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

Optimal. Leaf size=241 \[ -\frac{\sqrt{\pi } d e^{\frac{4 a}{b}} \text{Erf}\left (\frac{2 \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c^2}+\frac{\sqrt{\frac{\pi }{2}} d e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{3/2} c^2}-\frac{\sqrt{\pi } d e^{-\frac{4 a}{b}} \text{Erfi}\left (\frac{2 \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c^2}+\frac{\sqrt{\frac{\pi }{2}} d e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{3/2} c^2}+\frac{2 d x (c x-1)^{3/2} (c x+1)^{3/2}}{b c \sqrt{a+b \cosh ^{-1}(c x)}} \]

[Out]

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

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Rubi [A]  time = 1.15412, antiderivative size = 251, normalized size of antiderivative = 1.04, number of steps used = 17, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {5776, 5701, 3312, 3307, 2180, 2204, 2205, 5781, 5448} \[ -\frac{\sqrt{\pi } d e^{\frac{4 a}{b}} \text{Erf}\left (\frac{2 \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c^2}+\frac{\sqrt{\frac{\pi }{2}} d e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{3/2} c^2}-\frac{\sqrt{\pi } d e^{-\frac{4 a}{b}} \text{Erfi}\left (\frac{2 \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c^2}+\frac{\sqrt{\frac{\pi }{2}} d e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{3/2} c^2}-\frac{2 d x \sqrt{c x-1} \sqrt{c x+1} \left (1-c^2 x^2\right )}{b c \sqrt{a+b \cosh ^{-1}(c x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5776

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp
[((f*x)^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] + (Dist[(
f*m*(-d)^p)/(b*c*(n + 1)), Int[(f*x)^(m - 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n
+ 1), x], x] - Dist[(c*(-d)^p*(m + 2*p + 1))/(b*f*(n + 1)), Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(
p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && LtQ[n,
-1] && IGtQ[m, -3] && IGtQ[p, 0]

Rule 5701

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbo
l] :> Dist[(-(d1*d2))^p/c, Subst[Int[(a + b*x)^n*Sinh[x]^(2*p + 1), x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c
, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && IGtQ[p + 1/2, 0] && (GtQ[d1, 0] && LtQ[d2, 0]
)

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 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 5781

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(-(d1*d2))^p/c^(m + 1), Subst[Int[(a + b*x)^n*Cosh[x]^m*Sinh[x]^(2*p + 1), x], x, ArcCos
h[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p
+ 1/2] && GtQ[p, -1] && IGtQ[m, 0] && (GtQ[d1, 0] && LtQ[d2, 0])

Rule 5448

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]

Rubi steps

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

Mathematica [A]  time = 4.05243, size = 331, normalized size = 1.37 \[ \frac{d e^{-\frac{4 a}{b}} \left (\frac{\sqrt{b} \left (-\sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-\sqrt{2} e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+e^{\frac{4 a}{b}} \left (\sqrt{2} e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \text{Gamma}\left (\frac{1}{2},\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+e^{\frac{4 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \text{Gamma}\left (\frac{1}{2},\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+8 c x \left (\frac{c x-1}{c x+1}\right )^{3/2} (c x+1)^3\right )\right )}{\sqrt{a+b \cosh ^{-1}(c x)}}+2 \sqrt{2 \pi } e^{\frac{6 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )+2 \sqrt{2 \pi } e^{\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )\right )}{4 b^{3/2} c^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d*(2*E^((6*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]] + 2*E^((2*a)/b)*Sqrt[2*Pi]*Erfi[(
Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]] + (Sqrt[b]*(-(Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-4*(a + b
*ArcCosh[c*x]))/b]) - Sqrt[2]*E^((2*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-2*(a + b*ArcCosh[c*x]))
/b] + E^((4*a)/b)*(8*c*x*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3 + Sqrt[2]*E^((2*a)/b)*Sqrt[a/b + ArcCosh[c*x
]]*Gamma[1/2, (2*(a + b*ArcCosh[c*x]))/b] + E^((4*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (4*(a + b*ArcCosh[
c*x]))/b])))/Sqrt[a + b*ArcCosh[c*x]]))/(4*b^(3/2)*c^2*E^((4*a)/b))

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Maple [F]  time = 0.205, size = 0, normalized size = 0. \begin{align*} \int{x \left ( -{c}^{2}d{x}^{2}+d \right ) \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (c^{2} d x^{2} - d\right )} x}{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - d \left (\int - \frac{x}{a \sqrt{a + b \operatorname{acosh}{\left (c x \right )}} + b \sqrt{a + b \operatorname{acosh}{\left (c x \right )}} \operatorname{acosh}{\left (c x \right )}}\, dx + \int \frac{c^{2} x^{3}}{a \sqrt{a + b \operatorname{acosh}{\left (c x \right )}} + b \sqrt{a + b \operatorname{acosh}{\left (c x \right )}} \operatorname{acosh}{\left (c x \right )}}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d*(Integral(-x/(a*sqrt(a + b*acosh(c*x)) + b*sqrt(a + b*acosh(c*x))*acosh(c*x)), x) + Integral(c**2*x**3/(a*s
qrt(a + b*acosh(c*x)) + b*sqrt(a + b*acosh(c*x))*acosh(c*x)), x))

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

[Out]

sage0*x