3.7.28 \(\int \frac {1}{(d+e x^2)^2 (a+b \sinh ^{-1}(c x))^2} \, dx\) [628]

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {1}{\left (d+e x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2},x\right ) \]

[Out]

Unintegrable(1/(e*x^2+d)^2/(a+b*arcsinh(c*x))^2,x)

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Rubi [A]
time = 0.02, 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 {1}{\left (d+e x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/((d + e*x^2)^2*(a + b*ArcSinh[c*x])^2),x]

[Out]

Defer[Int][1/((d + e*x^2)^2*(a + b*ArcSinh[c*x])^2), x]

Rubi steps

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

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Mathematica [A]
time = 20.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d+e x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/((d + e*x^2)^2*(a + b*ArcSinh[c*x])^2),x]

[Out]

Integrate[1/((d + e*x^2)^2*(a + b*ArcSinh[c*x])^2), x]

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (e \,x^{2}+d \right )^{2} \left (a +b \arcsinh \left (c x \right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(e*x^2+d)^2/(a+b*arcsinh(c*x))^2,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(1/(e*x^2+d)^2/(a+b*arcsinh(c*x))^2,x, algorithm="maxima")

[Out]

-(c^3*x^3 + c*x + (c^2*x^2 + 1)^(3/2))/(a*b*c^3*x^6*e^2 + a*b*c*d^2 + (2*a*b*c^3*d*e + a*b*c*e^2)*x^4 + (a*b*c
^3*d^2 + 2*a*b*c*d*e)*x^2 + (b^2*c^3*x^6*e^2 + b^2*c*d^2 + (2*b^2*c^3*d*e + b^2*c*e^2)*x^4 + (b^2*c^3*d^2 + 2*
b^2*c*d*e)*x^2 + (b^2*c^2*x^5*e^2 + 2*b^2*c^2*d*x^3*e + b^2*c^2*d^2*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x
^2 + 1)) + (a*b*c^2*x^5*e^2 + 2*a*b*c^2*d*x^3*e + a*b*c^2*d^2*x)*sqrt(c^2*x^2 + 1)) - integrate((3*c^5*x^6*e -
(c^5*d - 6*c^3*e)*x^4 - (2*c^3*d - 3*c*e)*x^2 + (3*c^3*x^4*e - (c^3*d - 5*c*e)*x^2 + c*d)*(c^2*x^2 + 1) - c*d
+ (6*c^4*x^5*e - (2*c^4*d - 11*c^2*e)*x^3 - (c^2*d - 4*e)*x)*sqrt(c^2*x^2 + 1))/(a*b*c^5*x^10*e^3 + (3*a*b*c^
5*d*e^2 + 2*a*b*c^3*e^3)*x^8 + (3*a*b*c^5*d^2*e + 6*a*b*c^3*d*e^2 + a*b*c*e^3)*x^6 + a*b*c*d^3 + (a*b*c^5*d^3
+ 6*a*b*c^3*d^2*e + 3*a*b*c*d*e^2)*x^4 + (2*a*b*c^3*d^3 + 3*a*b*c*d^2*e)*x^2 + (a*b*c^3*x^8*e^3 + 3*a*b*c^3*d*
x^6*e^2 + 3*a*b*c^3*d^2*x^4*e + a*b*c^3*d^3*x^2)*(c^2*x^2 + 1) + (b^2*c^5*x^10*e^3 + (3*b^2*c^5*d*e^2 + 2*b^2*
c^3*e^3)*x^8 + (3*b^2*c^5*d^2*e + 6*b^2*c^3*d*e^2 + b^2*c*e^3)*x^6 + b^2*c*d^3 + (b^2*c^5*d^3 + 6*b^2*c^3*d^2*
e + 3*b^2*c*d*e^2)*x^4 + (2*b^2*c^3*d^3 + 3*b^2*c*d^2*e)*x^2 + (b^2*c^3*x^8*e^3 + 3*b^2*c^3*d*x^6*e^2 + 3*b^2*
c^3*d^2*x^4*e + b^2*c^3*d^3*x^2)*(c^2*x^2 + 1) + 2*(b^2*c^4*x^9*e^3 + b^2*c^2*d^3*x + (3*b^2*c^4*d*e^2 + b^2*c
^2*e^3)*x^7 + 3*(b^2*c^4*d^2*e + b^2*c^2*d*e^2)*x^5 + (b^2*c^4*d^3 + 3*b^2*c^2*d^2*e)*x^3)*sqrt(c^2*x^2 + 1))*
log(c*x + sqrt(c^2*x^2 + 1)) + 2*(a*b*c^4*x^9*e^3 + a*b*c^2*d^3*x + (3*a*b*c^4*d*e^2 + a*b*c^2*e^3)*x^7 + 3*(a
*b*c^4*d^2*e + a*b*c^2*d*e^2)*x^5 + (a*b*c^4*d^3 + 3*a*b*c^2*d^2*e)*x^3)*sqrt(c^2*x^2 + 1)), 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(1/(e*x^2+d)^2/(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

integral(1/(a^2*x^4*e^2 + 2*a^2*d*x^2*e + a^2*d^2 + (b^2*x^4*e^2 + 2*b^2*d*x^2*e + b^2*d^2)*arcsinh(c*x)^2 + 2
*(a*b*x^4*e^2 + 2*a*b*d*x^2*e + a*b*d^2)*arcsinh(c*x)), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x**2+d)**2/(a+b*asinh(c*x))**2,x)

[Out]

Timed out

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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(1/(e*x^2+d)^2/(a+b*arcsinh(c*x))^2,x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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