3.4.69 \(\int \frac {(1+c^2 x^2)^{3/2}}{x^2 (a+b \sinh ^{-1}(c x))} \, dx\) [369]

Optimal. Leaf size=106 \[ \frac {c \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{2 b}+\frac {3 c \log \left (a+b \sinh ^{-1}(c x)\right )}{2 b}-\frac {c \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{2 b}+\text {Int}\left (\frac {1}{x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.38, 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^2 \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^2*(a + b*ArcSinh[c*x])),x]

[Out]

(c*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(2*b) + (3*c*Log[a + b*ArcSinh[c*x]])/(2*b) - (c*Si
nh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(2*b) + Defer[Int][1/(x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSi
nh[c*x])), x]

Rubi steps

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

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Mathematica [A]
time = 1.03, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x^2 \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^2*(a + b*ArcSinh[c*x])),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

[Out]

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

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

[Out]

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

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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^2\,\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^2*(a + b*asinh(c*x))),x)

[Out]

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

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