3.4.68 \(\int \frac {(1+c^2 x^2)^{3/2}}{x (a+b \sinh ^{-1}(c x))} \, dx\) [368]

Optimal. Leaf size=139 \[ -\frac {5 \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{4 b}-\frac {\text {Chi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b}+\frac {5 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{4 b}+\frac {\cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{4 b}+\text {Int}\left (\frac {1}{x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )},x\right ) \]

[Out]

5/4*cosh(a/b)*Shi((a+b*arcsinh(c*x))/b)/b+1/4*cosh(3*a/b)*Shi(3*(a+b*arcsinh(c*x))/b)/b-5/4*Chi((a+b*arcsinh(c
*x))/b)*sinh(a/b)/b-1/4*Chi(3*(a+b*arcsinh(c*x))/b)*sinh(3*a/b)/b+Unintegrable(1/x/(a+b*arcsinh(c*x))/(c^2*x^2
+1)^(1/2),x)

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Rubi [A]
time = 0.50, 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 = {} \begin {gather*} \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x \left (a+b \sinh ^{-1}(c x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-5*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(4*b) - (CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b]*Sinh[(3*
a)/b])/(4*b) + (5*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(4*b) + (Cosh[(3*a)/b]*SinhIntegral[(3*(a +
b*ArcSinh[c*x]))/b])/(4*b) + Defer[Int][1/(x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x]

Rubi steps

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

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Mathematica [A]
time = 1.40, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x \left (a+b \sinh ^{-1}(c x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{x \left (a +b \arcsinh \left (c x \right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c^2\,x^2+1\right )}^{3/2}}{x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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