3.454 \(\int \frac {1}{x^2 (1+c^2 x^2)^{5/2} (a+b \sinh ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=30 \[ \text {Int}\left (\frac {1}{x^2 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2},x\right ) \]

[Out]

Unintegrable(1/x^2/(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x))^2,x)

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Rubi [A]  time = 0.13, 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 = {} \[ \int \frac {1}{x^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x^2*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2),x]

[Out]

Defer[Int][1/(x^2*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x]

Rubi steps

\begin {align*} \int \frac {1}{x^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=\int \frac {1}{x^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx\\ \end {align*}

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Mathematica [A]  time = 12.09, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x^2*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2),x]

[Out]

Integrate[1/(x^2*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x]

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fricas [A]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c^{2} x^{2} + 1}}{a^{2} c^{6} x^{8} + 3 \, a^{2} c^{4} x^{6} + 3 \, a^{2} c^{2} x^{4} + a^{2} x^{2} + {\left (b^{2} c^{6} x^{8} + 3 \, b^{2} c^{4} x^{6} + 3 \, b^{2} c^{2} x^{4} + b^{2} x^{2}\right )} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, {\left (a b c^{6} x^{8} + 3 \, a b c^{4} x^{6} + 3 \, a b c^{2} x^{4} + a b x^{2}\right )} \operatorname {arsinh}\left (c x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

integral(sqrt(c^2*x^2 + 1)/(a^2*c^6*x^8 + 3*a^2*c^4*x^6 + 3*a^2*c^2*x^4 + a^2*x^2 + (b^2*c^6*x^8 + 3*b^2*c^4*x
^6 + 3*b^2*c^2*x^4 + b^2*x^2)*arcsinh(c*x)^2 + 2*(a*b*c^6*x^8 + 3*a*b*c^4*x^6 + 3*a*b*c^2*x^4 + a*b*x^2)*arcsi
nh(c*x)), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x))^2,x, algorithm="giac")

[Out]

integrate(1/((c^2*x^2 + 1)^(5/2)*(b*arcsinh(c*x) + a)^2*x^2), x)

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maple [A]  time = 1.19, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (c^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a +b \arcsinh \left (c x \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x))^2,x)

[Out]

int(1/x^2/(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x))^2,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {c x + \sqrt {c^{2} x^{2} + 1}}{{\left (a b c^{4} x^{5} + a b c^{2} x^{3}\right )} {\left (c^{2} x^{2} + 1\right )} + {\left ({\left (b^{2} c^{4} x^{5} + b^{2} c^{2} x^{3}\right )} {\left (c^{2} x^{2} + 1\right )} + {\left (b^{2} c^{5} x^{6} + 2 \, b^{2} c^{3} x^{4} + b^{2} c x^{2}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (a b c^{5} x^{6} + 2 \, a b c^{3} x^{4} + a b c x^{2}\right )} \sqrt {c^{2} x^{2} + 1}} - \int \frac {6 \, c^{5} x^{5} + 7 \, c^{3} x^{3} + 3 \, {\left (2 \, c^{3} x^{3} + c x\right )} {\left (c^{2} x^{2} + 1\right )} + c x + 2 \, {\left (6 \, c^{4} x^{4} + 5 \, c^{2} x^{2} + 1\right )} \sqrt {c^{2} x^{2} + 1}}{{\left (a b c^{7} x^{9} + 2 \, a b c^{5} x^{7} + a b c^{3} x^{5}\right )} {\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} + 2 \, {\left (a b c^{8} x^{10} + 3 \, a b c^{6} x^{8} + 3 \, a b c^{4} x^{6} + a b c^{2} x^{4}\right )} {\left (c^{2} x^{2} + 1\right )} + {\left ({\left (b^{2} c^{7} x^{9} + 2 \, b^{2} c^{5} x^{7} + b^{2} c^{3} x^{5}\right )} {\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} + 2 \, {\left (b^{2} c^{8} x^{10} + 3 \, b^{2} c^{6} x^{8} + 3 \, b^{2} c^{4} x^{6} + b^{2} c^{2} x^{4}\right )} {\left (c^{2} x^{2} + 1\right )} + {\left (b^{2} c^{9} x^{11} + 4 \, b^{2} c^{7} x^{9} + 6 \, b^{2} c^{5} x^{7} + 4 \, b^{2} c^{3} x^{5} + b^{2} c x^{3}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (a b c^{9} x^{11} + 4 \, a b c^{7} x^{9} + 6 \, a b c^{5} x^{7} + 4 \, a b c^{3} x^{5} + a b c x^{3}\right )} \sqrt {c^{2} x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x))^2,x, algorithm="maxima")

[Out]

-(c*x + sqrt(c^2*x^2 + 1))/((a*b*c^4*x^5 + a*b*c^2*x^3)*(c^2*x^2 + 1) + ((b^2*c^4*x^5 + b^2*c^2*x^3)*(c^2*x^2
+ 1) + (b^2*c^5*x^6 + 2*b^2*c^3*x^4 + b^2*c*x^2)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + (a*b*c^5*x^
6 + 2*a*b*c^3*x^4 + a*b*c*x^2)*sqrt(c^2*x^2 + 1)) - integrate((6*c^5*x^5 + 7*c^3*x^3 + 3*(2*c^3*x^3 + c*x)*(c^
2*x^2 + 1) + c*x + 2*(6*c^4*x^4 + 5*c^2*x^2 + 1)*sqrt(c^2*x^2 + 1))/((a*b*c^7*x^9 + 2*a*b*c^5*x^7 + a*b*c^3*x^
5)*(c^2*x^2 + 1)^(3/2) + 2*(a*b*c^8*x^10 + 3*a*b*c^6*x^8 + 3*a*b*c^4*x^6 + a*b*c^2*x^4)*(c^2*x^2 + 1) + ((b^2*
c^7*x^9 + 2*b^2*c^5*x^7 + b^2*c^3*x^5)*(c^2*x^2 + 1)^(3/2) + 2*(b^2*c^8*x^10 + 3*b^2*c^6*x^8 + 3*b^2*c^4*x^6 +
b^2*c^2*x^4)*(c^2*x^2 + 1) + (b^2*c^9*x^11 + 4*b^2*c^7*x^9 + 6*b^2*c^5*x^7 + 4*b^2*c^3*x^5 + b^2*c*x^3)*sqrt(
c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + (a*b*c^9*x^11 + 4*a*b*c^7*x^9 + 6*a*b*c^5*x^7 + 4*a*b*c^3*x^5 + a
*b*c*x^3)*sqrt(c^2*x^2 + 1)), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (c^2\,x^2+1\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b*asinh(c*x))^2*(c^2*x^2 + 1)^(5/2)),x)

[Out]

int(1/(x^2*(a + b*asinh(c*x))^2*(c^2*x^2 + 1)^(5/2)), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \left (c^{2} x^{2} + 1\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(c**2*x**2+1)**(5/2)/(a+b*asinh(c*x))**2,x)

[Out]

Integral(1/(x**2*(a + b*asinh(c*x))**2*(c**2*x**2 + 1)**(5/2)), x)

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