3.161 \(\int \frac{x}{(a+b \cosh ^{-1}(c x))^{7/2}} \, dx\)

Optimal. Leaf size=229 \[ \frac{8 \sqrt{2 \pi } e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{15 b^{7/2} c^2}+\frac{8 \sqrt{2 \pi } e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{15 b^{7/2} c^2}+\frac{4}{15 b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac{8 x^2}{15 b^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac{32 x \sqrt{c x-1} \sqrt{c x+1}}{15 b^3 c \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{2 x \sqrt{c x-1} \sqrt{c x+1}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}} \]

[Out]

(-2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(5*b*c*(a + b*ArcCosh[c*x])^(5/2)) + 4/(15*b^2*c^2*(a + b*ArcCosh[c*x])^(3
/2)) - (8*x^2)/(15*b^2*(a + b*ArcCosh[c*x])^(3/2)) - (32*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(15*b^3*c*Sqrt[a + b*
ArcCosh[c*x]]) + (8*E^((2*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(15*b^(7/2)*c^2) +
(8*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(15*b^(7/2)*c^2*E^((2*a)/b))

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Rubi [A]  time = 0.870413, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {5668, 5775, 5666, 3307, 2180, 2204, 2205, 5676} \[ \frac{8 \sqrt{2 \pi } e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{15 b^{7/2} c^2}+\frac{8 \sqrt{2 \pi } e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{15 b^{7/2} c^2}+\frac{4}{15 b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac{8 x^2}{15 b^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac{32 x \sqrt{c x-1} \sqrt{c x+1}}{15 b^3 c \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{2 x \sqrt{c x-1} \sqrt{c x+1}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*ArcCosh[c*x])^(7/2),x]

[Out]

(-2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(5*b*c*(a + b*ArcCosh[c*x])^(5/2)) + 4/(15*b^2*c^2*(a + b*ArcCosh[c*x])^(3
/2)) - (8*x^2)/(15*b^2*(a + b*ArcCosh[c*x])^(3/2)) - (32*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(15*b^3*c*Sqrt[a + b*
ArcCosh[c*x]]) + (8*E^((2*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(15*b^(7/2)*c^2) +
(8*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(15*b^(7/2)*c^2*E^((2*a)/b))

Rule 5668

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(
a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] + (-Dist[(c*(m + 1))/(b*(n + 1)), Int[(x^(m + 1)*(a + b*ArcCosh
[c*x])^(n + 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] + Dist[m/(b*c*(n + 1)), Int[(x^(m - 1)*(a + b*ArcCosh[c
*x])^(n + 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 5775

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[((f*x)^m*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] - Dist[(f
*m)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1
, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && LtQ[n, -1] && GtQ[d1, 0] && LtQ[d2, 0]

Rule 5666

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(
a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] + Dist[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a +
b*x)^(n + 1)*Cosh[x]^(m - 1)*(m - (m + 1)*Cosh[x]^2), x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rule 3307

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d
, e, f, m}, x] && IntegerQ[2*k]

Rule 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]

Rubi steps

\begin{align*} \int \frac{x}{\left (a+b \cosh ^{-1}(c x)\right )^{7/2}} \, dx &=-\frac{2 x \sqrt{-1+c x} \sqrt{1+c x}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}}-\frac{2 \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{5/2}} \, dx}{5 b c}+\frac{(4 c) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{5/2}} \, dx}{5 b}\\ &=-\frac{2 x \sqrt{-1+c x} \sqrt{1+c x}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}}+\frac{4}{15 b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac{8 x^2}{15 b^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac{16 \int \frac{x}{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx}{15 b^2}\\ &=-\frac{2 x \sqrt{-1+c x} \sqrt{1+c x}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}}+\frac{4}{15 b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac{8 x^2}{15 b^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac{32 x \sqrt{-1+c x} \sqrt{1+c x}}{15 b^3 c \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{32 \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{15 b^3 c^2}\\ &=-\frac{2 x \sqrt{-1+c x} \sqrt{1+c x}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}}+\frac{4}{15 b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac{8 x^2}{15 b^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac{32 x \sqrt{-1+c x} \sqrt{1+c x}}{15 b^3 c \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{16 \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{15 b^3 c^2}+\frac{16 \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{15 b^3 c^2}\\ &=-\frac{2 x \sqrt{-1+c x} \sqrt{1+c x}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}}+\frac{4}{15 b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac{8 x^2}{15 b^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac{32 x \sqrt{-1+c x} \sqrt{1+c x}}{15 b^3 c \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{32 \operatorname{Subst}\left (\int e^{\frac{2 a}{b}-\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{15 b^4 c^2}+\frac{32 \operatorname{Subst}\left (\int e^{-\frac{2 a}{b}+\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{15 b^4 c^2}\\ &=-\frac{2 x \sqrt{-1+c x} \sqrt{1+c x}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}}+\frac{4}{15 b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac{8 x^2}{15 b^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac{32 x \sqrt{-1+c x} \sqrt{1+c x}}{15 b^3 c \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{8 e^{\frac{2 a}{b}} \sqrt{2 \pi } \text{erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{15 b^{7/2} c^2}+\frac{8 e^{-\frac{2 a}{b}} \sqrt{2 \pi } \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{15 b^{7/2} c^2}\\ \end{align*}

Mathematica [A]  time = 1.60954, size = 175, normalized size = 0.76 \[ \frac{\frac{\sqrt{b} \left (-\sinh \left (2 \cosh ^{-1}(c x)\right ) \left (16 \left (a+b \cosh ^{-1}(c x)\right )^2+3 b^2\right )-4 b \cosh \left (2 \cosh ^{-1}(c x)\right ) \left (a+b \cosh ^{-1}(c x)\right )\right )}{\left (a+b \cosh ^{-1}(c x)\right )^{5/2}}+8 \sqrt{2 \pi } \left (\sinh \left (\frac{2 a}{b}\right )+\cosh \left (\frac{2 a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )+8 \sqrt{2 \pi } \left (\cosh \left (\frac{2 a}{b}\right )-\sinh \left (\frac{2 a}{b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{15 b^{7/2} c^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x/(a + b*ArcCosh[c*x])^(7/2),x]

[Out]

(8*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]]*(Cosh[(2*a)/b] - Sinh[(2*a)/b]) + 8*Sqrt[2*Pi]*
Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) + (Sqrt[b]*(-4*b*(a + b*ArcCos
h[c*x])*Cosh[2*ArcCosh[c*x]] - (3*b^2 + 16*(a + b*ArcCosh[c*x])^2)*Sinh[2*ArcCosh[c*x]]))/(a + b*ArcCosh[c*x])
^(5/2))/(15*b^(7/2)*c^2)

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Maple [F]  time = 0.111, size = 0, normalized size = 0. \begin{align*} \int{x \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{-{\frac{7}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a+b*arccosh(c*x))^(7/2),x)

[Out]

int(x/(a+b*arccosh(c*x))^(7/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*arccosh(c*x))^(7/2),x, algorithm="maxima")

[Out]

integrate(x/(b*arccosh(c*x) + a)^(7/2), 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(x/(a+b*arccosh(c*x))^(7/2),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(x/(a+b*acosh(c*x))**(7/2),x)

[Out]

Timed out

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Giac [F]  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(x/(a+b*arccosh(c*x))^(7/2),x, algorithm="giac")

[Out]

sage0*x