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

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 = 0.04, 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 = {} \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

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 = 28.65, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

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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fricas [A]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a^{2} e^{2} x^{4} + 2 \, a^{2} d e x^{2} + a^{2} d^{2} + {\left (b^{2} e^{2} x^{4} + 2 \, b^{2} d e x^{2} + b^{2} d^{2}\right )} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, {\left (a b e^{2} x^{4} + 2 \, a b d e x^{2} + a b d^{2}\right )} \operatorname {arsinh}\left (c x\right )}, x\right ) \]

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

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

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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maple [A]  time = 0.56, 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 \[ -\frac {c^{3} x^{3} + c x + {\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a b c^{3} e^{2} x^{6} + {\left (2 \, c^{3} d e + c e^{2}\right )} a b x^{4} + a b c d^{2} + {\left (c^{3} d^{2} + 2 \, c d e\right )} a b x^{2} + {\left (b^{2} c^{3} e^{2} x^{6} + {\left (2 \, c^{3} d e + c e^{2}\right )} b^{2} x^{4} + b^{2} c d^{2} + {\left (c^{3} d^{2} + 2 \, c d e\right )} b^{2} x^{2} + {\left (b^{2} c^{2} e^{2} x^{5} + 2 \, b^{2} c^{2} d e x^{3} + b^{2} c^{2} d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (a b c^{2} e^{2} x^{5} + 2 \, a b c^{2} d e x^{3} + a b c^{2} d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}} - \int \frac {3 \, c^{5} e x^{6} - {\left (c^{5} d - 6 \, c^{3} e\right )} x^{4} - {\left (2 \, c^{3} d - 3 \, c e\right )} x^{2} + {\left (3 \, c^{3} e x^{4} - {\left (c^{3} d - 5 \, c e\right )} x^{2} + c d\right )} {\left (c^{2} x^{2} + 1\right )} - c d + {\left (6 \, c^{4} e x^{5} - {\left (2 \, c^{4} d - 11 \, c^{2} e\right )} x^{3} - {\left (c^{2} d - 4 \, e\right )} x\right )} \sqrt {c^{2} x^{2} + 1}}{a b c^{5} e^{3} x^{10} + {\left (3 \, c^{5} d e^{2} + 2 \, c^{3} e^{3}\right )} a b x^{8} + {\left (3 \, c^{5} d^{2} e + 6 \, c^{3} d e^{2} + c e^{3}\right )} a b x^{6} + {\left (c^{5} d^{3} + 6 \, c^{3} d^{2} e + 3 \, c d e^{2}\right )} a b x^{4} + a b c d^{3} + {\left (2 \, c^{3} d^{3} + 3 \, c d^{2} e\right )} a b x^{2} + {\left (a b c^{3} e^{3} x^{8} + 3 \, a b c^{3} d e^{2} x^{6} + 3 \, a b c^{3} d^{2} e x^{4} + a b c^{3} d^{3} x^{2}\right )} {\left (c^{2} x^{2} + 1\right )} + {\left (b^{2} c^{5} e^{3} x^{10} + {\left (3 \, c^{5} d e^{2} + 2 \, c^{3} e^{3}\right )} b^{2} x^{8} + {\left (3 \, c^{5} d^{2} e + 6 \, c^{3} d e^{2} + c e^{3}\right )} b^{2} x^{6} + {\left (c^{5} d^{3} + 6 \, c^{3} d^{2} e + 3 \, c d e^{2}\right )} b^{2} x^{4} + b^{2} c d^{3} + {\left (2 \, c^{3} d^{3} + 3 \, c d^{2} e\right )} b^{2} x^{2} + {\left (b^{2} c^{3} e^{3} x^{8} + 3 \, b^{2} c^{3} d e^{2} x^{6} + 3 \, b^{2} c^{3} d^{2} e x^{4} + b^{2} c^{3} d^{3} x^{2}\right )} {\left (c^{2} x^{2} + 1\right )} + 2 \, {\left (b^{2} c^{4} e^{3} x^{9} + {\left (3 \, c^{4} d e^{2} + c^{2} e^{3}\right )} b^{2} x^{7} + b^{2} c^{2} d^{3} x + 3 \, {\left (c^{4} d^{2} e + c^{2} d e^{2}\right )} b^{2} x^{5} + {\left (c^{4} d^{3} + 3 \, c^{2} d^{2} e\right )} b^{2} x^{3}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, {\left (a b c^{4} e^{3} x^{9} + {\left (3 \, c^{4} d e^{2} + c^{2} e^{3}\right )} a b x^{7} + a b c^{2} d^{3} x + 3 \, {\left (c^{4} d^{2} e + c^{2} d e^{2}\right )} a b x^{5} + {\left (c^{4} d^{3} + 3 \, c^{2} d^{2} e\right )} a b x^{3}\right )} \sqrt {c^{2} x^{2} + 1}}\,{d x} \]

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*e^2*x^6 + (2*c^3*d*e + c*e^2)*a*b*x^4 + a*b*c*d^2 + (c^3*d^2 +
2*c*d*e)*a*b*x^2 + (b^2*c^3*e^2*x^6 + (2*c^3*d*e + c*e^2)*b^2*x^4 + b^2*c*d^2 + (c^3*d^2 + 2*c*d*e)*b^2*x^2 +
(b^2*c^2*e^2*x^5 + 2*b^2*c^2*d*e*x^3 + 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*e^2*x^5 + 2*a*b*c^2*d*e*x^3 + a*b*c^2*d^2*x)*sqrt(c^2*x^2 + 1)) - integrate((3*c^5*e*x^6 - (c^5*d - 6*c^3*
e)*x^4 - (2*c^3*d - 3*c*e)*x^2 + (3*c^3*e*x^4 - (c^3*d - 5*c*e)*x^2 + c*d)*(c^2*x^2 + 1) - c*d + (6*c^4*e*x^5
- (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*e^3*x^10 + (3*c^5*d*e^2 + 2*c^3*e^3)
*a*b*x^8 + (3*c^5*d^2*e + 6*c^3*d*e^2 + c*e^3)*a*b*x^6 + (c^5*d^3 + 6*c^3*d^2*e + 3*c*d*e^2)*a*b*x^4 + a*b*c*d
^3 + (2*c^3*d^3 + 3*c*d^2*e)*a*b*x^2 + (a*b*c^3*e^3*x^8 + 3*a*b*c^3*d*e^2*x^6 + 3*a*b*c^3*d^2*e*x^4 + a*b*c^3*
d^3*x^2)*(c^2*x^2 + 1) + (b^2*c^5*e^3*x^10 + (3*c^5*d*e^2 + 2*c^3*e^3)*b^2*x^8 + (3*c^5*d^2*e + 6*c^3*d*e^2 +
c*e^3)*b^2*x^6 + (c^5*d^3 + 6*c^3*d^2*e + 3*c*d*e^2)*b^2*x^4 + b^2*c*d^3 + (2*c^3*d^3 + 3*c*d^2*e)*b^2*x^2 + (
b^2*c^3*e^3*x^8 + 3*b^2*c^3*d*e^2*x^6 + 3*b^2*c^3*d^2*e*x^4 + b^2*c^3*d^3*x^2)*(c^2*x^2 + 1) + 2*(b^2*c^4*e^3*
x^9 + (3*c^4*d*e^2 + c^2*e^3)*b^2*x^7 + b^2*c^2*d^3*x + 3*(c^4*d^2*e + c^2*d*e^2)*b^2*x^5 + (c^4*d^3 + 3*c^2*d
^2*e)*b^2*x^3)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + 2*(a*b*c^4*e^3*x^9 + (3*c^4*d*e^2 + c^2*e^3)*
a*b*x^7 + a*b*c^2*d^3*x + 3*(c^4*d^2*e + c^2*d*e^2)*a*b*x^5 + (c^4*d^3 + 3*c^2*d^2*e)*a*b*x^3)*sqrt(c^2*x^2 +
1)), x)

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

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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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