3.1.4 \(\int x^2 (a+b \text {csch}(c+d x^2)) \, dx\) [4]

Optimal. Leaf size=26 \[ \frac {a x^3}{3}+b \text {Int}\left (x^2 \text {csch}\left (c+d x^2\right ),x\right ) \]

[Out]

1/3*a*x^3+b*Unintegrable(x^2*csch(d*x^2+c),x)

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Rubi [A]
time = 0.01, 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 x^2 \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[x^2*(a + b*Csch[c + d*x^2]),x]

[Out]

(a*x^3)/3 + b*Defer[Int][x^2*Csch[c + d*x^2], x]

Rubi steps

\begin {align*} \int x^2 \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx &=\int \left (a x^2+b x^2 \text {csch}\left (c+d x^2\right )\right ) \, dx\\ &=\frac {a x^3}{3}+b \int x^2 \text {csch}\left (c+d x^2\right ) \, dx\\ \end {align*}

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Mathematica [A]
time = 6.66, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[x^2*(a + b*Csch[c + d*x^2]),x]

[Out]

Integrate[x^2*(a + b*Csch[c + d*x^2]), x]

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Maple [A]
time = 0.75, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \,\mathrm {csch}\left (d \,x^{2}+c \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*csch(d*x^2+c)),x)

[Out]

int(x^2*(a+b*csch(d*x^2+c)),x)

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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.

[In]

integrate(x^2*(a+b*csch(d*x^2+c)),x, algorithm="maxima")

[Out]

1/3*a*x^3 + 2*b*integrate(x^2/(e^(d*x^2 + c) - e^(-d*x^2 - c)), x)

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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.

[In]

integrate(x^2*(a+b*csch(d*x^2+c)),x, algorithm="fricas")

[Out]

integral(b*x^2*csch(d*x^2 + c) + a*x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*csch(d*x**2+c)),x)

[Out]

Integral(x**2*(a + b*csch(c + d*x**2)), x)

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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.

[In]

integrate(x^2*(a+b*csch(d*x^2+c)),x, algorithm="giac")

[Out]

integrate((b*csch(d*x^2 + c) + a)*x^2, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int x^2\,\left (a+\frac {b}{\mathrm {sinh}\left (d\,x^2+c\right )}\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b/sinh(c + d*x^2)),x)

[Out]

int(x^2*(a + b/sinh(c + d*x^2)), x)

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