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

Optimal. Leaf size=297 \[ -\frac{\sqrt{1-c x} \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{8 b c^2 \sqrt{c x-1}}+\frac{3 \sqrt{1-c x} \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b c^2 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \cosh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b c^2 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{8 b c^2 \sqrt{c x-1}}-\frac{3 \sqrt{1-c x} \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b c^2 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \sinh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b c^2 \sqrt{c x-1}} \]

[Out]

-(Sqrt[1 - c*x]*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(8*b*c^2*Sqrt[-1 + c*x]) + (3*Sqrt[1 - c*x]*Co
sh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(16*b*c^2*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Cosh[(5*a)/b]
*CoshIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(16*b*c^2*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Sinh[a/b]*SinhIntegral[(
a + b*ArcCosh[c*x])/b])/(8*b*c^2*Sqrt[-1 + c*x]) - (3*Sqrt[1 - c*x]*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCo
sh[c*x]))/b])/(16*b*c^2*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b
])/(16*b*c^2*Sqrt[-1 + c*x])

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Rubi [A]  time = 0.675342, antiderivative size = 371, normalized size of antiderivative = 1.25, number of steps used = 13, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {5798, 5781, 5448, 3303, 3298, 3301} \[ -\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{8 b c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 \sqrt{1-c^2 x^2} \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{16 b c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c x)\right )}{16 b c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{8 b c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 \sqrt{1-c^2 x^2} \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{16 b c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c x)\right )}{16 b c^2 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

Int[(x*(1 - c^2*x^2)^(3/2))/(a + b*ArcCosh[c*x]),x]

[Out]

-(Sqrt[1 - c^2*x^2]*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]])/(8*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*Sq
rt[1 - c^2*x^2]*Cosh[(3*a)/b]*CoshIntegral[(3*a)/b + 3*ArcCosh[c*x]])/(16*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
- (Sqrt[1 - c^2*x^2]*Cosh[(5*a)/b]*CoshIntegral[(5*a)/b + 5*ArcCosh[c*x]])/(16*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c
*x]) + (Sqrt[1 - c^2*x^2]*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]])/(8*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) -
(3*Sqrt[1 - c^2*x^2]*Sinh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcCosh[c*x]])/(16*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 +
c*x]) + (Sqrt[1 - c^2*x^2]*Sinh[(5*a)/b]*SinhIntegral[(5*a)/b + 5*ArcCosh[c*x]])/(16*b*c^2*Sqrt[-1 + c*x]*Sqrt
[1 + c*x])

Rule 5798

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

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 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3298

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(I*SinhIntegral[(c*f*fz)
/d + f*fz*x])/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rubi steps

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

Mathematica [A]  time = 0.663466, size = 172, normalized size = 0.58 \[ \frac{\sqrt{1-c^2 x^2} \left (-2 \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )+3 \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-\cosh \left (\frac{5 a}{b}\right ) \text{Chi}\left (5 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+2 \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )-3 \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+\sinh \left (\frac{5 a}{b}\right ) \text{Shi}\left (5 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )\right )}{16 c^2 \sqrt{\frac{c x-1}{c x+1}} (b c x+b)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x*(1 - c^2*x^2)^(3/2))/(a + b*ArcCosh[c*x]),x]

[Out]

(Sqrt[1 - c^2*x^2]*(-2*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]] + 3*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcC
osh[c*x])] - Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcCosh[c*x])] + 2*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]]
- 3*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])]))
/(16*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(b + b*c*x))

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Maple [B]  time = 0.185, size = 543, normalized size = 1.8 \begin{align*} -{\frac{1}{ \left ( 32\,cx+32 \right ){c}^{2} \left ( cx-1 \right ) b}\sqrt{-{c}^{2}{x}^{2}+1} \left ( -\sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ){\it Ei} \left ( 1,5\,{\rm arccosh} \left (cx\right )+5\,{\frac{a}{b}} \right ){{\rm e}^{{\frac{b{\rm arccosh} \left (cx\right )+5\,a}{b}}}}}-{\frac{1}{ \left ( 32\,cx+32 \right ){c}^{2} \left ( cx-1 \right ) b}\sqrt{-{c}^{2}{x}^{2}+1} \left ( -\sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ){\it Ei} \left ( 1,-5\,{\rm arccosh} \left (cx\right )-5\,{\frac{a}{b}} \right ){{\rm e}^{{\frac{b{\rm arccosh} \left (cx\right )-5\,a}{b}}}}}+{\frac{3}{ \left ( 32\,cx+32 \right ){c}^{2} \left ( cx-1 \right ) b}\sqrt{-{c}^{2}{x}^{2}+1} \left ( -\sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ){\it Ei} \left ( 1,3\,{\rm arccosh} \left (cx\right )+3\,{\frac{a}{b}} \right ){{\rm e}^{{\frac{b{\rm arccosh} \left (cx\right )+3\,a}{b}}}}}-{\frac{1}{ \left ( 16\,cx+16 \right ){c}^{2} \left ( cx-1 \right ) b}\sqrt{-{c}^{2}{x}^{2}+1} \left ( -\sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ){\it Ei} \left ( 1,-{\rm arccosh} \left (cx\right )-{\frac{a}{b}} \right ){{\rm e}^{{\frac{b{\rm arccosh} \left (cx\right )-a}{b}}}}}+{\frac{3}{ \left ( 32\,cx+32 \right ){c}^{2} \left ( cx-1 \right ) b}\sqrt{-{c}^{2}{x}^{2}+1} \left ( -\sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ){\it Ei} \left ( 1,-3\,{\rm arccosh} \left (cx\right )-3\,{\frac{a}{b}} \right ){{\rm e}^{{\frac{b{\rm arccosh} \left (cx\right )-3\,a}{b}}}}}-{\frac{1}{ \left ( 16\,cx+16 \right ){c}^{2} \left ( cx-1 \right ) b}\sqrt{-{c}^{2}{x}^{2}+1} \left ( -\sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ){\it Ei} \left ( 1,{\rm arccosh} \left (cx\right )+{\frac{a}{b}} \right ){{\rm e}^{{\frac{a+b{\rm arccosh} \left (cx\right )}{b}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x)),x)

[Out]

-1/32*(-c^2*x^2+1)^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*Ei(1,5*arccosh(c*x)+5*a/b)*exp((b*arccos
h(c*x)+5*a)/b)/(c*x+1)/c^2/(c*x-1)/b-1/32*(-c^2*x^2+1)^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*Ei(1
,-5*arccosh(c*x)-5*a/b)*exp((b*arccosh(c*x)-5*a)/b)/(c*x+1)/c^2/(c*x-1)/b+3/32*(-c^2*x^2+1)^(1/2)*(-(c*x+1)^(1
/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*Ei(1,3*arccosh(c*x)+3*a/b)*exp((b*arccosh(c*x)+3*a)/b)/(c*x+1)/c^2/(c*x-1)/b-
1/16*(-c^2*x^2+1)^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*Ei(1,-arccosh(c*x)-a/b)*exp((b*arccosh(c*
x)-a)/b)/(c*x+1)/c^2/(c*x-1)/b+3/32*(-c^2*x^2+1)^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*Ei(1,-3*ar
ccosh(c*x)-3*a/b)*exp((b*arccosh(c*x)-3*a)/b)/(c*x+1)/c^2/(c*x-1)/b-1/16*(-c^2*x^2+1)^(1/2)*(-(c*x+1)^(1/2)*(c
*x-1)^(1/2)*x*c+c^2*x^2-1)*Ei(1,arccosh(c*x)+a/b)*exp((a+b*arccosh(c*x))/b)/(c*x+1)/c^2/(c*x-1)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} x}{b \operatorname{arcosh}\left (c x\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

integrate((-c^2*x^2 + 1)^(3/2)*x/(b*arccosh(c*x) + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (c^{2} x^{3} - x\right )} \sqrt{-c^{2} x^{2} + 1}}{b \operatorname{arcosh}\left (c x\right ) + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

integral(-(c^2*x^3 - x)*sqrt(-c^2*x^2 + 1)/(b*arccosh(c*x) + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{2}}}{a + b \operatorname{acosh}{\left (c x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-c**2*x**2+1)**(3/2)/(a+b*acosh(c*x)),x)

[Out]

Integral(x*(-(c*x - 1)*(c*x + 1))**(3/2)/(a + b*acosh(c*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} x}{b \operatorname{arcosh}\left (c x\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x)),x, algorithm="giac")

[Out]

integrate((-c^2*x^2 + 1)^(3/2)*x/(b*arccosh(c*x) + a), x)