3.2.68 \(\int \frac {(f x)^m (a+b \text {csch}^{-1}(c x))}{d+e x^2} \, dx\) [168]

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {(f x)^m \left (a+b \text {csch}^{-1}(c x)\right )}{d+e x^2},x\right ) \]

[Out]

Unintegrable((f*x)^m*(a+b*arccsch(c*x))/(e*x^2+d),x)

________________________________________________________________________________________

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 \text {csch}^{-1}(c x)\right )}{d+e x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((f*x)^m*(a + b*ArcCsch[c*x]))/(d + e*x^2),x]

[Out]

Defer[Int][((f*x)^m*(a + b*ArcCsch[c*x]))/(d + e*x^2), x]

Rubi steps

\begin {align*} \int \frac {(f x)^m \left (a+b \text {csch}^{-1}(c x)\right )}{d+e x^2} \, dx &=\int \frac {(f x)^m \left (a+b \text {csch}^{-1}(c x)\right )}{d+e x^2} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.23, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(f x)^m \left (a+b \text {csch}^{-1}(c x)\right )}{d+e x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((f*x)^m*(a + b*ArcCsch[c*x]))/(d + e*x^2),x]

[Out]

Integrate[((f*x)^m*(a + b*ArcCsch[c*x]))/(d + e*x^2), x]

________________________________________________________________________________________

Maple [A]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {\left (f x \right )^{m} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{e \,x^{2}+d}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^m*(a+b*arccsch(c*x))/(e*x^2+d),x)

[Out]

int((f*x)^m*(a+b*arccsch(c*x))/(e*x^2+d),x)

________________________________________________________________________________________

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.

[In]

integrate((f*x)^m*(a+b*arccsch(c*x))/(e*x^2+d),x, algorithm="maxima")

[Out]

integrate((b*arccsch(c*x) + a)*(f*x)^m/(x^2*e + d), x)

________________________________________________________________________________________

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.

[In]

integrate((f*x)^m*(a+b*arccsch(c*x))/(e*x^2+d),x, algorithm="fricas")

[Out]

integral((b*arccsch(c*x) + a)*(f*x)^m/(x^2*e + d), x)

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (f x\right )^{m} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{d + e x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)**m*(a+b*acsch(c*x))/(e*x**2+d),x)

[Out]

Integral((f*x)**m*(a + b*acsch(c*x))/(d + e*x**2), x)

________________________________________________________________________________________

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.

[In]

integrate((f*x)^m*(a+b*arccsch(c*x))/(e*x^2+d),x, algorithm="giac")

[Out]

integrate((b*arccsch(c*x) + a)*(f*x)^m/(e*x^2 + d), x)

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\left (f\,x\right )}^m\,\left (a+b\,\mathrm {asinh}\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.

[In]

int(((f*x)^m*(a + b*asinh(1/(c*x))))/(d + e*x^2),x)

[Out]

int(((f*x)^m*(a + b*asinh(1/(c*x))))/(d + e*x^2), x)

________________________________________________________________________________________