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

Optimal. Leaf size=280 \[ \frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )}{d x}-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )}{d x}+\frac{30 b^2 \sinh ^2\left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{d x}-\frac{5 b \left (d x^4+2 x^2\right ) \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^{3/2}}{x \sqrt{d x^2} \sqrt{d x^2+2}}+x \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^{5/2} \]

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

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Rubi [A]  time = 0.108965, antiderivative size = 280, 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, 5878} \[ \frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )}{d x}-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )}{d x}+\frac{30 b^2 \sinh ^2\left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{d x}-\frac{5 b \left (d x^4+2 x^2\right ) \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^{3/2}}{x \sqrt{d x^2} \sqrt{d x^2+2}}+x \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^{5/2} \]

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) - (15*b^(5/2)*Sqrt[Pi/2]*Erfi[Sqrt[a + b*ArcCosh[1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)]
- Sinh[a/(2*b)])*Sinh[ArcCosh[1 + d*x^2]/2])/(d*x) + (15*b^(5/2)*Sqrt[Pi/2]*Erf[Sqrt[a + b*ArcCosh[1 + d*x^2]]
/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])*Sinh[ArcCosh[1 + d*x^2]/2])/(d*x) + (30*b^2*Sqrt[a + b*Arc
Cosh[1 + d*x^2]]*Sinh[ArcCosh[1 + d*x^2]/2]^2)/(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 5878

Int[Sqrt[(a_.) + ArcCosh[1 + (d_.)*(x_)^2]*(b_.)], x_Symbol] :> Simp[(2*Sqrt[a + b*ArcCosh[1 + d*x^2]]*Sinh[(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)])*Sinh[(1/2)*Ar
cCosh[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]*(Cos
h[a/(2*b)] - Sinh[a/(2*b)])*Sinh[(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{15 b^{5/2} \sqrt{\frac{\pi }{2}} \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 ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (1+d x^2\right )\right )}{d x}+\frac{15 b^{5/2} \sqrt{\frac{\pi }{2}} \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 ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (1+d x^2\right )\right )}{d x}+\frac{30 b^2 \sqrt{a+b \cosh ^{-1}\left (1+d x^2\right )} \sinh ^2\left (\frac{1}{2} \cosh ^{-1}\left (1+d x^2\right )\right )}{d x}\\ \end{align*}

Mathematica [A]  time = 3.30582, size = 311, normalized size = 1.11 \[ \frac{x \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \left (4 \sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )} \left (\left (a^2+15 b^2\right ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right )-5 a b \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right )-b \cosh ^{-1}\left (d x^2+1\right ) \left (5 b \cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right )-2 a \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right )\right )+b^2 \cosh ^{-1}\left (d x^2+1\right )^2 \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right )\right )+15 \sqrt{2 \pi } b^{5/2} \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )-15 \sqrt{2 \pi } b^{5/2} \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )\right )}{2 \sqrt{d x^2} \sqrt{\frac{d x^2}{d x^2+2}} \sqrt{d x^2+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.075, 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