Optimal. Leaf size=26 \[ \text {Int}\left (\frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2},x\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx &=\int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx\\ \end {align*}
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Mathematica [A]
time = 1.29, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(f x)^m \left (a+b \csc ^{-1}(c x)\right )}{d+e x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (f x \right )^{m} \left (a +b \,\mathrm {arccsc}\left (c x \right )\right )}{e \,x^{2}+d}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (f x\right )^{m} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{d + e x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\left (f\,x\right )}^m\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{e\,x^2+d} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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