3.3.48 \(\int \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3 \, dx\) [248]

Optimal. Leaf size=231 \[ -\frac {3 a x^2 \sqrt {c-a^2 c x^2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{4} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)+\frac {3 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{8 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{4 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^4}{8 a \sqrt {-1+a x} \sqrt {1+a x}} \]

[Out]

3/4*x*arccosh(a*x)*(-a^2*c*x^2+c)^(1/2)+1/2*x*arccosh(a*x)^3*(-a^2*c*x^2+c)^(1/2)-3/8*a*x^2*(-a^2*c*x^2+c)^(1/
2)/(a*x-1)^(1/2)/(a*x+1)^(1/2)+3/8*arccosh(a*x)^2*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)-3/4*a*x^2
*arccosh(a*x)^2*(-a^2*c*x^2+c)^(1/2)/(a*x-1)^(1/2)/(a*x+1)^(1/2)-1/8*arccosh(a*x)^4*(-a^2*c*x^2+c)^(1/2)/a/(a*
x-1)^(1/2)/(a*x+1)^(1/2)

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Rubi [A]
time = 0.27, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5895, 5893, 5883, 5939, 30} \begin {gather*} -\frac {3 a x^2 \sqrt {c-a^2 c x^2}}{8 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^4}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{4 \sqrt {a x-1} \sqrt {a x+1}}+\frac {3 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {3}{4} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^3,x]

[Out]

(-3*a*x^2*Sqrt[c - a^2*c*x^2])/(8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x])/4 + (
3*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^2)/(8*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (3*a*x^2*Sqrt[c - a^2*c*x^2]*ArcCos
h[a*x]^2)/(4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^3)/2 - (Sqrt[c - a^2*c*x^2]*A
rcCosh[a*x]^4)/(8*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 5883

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcC
osh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5893

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*Arc
Cosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && NeQ[n
, -1]

Rule 5895

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[x*Sqrt[d + e*x^2]*(
(a + b*ArcCosh[c*x])^n/2), x] + (-Dist[(1/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])], Int[(a + b*
ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sq
rt[-1 + c*x])], Int[x*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
&& GtQ[n, 0]

Rule 5939

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1
*e2*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m + 2*p + 1))), Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)
^p*(a + b*ArcCosh[c*x])^n, x], x] - Dist[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 +
e2*x)^p/(-1 + c*x)^p], 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, d1, e1, d2, e2, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IG
tQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rubi steps

\begin {align*} \int \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3 \, dx &=\frac {\sqrt {c-a^2 c x^2} \int \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3 \, dx}{\sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac {\sqrt {c-a^2 c x^2} \int \frac {\cosh ^{-1}(a x)^3}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{2 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (3 a \sqrt {c-a^2 c x^2}\right ) \int x \cosh ^{-1}(a x)^2 \, dx}{2 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{4 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^4}{8 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 a^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {x^2 \cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{2 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {3}{4} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{4 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^4}{8 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 \sqrt {c-a^2 c x^2}\right ) \int \frac {\cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{4 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (3 a \sqrt {c-a^2 c x^2}\right ) \int x \, dx}{4 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=-\frac {3 a x^2 \sqrt {c-a^2 c x^2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{4} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)+\frac {3 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{8 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{4 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^4}{8 a \sqrt {-1+a x} \sqrt {1+a x}}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 98, normalized size = 0.42 \begin {gather*} -\frac {\sqrt {-c (-1+a x) (1+a x)} \left (2 \cosh ^{-1}(a x)^4+\left (3+6 \cosh ^{-1}(a x)^2\right ) \cosh \left (2 \cosh ^{-1}(a x)\right )-2 \cosh ^{-1}(a x) \left (3+2 \cosh ^{-1}(a x)^2\right ) \sinh \left (2 \cosh ^{-1}(a x)\right )\right )}{16 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^3,x]

[Out]

-1/16*(Sqrt[-(c*(-1 + a*x)*(1 + a*x))]*(2*ArcCosh[a*x]^4 + (3 + 6*ArcCosh[a*x]^2)*Cosh[2*ArcCosh[a*x]] - 2*Arc
Cosh[a*x]*(3 + 2*ArcCosh[a*x]^2)*Sinh[2*ArcCosh[a*x]]))/(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))

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Maple [A]
time = 2.77, size = 256, normalized size = 1.11

method	result	size
default	\(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (a x \right )^{4}}{8 \sqrt {a x -1}\, \sqrt {a x +1}\, a}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 a^{3} x^{3}-2 a x +2 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \mathrm {arccosh}\left (a x \right )^{3}-6 \mathrm {arccosh}\left (a x \right )^{2}+6 \,\mathrm {arccosh}\left (a x \right )-3\right )}{32 \left (a x -1\right ) \left (a x +1\right ) a}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 a^{3} x^{3}-2 a x -2 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \mathrm {arccosh}\left (a x \right )^{3}+6 \mathrm {arccosh}\left (a x \right )^{2}+6 \,\mathrm {arccosh}\left (a x \right )+3\right )}{32 \left (a x -1\right ) \left (a x +1\right ) a}\)	\(256\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(-c*(a^2*x^2-1))^(1/2)/(a*x-1)^(1/2)/(a*x+1)^(1/2)/a*arccosh(a*x)^4+1/32*(-c*(a^2*x^2-1))^(1/2)*(2*a^3*x^
3-2*a*x+2*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a^2*x^2-(a*x-1)^(1/2)*(a*x+1)^(1/2))*(4*arccosh(a*x)^3-6*arccosh(a*x)^2+
6*arccosh(a*x)-3)/(a*x-1)/(a*x+1)/a+1/32*(-c*(a^2*x^2-1))^(1/2)*(2*a^3*x^3-2*a*x-2*(a*x+1)^(1/2)*(a*x-1)^(1/2)
*a^2*x^2+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(4*arccosh(a*x)^3+6*arccosh(a*x)^2+6*arccosh(a*x)+3)/(a*x-1)/(a*x+1)/a

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {acosh}^{3}{\left (a x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*acosh(a*x)**3, x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {acosh}\left (a\,x\right )}^3\,\sqrt {c-a^2\,c\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(a*x)^3*(c - a^2*c*x^2)^(1/2),x)

[Out]

int(acosh(a*x)^3*(c - a^2*c*x^2)^(1/2), x)

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