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

Optimal. Leaf size=62 \[ -\frac {4 c \text {Int}\left (\frac {x}{\left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )},x\right )}{b}-\frac {1}{b c \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )} \]

[Out]

-1/b/c/(c^2*x^2+1)^2/(a+b*arcsinh(c*x))-4*c*Unintegrable(x/(c^2*x^2+1)^3/(a+b*arcsinh(c*x)),x)/b

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Rubi [A]  time = 0.11, 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 = {} \[ \int \frac {1}{\left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-(1/(b*c*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))) - (4*c*Defer[Int][x/((1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])), x]
)/b

Rubi steps

\begin {align*} \int \frac {1}{\left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=-\frac {1}{b c \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}-\frac {(4 c) \int \frac {x}{\left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )} \, dx}{b}\\ \end {align*}

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Mathematica [A]  time = 2.86, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c^{2} x^{2} + 1}}{a^{2} c^{6} x^{6} + 3 \, a^{2} c^{4} x^{4} + 3 \, a^{2} c^{2} x^{2} + {\left (b^{2} c^{6} x^{6} + 3 \, b^{2} c^{4} x^{4} + 3 \, b^{2} c^{2} x^{2} + b^{2}\right )} \operatorname {arsinh}\left (c x\right )^{2} + a^{2} + 2 \, {\left (a b c^{6} x^{6} + 3 \, a b c^{4} x^{4} + 3 \, a b c^{2} x^{2} + a b\right )} \operatorname {arsinh}\left (c x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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^6 + 3*a^2*c^4*x^4 + 3*a^2*c^2*x^2 + (b^2*c^6*x^6 + 3*b^2*c^4*x^4 + 3*b^2
*c^2*x^2 + b^2)*arcsinh(c*x)^2 + a^2 + 2*(a*b*c^6*x^6 + 3*a*b*c^4*x^4 + 3*a*b*c^2*x^2 + a*b)*arcsinh(c*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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)

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maple [A]  time = 0.25, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a +b \arcsinh \left (c x \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(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 \[ -\frac {c x + \sqrt {c^{2} x^{2} + 1}}{{\left (a b c^{4} x^{3} + a b c^{2} x\right )} {\left (c^{2} x^{2} + 1\right )} + {\left ({\left (b^{2} c^{4} x^{3} + b^{2} c^{2} x\right )} {\left (c^{2} x^{2} + 1\right )} + {\left (b^{2} c^{5} x^{4} + 2 \, b^{2} c^{3} x^{2} + b^{2} c\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (a b c^{5} x^{4} + 2 \, a b c^{3} x^{2} + a b c\right )} \sqrt {c^{2} x^{2} + 1}} - \int \frac {4 \, c^{4} x^{4} + 3 \, c^{2} x^{2} + {\left (4 \, c^{2} x^{2} + 1\right )} {\left (c^{2} x^{2} + 1\right )} + 4 \, {\left (2 \, c^{3} x^{3} + c x\right )} \sqrt {c^{2} x^{2} + 1} - 1}{{\left (a b c^{6} x^{6} + 2 \, a b c^{4} x^{4} + a b c^{2} x^{2}\right )} {\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} + 2 \, {\left (a b c^{7} x^{7} + 3 \, a b c^{5} x^{5} + 3 \, a b c^{3} x^{3} + a b c x\right )} {\left (c^{2} x^{2} + 1\right )} + {\left ({\left (b^{2} c^{6} x^{6} + 2 \, b^{2} c^{4} x^{4} + b^{2} c^{2} x^{2}\right )} {\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} + 2 \, {\left (b^{2} c^{7} x^{7} + 3 \, b^{2} c^{5} x^{5} + 3 \, b^{2} c^{3} x^{3} + b^{2} c x\right )} {\left (c^{2} x^{2} + 1\right )} + {\left (b^{2} c^{8} x^{8} + 4 \, b^{2} c^{6} x^{6} + 6 \, b^{2} c^{4} x^{4} + 4 \, b^{2} c^{2} x^{2} + b^{2}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (a b c^{8} x^{8} + 4 \, a b c^{6} x^{6} + 6 \, a b c^{4} x^{4} + 4 \, a b c^{2} x^{2} + a b\right )} \sqrt {c^{2} x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (c^2\,x^2+1\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/((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 \[ \int \frac {1}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \left (c^{2} x^{2} + 1\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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