3.399 \(\int \frac {1}{x (1+c^2 x^2)^{3/2} (a+b \sinh ^{-1}(c x))} \, dx\)

Optimal. Leaf size=30 \[ \text {Int}\left (\frac {1}{x \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.14, 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 = {} \[ \int \frac {1}{x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 1.50, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])),x]

[Out]

Integrate[1/(x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x]

________________________________________________________________________________________

fricas [A]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c^{2} x^{2} + 1}}{a c^{4} x^{5} + 2 \, a c^{2} x^{3} + a x + {\left (b c^{4} x^{5} + 2 \, b c^{2} x^{3} + b x\right )} \operatorname {arsinh}\left (c x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

integral(sqrt(c^2*x^2 + 1)/(a*c^4*x^5 + 2*a*c^2*x^3 + a*x + (b*c^4*x^5 + 2*b*c^2*x^3 + b*x)*arcsinh(c*x)), x)

________________________________________________________________________________________

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

________________________________________________________________________________________

maple [A]  time = 0.50, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \arcsinh \left (c x \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (c^2\,x^2+1\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*asinh(c*x))*(c^2*x^2 + 1)^(3/2)),x)

[Out]

int(1/(x*(a + b*asinh(c*x))*(c^2*x^2 + 1)^(3/2)), x)

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \left (c^{2} x^{2} + 1\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(c**2*x**2+1)**(3/2)/(a+b*asinh(c*x)),x)

[Out]

Integral(1/(x*(a + b*asinh(c*x))*(c**2*x**2 + 1)**(3/2)), x)

________________________________________________________________________________________