3.2.58 \(\int \frac {x}{(a+b \cosh ^{-1}(c x))^{5/2}} \, dx\) [158]

Optimal. Leaf size=188 \[ -\frac {2 x \sqrt {-1+c x} \sqrt {1+c x}}{3 b c \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac {4}{3 b^2 c^2 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {8 x^2}{3 b^2 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {2 e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^2}+\frac {2 e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^2} \]

[Out]

-2/3*exp(2*a/b)*erf(2^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(5/2)/c^2+2/3*erfi(2^(1/2)*(a
+b*arccosh(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(5/2)/c^2/exp(2*a/b)-2/3*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/
(a+b*arccosh(c*x))^(3/2)+4/3/b^2/c^2/(a+b*arccosh(c*x))^(1/2)-8/3*x^2/b^2/(a+b*arccosh(c*x))^(1/2)

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Rubi [A]
time = 0.56, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5886, 5951, 5887, 5556, 12, 3389, 2211, 2236, 2235, 5893} \begin {gather*} -\frac {2 \sqrt {2 \pi } e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^2}+\frac {2 \sqrt {2 \pi } e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^2}+\frac {4}{3 b^2 c^2 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {8 x^2}{3 b^2 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{3 b c \left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(a + b*ArcCosh[c*x])^(5/2),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2211

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] &&  !TrueQ[$UseGamma]

Rule 2235

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 2236

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 3389

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

Rule 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5886

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 5887

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

Rule 5893

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

Rule 5951

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*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 +
e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], x] - Dist[f*(m/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]
]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], 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] && EqQ[e2, (-c)*d2] && LtQ[n, -1]

Rubi steps

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

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Mathematica [A]
time = 1.08, size = 157, normalized size = 0.84 \begin {gather*} \frac {2 \sqrt {2 \pi } \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )-\sinh \left (\frac {2 a}{b}\right )\right )-2 \sqrt {2 \pi } \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )-\frac {\sqrt {b} \left (4 \left (a+b \cosh ^{-1}(c x)\right ) \cosh \left (2 \cosh ^{-1}(c x)\right )+b \sinh \left (2 \cosh ^{-1}(c x)\right )\right )}{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}}}{3 b^{5/2} c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]]*(Cosh[(2*a)/b] - Sinh[(2*a)/b]) - 2*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*(a + b*ArcCosh[c
*x])*Cosh[2*ArcCosh[c*x]] + b*Sinh[2*ArcCosh[c*x]]))/(a + b*ArcCosh[c*x])^(3/2))/(3*b^(5/2)*c^2)

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x}{\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{\frac {5}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/(a + b*acosh(c*x))**(5/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*acosh(c*x))^(5/2),x)

[Out]

int(x/(a + b*acosh(c*x))^(5/2), x)

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