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

Optimal. Leaf size=262 \[ \frac{\sqrt{\pi } e^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 b^{3/2} d}+\frac{\sqrt{3 \pi } e^2 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 b^{3/2} d}+\frac{\sqrt{\pi } e^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 b^{3/2} d}+\frac{\sqrt{3 \pi } e^2 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 b^{3/2} d}-\frac{2 e^2 \sqrt{c+d x-1} (c+d x)^2 \sqrt{c+d x+1}}{b d \sqrt{a+b \cosh ^{-1}(c+d x)}} \]

[Out]

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

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Rubi [A]  time = 0.449567, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {5866, 12, 5666, 3307, 2180, 2204, 2205} \[ \frac{\sqrt{\pi } e^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 b^{3/2} d}+\frac{\sqrt{3 \pi } e^2 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 b^{3/2} d}+\frac{\sqrt{\pi } e^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 b^{3/2} d}+\frac{\sqrt{3 \pi } e^2 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 b^{3/2} d}-\frac{2 e^2 \sqrt{c+d x-1} (c+d x)^2 \sqrt{c+d x+1}}{b d \sqrt{a+b \cosh ^{-1}(c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5866

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rule 12

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

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]

Rubi steps

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

Mathematica [A]  time = 1.54405, size = 265, normalized size = 1.01 \[ \frac{e^2 e^{-\frac{3 a}{b}} \left (-e^{\frac{4 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c+d x)} \text{Gamma}\left (\frac{1}{2},\frac{a}{b}+\cosh ^{-1}(c+d x)\right )+\sqrt{3} \sqrt{-\frac{a+b \cosh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )+e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )-\sqrt{3} e^{\frac{6 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c+d x)} \text{Gamma}\left (\frac{1}{2},\frac{3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )-2 e^{\frac{3 a}{b}} \left (\sqrt{\frac{c+d x-1}{c+d x+1}} (c+d x+1)+\sinh \left (3 \cosh ^{-1}(c+d x)\right )\right )\right )}{4 b d \sqrt{a+b \cosh ^{-1}(c+d x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(e^2*(-(E^((4*a)/b)*Sqrt[a/b + ArcCosh[c + d*x]]*Gamma[1/2, a/b + ArcCosh[c + d*x]]) + Sqrt[3]*Sqrt[-((a + b*A
rcCosh[c + d*x])/b)]*Gamma[1/2, (-3*(a + b*ArcCosh[c + d*x]))/b] + E^((2*a)/b)*Sqrt[-((a + b*ArcCosh[c + d*x])
/b)]*Gamma[1/2, -((a + b*ArcCosh[c + d*x])/b)] - Sqrt[3]*E^((6*a)/b)*Sqrt[a/b + ArcCosh[c + d*x]]*Gamma[1/2, (
3*(a + b*ArcCosh[c + d*x]))/b] - 2*E^((3*a)/b)*(Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x) + Sinh[3*ArcC
osh[c + d*x]])))/(4*b*d*E^((3*a)/b)*Sqrt[a + b*ArcCosh[c + d*x]])

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Maple [F]  time = 0.237, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dex+ce \right ) ^{2} \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e**2*(Integral(c**2/(a*sqrt(a + b*acosh(c + d*x)) + b*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x)), x) + Integra
l(d**2*x**2/(a*sqrt(a + b*acosh(c + d*x)) + b*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x)), x) + Integral(2*c*d*
x/(a*sqrt(a + b*acosh(c + d*x)) + b*sqrt(a + b*acosh(c + d*x))*acosh(c + d*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((d*e*x+c*e)^2/(a+b*arccosh(d*x+c))^(3/2),x, algorithm="giac")

[Out]

sage0*x