3.441 \(\int \frac {1}{x \sqrt {1+c^2 x^2} (a+b \sinh ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=47 \[ -\frac {\text {Int}\left (\frac {1}{x^2 \left (a+b \sinh ^{-1}(c x)\right )},x\right )}{b c}-\frac {1}{b c x \left (a+b \sinh ^{-1}(c x)\right )} \]

[Out]

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

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Rubi [A]  time = 0.15, 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 = {} \[ \int \frac {1}{x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 5.44, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c^{2} x^{2} + 1}}{a^{2} c^{2} x^{3} + a^{2} x + {\left (b^{2} c^{2} x^{3} + b^{2} x\right )} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, {\left (a b c^{2} x^{3} + a b x\right )} \operatorname {arsinh}\left (c x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c^2*x^2 + 1)/(a^2*c^2*x^3 + a^2*x + (b^2*c^2*x^3 + b^2*x)*arcsinh(c*x)^2 + 2*(a*b*c^2*x^3 + a*b*
x)*arcsinh(c*x)), x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a +b \arcsinh \left (c x \right )\right )^{2} \sqrt {c^{2} x^{2}+1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c^3*x^3 + c*x + (c^2*x^2 + 1)^(3/2))/((c^2*x^2 + 1)*a*b*c^2*x^2 + ((c^2*x^2 + 1)*b^2*c^2*x^2 + (b^2*c^3*x^3
+ b^2*c*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + (a*b*c^3*x^3 + a*b*c*x)*sqrt(c^2*x^2 + 1)) - inte
grate((c^5*x^5 + c^3*x^3 + (c^3*x^3 + 2*c*x)*(c^2*x^2 + 1) + (2*c^4*x^4 + 3*c^2*x^2 + 1)*sqrt(c^2*x^2 + 1))/((
c^2*x^2 + 1)^(3/2)*a*b*c^3*x^4 + 2*(a*b*c^4*x^5 + a*b*c^2*x^3)*(c^2*x^2 + 1) + ((c^2*x^2 + 1)^(3/2)*b^2*c^3*x^
4 + 2*(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)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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