3.433 \(\int \frac{(d-c^2 d x^2)^{5/2} (a+b \cosh ^{-1}(c x))^n}{x^2} \, dx\)

Optimal. Leaf size=485 \[ d^3 \text{Unintegrable}\left (\frac{\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \sqrt{d-c^2 d x^2}},x\right )-\frac{c d^3 2^{-2 (n+3)} e^{-\frac{4 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{\sqrt{d-c^2 d x^2}}+\frac{c d^3 2^{-n-2} e^{-\frac{2 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{\sqrt{d-c^2 d x^2}}-\frac{c d^3 2^{-n-2} e^{\frac{2 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{\sqrt{d-c^2 d x^2}}+\frac{c d^3 2^{-2 (n+3)} e^{\frac{4 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{\sqrt{d-c^2 d x^2}}-\frac{15 c d^3 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{8 b (n+1) \sqrt{d-c^2 d x^2}} \]

[Out]

(-15*c*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(1 + n))/(8*b*(1 + n)*Sqrt[d - c^2*d*x^2]) - (c*d
^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcCosh[c*x]))/b])/(2^(2*(3 + n
))*E^((4*a)/b)*Sqrt[d - c^2*d*x^2]*(-((a + b*ArcCosh[c*x])/b))^n) + (2^(-2 - n)*c*d^3*Sqrt[-1 + c*x]*Sqrt[1 +
c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]))/b])/(E^((2*a)/b)*Sqrt[d - c^2*d*x^2]*(-((a
+ b*ArcCosh[c*x])/b))^n) - (2^(-2 - n)*c*d^3*E^((2*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*G
amma[1 + n, (2*(a + b*ArcCosh[c*x]))/b])/(Sqrt[d - c^2*d*x^2]*((a + b*ArcCosh[c*x])/b)^n) + (c*d^3*E^((4*a)/b)
*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (4*(a + b*ArcCosh[c*x]))/b])/(2^(2*(3 + n))*
Sqrt[d - c^2*d*x^2]*((a + b*ArcCosh[c*x])/b)^n) + d^3*Unintegrable[(a + b*ArcCosh[c*x])^n/(x^2*Sqrt[d - c^2*d*
x^2]), x]

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Rubi [A]  time = 2.18818, 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 = {} \[ \int \frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^n}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^n)/x^2,x]

[Out]

(15*c*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^(1 + n))/(8*b*(1 + n)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (c*d^
2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcCosh[c*x]))/b])/(2^(2*(3 + n))*E^((4*a
)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) - (2^(-2 - n)*c*d^2*Sqrt[d - c^2*d*x^2]*(a +
b*ArcCosh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]))/b])/(E^((2*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a +
b*ArcCosh[c*x])/b))^n) + (2^(-2 - n)*c*d^2*E^((2*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n
, (2*(a + b*ArcCosh[c*x]))/b])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) - (c*d^2*E^((4*a)/b)*
Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (4*(a + b*ArcCosh[c*x]))/b])/(2^(2*(3 + n))*Sqrt[-1 +
c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) - (d^2*Sqrt[d - c^2*d*x^2]*Defer[Int][(a + b*ArcCosh[c*x])^n/(x
^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rubi steps

\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^n}{x^2} \, dx &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{(-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^n}{x^2} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{3 c^4 x^2 \left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{c^6 x^4 \left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt{-1+c x} \sqrt{1+c x}}\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 c^4 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c^6 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{3 c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{b (1+n) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \cosh ^4(x) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \cosh ^2(x) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{3 c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{b (1+n) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{3}{8} (a+b x)^n+\frac{1}{2} (a+b x)^n \cosh (2 x)+\frac{1}{8} (a+b x)^n \cosh (4 x)\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2} (a+b x)^n+\frac{1}{2} (a+b x)^n \cosh (2 x)\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{15 c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{8 b (1+n) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \cosh (4 x) \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \cosh (2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \cosh (2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{15 c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{8 b (1+n) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{-4 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{16 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{4 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{16 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{-2 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{2 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{-2 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{2 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{4 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{15 c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{8 b (1+n) \sqrt{-1+c x} \sqrt{1+c x}}+\frac{4^{-3-n} c d^2 e^{-\frac{4 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{2^{-2-n} c d^2 e^{-\frac{2 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{2^{-2-n} c d^2 e^{\frac{2 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{4^{-3-n} c d^2 e^{\frac{4 a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 0.584664, size = 0, normalized size = 0. \[ \int \frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^n}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^n)/x^2,x]

[Out]

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^n)/x^2, x]

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Maple [A]  time = 0.27, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{n}}{{x}^{2}} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^n/x^2,x)

[Out]

int((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^n/x^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{n}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^n/x^2,x, algorithm="maxima")

[Out]

integrate((-c^2*d*x^2 + d)^(5/2)*(b*arccosh(c*x) + a)^n/x^2, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}\right )} \sqrt{-c^{2} d x^{2} + d}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{n}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^n/x^2,x, algorithm="fricas")

[Out]

integral((c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2)*sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)^n/x^2, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x))**n/x**2,x)

[Out]

Timed out

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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)^(5/2)*(a+b*arccosh(c*x))^n/x^2,x, algorithm="giac")

[Out]

sage0*x