3.155 \(\int \frac {x}{(a+b \cosh ^{-1}(c x))^{3/2}} \, dx\)

Optimal. Leaf size=140 \[ \frac {\sqrt {\frac {\pi }{2}} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^2}+\frac {\sqrt {\frac {\pi }{2}} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^2}-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \cosh ^{-1}(c x)}} \]

[Out]

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

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Rubi [A]  time = 0.15, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5666, 3307, 2180, 2204, 2205} \[ \frac {\sqrt {\frac {\pi }{2}} e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^2}+\frac {\sqrt {\frac {\pi }{2}} e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^2}-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \cosh ^{-1}(c x)}} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*ArcCosh[c*x])^(3/2),x]

[Out]

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

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 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 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]

Rubi steps

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

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Mathematica [A]  time = 1.29, size = 135, normalized size = 0.96 \[ \frac {\sqrt {2 \pi } \left (\sinh \left (\frac {2 a}{b}\right )+\cosh \left (\frac {2 a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } \left (\cosh \left (\frac {2 a}{b}\right )-\sinh \left (\frac {2 a}{b}\right )\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )-\frac {2 \sqrt {b} \sinh \left (2 \cosh ^{-1}(c x)\right )}{\sqrt {a+b \cosh ^{-1}(c x)}}}{2 b^{3/2} c^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x/(a + b*ArcCosh[c*x])^(3/2),x]

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*arccosh(c*x))^(3/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*arccosh(c*x))^(3/2),x, algorithm="giac")

[Out]

integrate(x/(b*arccosh(c*x) + a)^(3/2), x)

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maple [F]  time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a+b*arccosh(c*x))^(3/2),x)

[Out]

int(x/(a+b*arccosh(c*x))^(3/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*arccosh(c*x))^(3/2),x, algorithm="maxima")

[Out]

integrate(x/(b*arccosh(c*x) + a)^(3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*acosh(c*x))^(3/2),x)

[Out]

int(x/(a + b*acosh(c*x))^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*acosh(c*x))**(3/2),x)

[Out]

Integral(x/(a + b*acosh(c*x))**(3/2), x)

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