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

Optimal. Leaf size=281 \[ -\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )}{d x}-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )}{d x}+\frac{30 b^2 \cosh ^2\left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{d x}+\frac{5 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^{3/2}}{x \sqrt{d x^2} \sqrt{d x^2-2}}+x \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^{5/2} \]

[Out]

(5*b*(2*x^2 - d*x^4)*(a + b*ArcCosh[-1 + d*x^2])^(3/2))/(x*Sqrt[d*x^2]*Sqrt[-2 + d*x^2]) + x*(a + b*ArcCosh[-1
+ d*x^2])^(5/2) + (30*b^2*Sqrt[a + b*ArcCosh[-1 + d*x^2]]*Cosh[ArcCosh[-1 + d*x^2]/2]^2)/(d*x) - (15*b^(5/2)*
Sqrt[Pi/2]*Cosh[ArcCosh[-1 + d*x^2]/2]*Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)]
- Sinh[a/(2*b)]))/(d*x) - (15*b^(5/2)*Sqrt[Pi/2]*Cosh[ArcCosh[-1 + d*x^2]/2]*Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2
]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(d*x)

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Rubi [A]  time = 0.0593825, antiderivative size = 281, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {5880, 5879} \[ -\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )}{d x}-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )}{d x}+\frac{30 b^2 \cosh ^2\left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{d x}+\frac{5 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^{3/2}}{x \sqrt{d x^2} \sqrt{d x^2-2}}+x \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[-1 + d*x^2])^(5/2),x]

[Out]

(5*b*(2*x^2 - d*x^4)*(a + b*ArcCosh[-1 + d*x^2])^(3/2))/(x*Sqrt[d*x^2]*Sqrt[-2 + d*x^2]) + x*(a + b*ArcCosh[-1
+ d*x^2])^(5/2) + (30*b^2*Sqrt[a + b*ArcCosh[-1 + d*x^2]]*Cosh[ArcCosh[-1 + d*x^2]/2]^2)/(d*x) - (15*b^(5/2)*
Sqrt[Pi/2]*Cosh[ArcCosh[-1 + d*x^2]/2]*Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)]
- Sinh[a/(2*b)]))/(d*x) - (15*b^(5/2)*Sqrt[Pi/2]*Cosh[ArcCosh[-1 + d*x^2]/2]*Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2
]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(d*x)

Rule 5880

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)^2]*(b_.))^(n_), x_Symbol] :> Simp[x*(a + b*ArcCosh[c + d*x^2])^n, x] +
(Dist[4*b^2*n*(n - 1), Int[(a + b*ArcCosh[c + d*x^2])^(n - 2), x], x] - Simp[(2*b*n*(2*c*d*x^2 + d^2*x^4)*(a +
b*ArcCosh[c + d*x^2])^(n - 1))/(d*x*Sqrt[-1 + c + d*x^2]*Sqrt[1 + c + d*x^2]), x]) /; FreeQ[{a, b, c, d}, x]
&& EqQ[c^2, 1] && GtQ[n, 1]

Rule 5879

Int[Sqrt[(a_.) + ArcCosh[-1 + (d_.)*(x_)^2]*(b_.)], x_Symbol] :> Simp[(2*Sqrt[a + b*ArcCosh[-1 + d*x^2]]*Cosh[
(1/2)*ArcCosh[-1 + d*x^2]]^2)/(d*x), x] + (-Simp[(Sqrt[b]*Sqrt[Pi/2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])*Cosh[(1/2
)*ArcCosh[-1 + d*x^2]]*Erf[(1/Sqrt[2*b])*Sqrt[a + b*ArcCosh[-1 + d*x^2]]])/(d*x), x] - Simp[(Sqrt[b]*Sqrt[Pi/2
]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])*Cosh[(1/2)*ArcCosh[-1 + d*x^2]]*Erfi[(1/Sqrt[2*b])*Sqrt[a + b*ArcCosh[-1 + d
*x^2]]])/(d*x), x]) /; FreeQ[{a, b, d}, x]

Rubi steps

\begin{align*} \int \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^{5/2} \, dx &=\frac{5 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^{3/2}}{x \sqrt{d x^2} \sqrt{-2+d x^2}}+x \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^{5/2}+\left (15 b^2\right ) \int \sqrt{a+b \cosh ^{-1}\left (-1+d x^2\right )} \, dx\\ &=\frac{5 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^{3/2}}{x \sqrt{d x^2} \sqrt{-2+d x^2}}+x \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^{5/2}+\frac{30 b^2 \sqrt{a+b \cosh ^{-1}\left (-1+d x^2\right )} \cosh ^2\left (\frac{1}{2} \cosh ^{-1}\left (-1+d x^2\right )\right )}{d x}-\frac{15 b^{5/2} \sqrt{\frac{\pi }{2}} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (-1+d x^2\right )\right ) \text{erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (-1+d x^2\right )}}{\sqrt{2} \sqrt{b}}\right ) \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right )}{d x}-\frac{15 b^{5/2} \sqrt{\frac{\pi }{2}} \cosh \left (\frac{1}{2} \cosh ^{-1}\left (-1+d x^2\right )\right ) \text{erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (-1+d x^2\right )}}{\sqrt{2} \sqrt{b}}\right ) \left (\cosh \left (\frac{a}{2 b}\right )+\sinh \left (\frac{a}{2 b}\right )\right )}{d x}\\ \end{align*}

Mathematica [A]  time = 1.56875, size = 277, normalized size = 0.99 \[ \frac{\cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (4 \sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )} \left (\left (a^2+15 b^2\right ) \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right )+b \cosh ^{-1}\left (d x^2-1\right ) \left (2 a \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right )-5 b \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right )\right )-5 a b \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right )+b^2 \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \cosh ^{-1}\left (d x^2-1\right )^2\right )-15 \sqrt{2 \pi } b^{5/2} \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )-15 \sqrt{2 \pi } b^{5/2} \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt{2} \sqrt{b}}\right )\right )}{2 d x} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCosh[-1 + d*x^2])^(5/2),x]

[Out]

(Cosh[ArcCosh[-1 + d*x^2]/2]*(-15*b^(5/2)*Sqrt[2*Pi]*Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(
Cosh[a/(2*b)] - Sinh[a/(2*b)]) - 15*b^(5/2)*Sqrt[2*Pi]*Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*
(Cosh[a/(2*b)] + Sinh[a/(2*b)]) + 4*Sqrt[a + b*ArcCosh[-1 + d*x^2]]*((a^2 + 15*b^2)*Cosh[ArcCosh[-1 + d*x^2]/2
] + b^2*ArcCosh[-1 + d*x^2]^2*Cosh[ArcCosh[-1 + d*x^2]/2] - 5*a*b*Sinh[ArcCosh[-1 + d*x^2]/2] + b*ArcCosh[-1 +
d*x^2]*(2*a*Cosh[ArcCosh[-1 + d*x^2]/2] - 5*b*Sinh[ArcCosh[-1 + d*x^2]/2]))))/(2*d*x)

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\rm arccosh} \left (d{x}^{2}-1\right ) \right ) ^{{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(d*x^2-1))^(5/2),x)

[Out]

int((a+b*arccosh(d*x^2-1))^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arcosh}\left (d x^{2} - 1\right ) + a\right )}^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x^2-1))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*arccosh(d*x^2 - 1) + a)^(5/2), 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((a+b*arccosh(d*x^2-1))^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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((a+b*acosh(d*x**2-1))**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x^2-1))^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError