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

Optimal. Leaf size=259 \[ \frac{3 \sqrt{\frac{\pi }{2}} d e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^4}-\frac{\sqrt{\frac{3 \pi }{2}} d e^{\frac{6 a}{b}} \text{Erf}\left (\frac{\sqrt{6} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^4}+\frac{3 \sqrt{\frac{\pi }{2}} d e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^4}-\frac{\sqrt{\frac{3 \pi }{2}} d e^{-\frac{6 a}{b}} \text{Erfi}\left (\frac{\sqrt{6} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^4}+\frac{2 d x^3 (c x-1)^{3/2} (c x+1)^{3/2}}{b c \sqrt{a+b \cosh ^{-1}(c x)}} \]

[Out]

(2*d*x^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(b*c*Sqrt[a + b*ArcCosh[c*x]]) + (3*d*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(S
qrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^4) - (d*E^((6*a)/b)*Sqrt[(3*Pi)/2]*Erf[(Sqrt[6]*Sqrt[
a + b*ArcCosh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^4) + (3*d*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt
[b]])/(16*b^(3/2)*c^4*E^((2*a)/b)) - (d*Sqrt[(3*Pi)/2]*Erfi[(Sqrt[6]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(16*b
^(3/2)*c^4*E^((6*a)/b))

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Rubi [A]  time = 1.74749, antiderivative size = 269, normalized size of antiderivative = 1.04, number of steps used = 27, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {5776, 5781, 5448, 3307, 2180, 2204, 2205} \[ \frac{3 \sqrt{\frac{\pi }{2}} d e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^4}-\frac{\sqrt{\frac{3 \pi }{2}} d e^{\frac{6 a}{b}} \text{Erf}\left (\frac{\sqrt{6} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^4}+\frac{3 \sqrt{\frac{\pi }{2}} d e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^4}-\frac{\sqrt{\frac{3 \pi }{2}} d e^{-\frac{6 a}{b}} \text{Erfi}\left (\frac{\sqrt{6} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^4}-\frac{2 d x^3 \sqrt{c x-1} \sqrt{c x+1} \left (1-c^2 x^2\right )}{b c \sqrt{a+b \cosh ^{-1}(c x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2))/(b*c*Sqrt[a + b*ArcCosh[c*x]]) + (3*d*E^((2*a)/b)*Sqrt[P
i/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^4) - (d*E^((6*a)/b)*Sqrt[(3*Pi)/2]*Erf[(Sq
rt[6]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^4) + (3*d*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[
c*x]])/Sqrt[b]])/(16*b^(3/2)*c^4*E^((2*a)/b)) - (d*Sqrt[(3*Pi)/2]*Erfi[(Sqrt[6]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt
[b]])/(16*b^(3/2)*c^4*E^((6*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 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]

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{x^3 \left (d-c^2 d x^2\right )}{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac{2 d x^3 \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{(6 d) \int \frac{x^2 \sqrt{-1+c x} \sqrt{1+c x}}{\sqrt{a+b \cosh ^{-1}(c x)}} \, dx}{b c}-\frac{(12 c d) \int \frac{x^4 \sqrt{-1+c x} \sqrt{1+c x}}{\sqrt{a+b \cosh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac{2 d x^3 \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{(6 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^4}-\frac{(12 d) \operatorname{Subst}\left (\int \frac{\cosh ^4(x) \sinh ^2(x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^4}\\ &=-\frac{2 d x^3 \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{(6 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^4}-\frac{(12 d) \operatorname{Subst}\left (\int \left (-\frac{1}{16 \sqrt{a+b x}}-\frac{\cosh (2 x)}{32 \sqrt{a+b x}}+\frac{\cosh (4 x)}{16 \sqrt{a+b x}}+\frac{\cosh (6 x)}{32 \sqrt{a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^4}\\ &=-\frac{2 d x^3 \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{(3 d) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^4}-\frac{(3 d) \operatorname{Subst}\left (\int \frac{\cosh (6 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^4}\\ &=-\frac{2 d x^3 \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{(3 d) \operatorname{Subst}\left (\int \frac{e^{-6 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^4}+\frac{(3 d) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^4}+\frac{(3 d) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^4}-\frac{(3 d) \operatorname{Subst}\left (\int \frac{e^{6 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^4}\\ &=-\frac{2 d x^3 \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{(3 d) \operatorname{Subst}\left (\int e^{\frac{6 a}{b}-\frac{6 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{8 b^2 c^4}+\frac{(3 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 )}{8 b^2 c^4}+\frac{(3 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 )}{8 b^2 c^4}-\frac{(3 d) \operatorname{Subst}\left (\int e^{-\frac{6 a}{b}+\frac{6 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{8 b^2 c^4}\\ &=-\frac{2 d x^3 \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{3 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 )}{16 b^{3/2} c^4}-\frac{d e^{\frac{6 a}{b}} \sqrt{\frac{3 \pi }{2}} \text{erf}\left (\frac{\sqrt{6} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^4}+\frac{3 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 )}{16 b^{3/2} c^4}-\frac{d e^{-\frac{6 a}{b}} \sqrt{\frac{3 \pi }{2}} \text{erfi}\left (\frac{\sqrt{6} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^4}\\ \end{align*}

Mathematica [A]  time = 2.4936, size = 300, normalized size = 1.16 \[ \frac{d e^{-\frac{6 a}{b}} \left (e^{\frac{6 a}{b}} \left (-3 \sqrt{2} e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \text{Gamma}\left (\frac{1}{2},\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+\sqrt{6} e^{\frac{6 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \text{Gamma}\left (\frac{1}{2},\frac{6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-64 c^4 x^4 \sqrt{\frac{c x-1}{c x+1}}-64 c^3 x^3 \sqrt{\frac{c x-1}{c x+1}}+10 \sinh \left (2 \cosh ^{-1}(c x)\right )+8 \sinh \left (4 \cosh ^{-1}(c x)\right )+2 \sinh \left (6 \cosh ^{-1}(c x)\right )\right )-\sqrt{6} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+3 \sqrt{2} e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )}{32 b c^4 \sqrt{a+b \cosh ^{-1}(c x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d*(-(Sqrt[6]*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-6*(a + b*ArcCosh[c*x]))/b]) + 3*Sqrt[2]*E^((4*a)/b)
*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-2*(a + b*ArcCosh[c*x]))/b] + E^((6*a)/b)*(-64*c^3*x^3*Sqrt[(-1 +
c*x)/(1 + c*x)] - 64*c^4*x^4*Sqrt[(-1 + c*x)/(1 + c*x)] - 3*Sqrt[2]*E^((2*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamm
a[1/2, (2*(a + b*ArcCosh[c*x]))/b] + Sqrt[6]*E^((6*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (6*(a + b*ArcCosh
[c*x]))/b] + 10*Sinh[2*ArcCosh[c*x]] + 8*Sinh[4*ArcCosh[c*x]] + 2*Sinh[6*ArcCosh[c*x]])))/(32*b*c^4*E^((6*a)/b
)*Sqrt[a + b*ArcCosh[c*x]])

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Maple [F]  time = 0.237, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \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^3*(-c^2*d*x^2+d)/(a+b*arccosh(c*x))^(3/2),x)

[Out]

int(x^3*(-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^{3}}{{\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^3*(-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^3/(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^3*(-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^{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 + \int \frac{c^{2} x^{5}}{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**3*(-c**2*d*x**2+d)/(a+b*acosh(c*x))**(3/2),x)

[Out]

-d*(Integral(-x**3/(a*sqrt(a + b*acosh(c*x)) + b*sqrt(a + b*acosh(c*x))*acosh(c*x)), x) + Integral(c**2*x**5/(
a*sqrt(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^3*(-c^2*d*x^2+d)/(a+b*arccosh(c*x))^(3/2),x, algorithm="giac")

[Out]

sage0*x