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

Optimal. Leaf size=186 \[ \frac{2 d \text{Unintegrable}\left (\frac{1}{x^2 \sqrt{c x-1} \sqrt{c x+1} \sqrt{a+b \cosh ^{-1}(c x)}},x\right )}{b c}-\frac{\sqrt{\frac{\pi }{2}} d e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} d e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2}}+\frac{2 d (c x-1)^{3/2} (c x+1)^{3/2}}{b c x \sqrt{a+b \cosh ^{-1}(c x)}} \]

[Out]

(2*d*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(b*c*x*Sqrt[a + b*ArcCosh[c*x]]) - (d*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[
2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/b^(3/2) - (d*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]
])/(b^(3/2)*E^((2*a)/b)) + (2*d*Unintegrable[1/(x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[a + b*ArcCosh[c*x]]), x]
)/(b*c)

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Rubi [A]  time = 1.6619, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{d-c^2 d x^2}{x \left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(d - c^2*d*x^2)/(x*(a + b*ArcCosh[c*x])^(3/2)),x]

[Out]

(-2*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2))/(b*c*x*Sqrt[a + b*ArcCosh[c*x]]) - (d*E^((2*a)/b)*Sqrt[Pi/2]
*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/b^(3/2) - (d*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x
]])/Sqrt[b]])/(b^(3/2)*E^((2*a)/b)) + (2*d*Defer[Int][1/(x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[a + b*ArcCosh[c
*x]]), x])/(b*c)

Rubi steps

\begin{align*} \int \frac{d-c^2 d x^2}{x \left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac{2 d \sqrt{-1+c x} \sqrt{1+c x} \left (1-c^2 x^2\right )}{b c x \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{(2 d) \int \frac{\sqrt{-1+c x} \sqrt{1+c x}}{x^2 \sqrt{a+b \cosh ^{-1}(c x)}} \, dx}{b c}-\frac{(4 c d) \int \frac{\sqrt{-1+c x} \sqrt{1+c x}}{\sqrt{a+b \cosh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac{2 d \sqrt{-1+c x} \sqrt{1+c x} \left (1-c^2 x^2\right )}{b c x \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{(4 d) \operatorname{Subst}\left (\int \frac{\sinh ^2(x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b}-\frac{(2 d) \int \left (\frac{c^2}{\sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{1}{x^2 \sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}}\right ) \, dx}{b c}\\ &=-\frac{2 d \sqrt{-1+c x} \sqrt{1+c x} \left (1-c^2 x^2\right )}{b c x \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{(4 d) \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{a+b x}}-\frac{\cosh (2 x)}{2 \sqrt{a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b}+\frac{(2 d) \int \frac{1}{x^2 \sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}} \, dx}{b c}-\frac{(2 c d) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac{2 d \sqrt{-1+c x} \sqrt{1+c x} \left (1-c^2 x^2\right )}{b c x \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{(2 d) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b}+\frac{(2 d) \int \frac{1}{x^2 \sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}} \, dx}{b c}\\ &=-\frac{2 d \sqrt{-1+c x} \sqrt{1+c x} \left (1-c^2 x^2\right )}{b c x \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{d \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b}-\frac{d \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b}+\frac{(2 d) \int \frac{1}{x^2 \sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}} \, dx}{b c}\\ &=-\frac{2 d \sqrt{-1+c x} \sqrt{1+c x} \left (1-c^2 x^2\right )}{b c x \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{(2 d) \operatorname{Subst}\left (\int e^{\frac{2 a}{b}-\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{b^2}-\frac{(2 d) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b}+\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{b^2}+\frac{(2 d) \int \frac{1}{x^2 \sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}} \, dx}{b c}\\ &=-\frac{2 d \sqrt{-1+c x} \sqrt{1+c x} \left (1-c^2 x^2\right )}{b c x \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{d e^{\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2}}-\frac{d e^{-\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2}}+\frac{(2 d) \int \frac{1}{x^2 \sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}} \, dx}{b c}\\ \end{align*}

Mathematica [A]  time = 4.12511, size = 0, normalized size = 0. \[ \int \frac{d-c^2 d x^2}{x \left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(d - c^2*d*x^2)/(x*(a + b*ArcCosh[c*x])^(3/2)),x]

[Out]

Integrate[(d - c^2*d*x^2)/(x*(a + b*ArcCosh[c*x])^(3/2)), x]

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Maple [A]  time = 0.222, size = 0, normalized size = 0. \begin{align*} \int{\frac{-{c}^{2}d{x}^{2}+d}{x} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)/x/(a+b*arccosh(c*x))^(3/2),x)

[Out]

int((-c^2*d*x^2+d)/x/(a+b*arccosh(c*x))^(3/2),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{c^{2} d x^{2} - d}{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)/x/(a+b*arccosh(c*x))^(3/2),x, algorithm="maxima")

[Out]

-integrate((c^2*d*x^2 - d)/((b*arccosh(c*x) + a)^(3/2)*x), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)/x/(a+b*arccosh(c*x))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} - d \left (\int \frac{c^{2} x^{2}}{a x \sqrt{a + b \operatorname{acosh}{\left (c x \right )}} + b x \sqrt{a + b \operatorname{acosh}{\left (c x \right )}} \operatorname{acosh}{\left (c x \right )}}\, dx + \int - \frac{1}{a x \sqrt{a + b \operatorname{acosh}{\left (c x \right )}} + b x \sqrt{a + b \operatorname{acosh}{\left (c x \right )}} \operatorname{acosh}{\left (c x \right )}}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)/x/(a+b*acosh(c*x))**(3/2),x)

[Out]

-d*(Integral(c**2*x**2/(a*x*sqrt(a + b*acosh(c*x)) + b*x*sqrt(a + b*acosh(c*x))*acosh(c*x)), x) + Integral(-1/
(a*x*sqrt(a + b*acosh(c*x)) + b*x*sqrt(a + b*acosh(c*x))*acosh(c*x)), x))

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)/x/(a+b*arccosh(c*x))^(3/2),x, algorithm="giac")

[Out]

sage0*x