{2} - 1\right )} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 26, normalized size = 1.62 \begin {gather*} -\frac {\cos \left (f x + e\right )}{f \cos \left (f x + e\right )^{2} - f} \end {gather*}
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 \begin {gather*} \int \left (\sin ^{2}{\left (e + f x \right )} - 2\right ) \csc ^{3}{\left (e + f x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs.
\(2 (18) = 36\).
time = 0.59, size = 49, normalized size = 3.06 \begin {gather*} -\frac {\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} - \frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1}}{4 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.23, size = 23, normalized size = 1.44 \begin {gather*} -\frac {\cos \left (e+f\,x\right )}{f\,\left ({\cos \left (e+f\,x\right )}^2-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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