3.3.26 \(\int \frac {x^3 \cosh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\) [226]

Optimal. Leaf size=177 \[ -\frac {40 \sqrt {1-a x} \sqrt {1+a x}}{27 a^4}-\frac {2 x^2 \sqrt {1-a x} \sqrt {1+a x}}{27 a^2}-\frac {4 x \sqrt {-1+a x} \cosh ^{-1}(a x)}{3 a^3 \sqrt {1-a x}}-\frac {2 x^3 \sqrt {-1+a x} \cosh ^{-1}(a x)}{9 a \sqrt {1-a x}}-\frac {2 \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^2}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^2}{3 a^2} \]

[Out]

-4/3*x*arccosh(a*x)*(a*x-1)^(1/2)/a^3/(-a*x+1)^(1/2)-2/9*x^3*arccosh(a*x)*(a*x-1)^(1/2)/a/(-a*x+1)^(1/2)-40/27
*(-a*x+1)^(1/2)*(a*x+1)^(1/2)/a^4-2/27*x^2*(-a*x+1)^(1/2)*(a*x+1)^(1/2)/a^2-2/3*arccosh(a*x)^2*(-a^2*x^2+1)^(1
/2)/a^4-1/3*x^2*arccosh(a*x)^2*(-a^2*x^2+1)^(1/2)/a^2

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Rubi [A]
time = 0.16, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {5938, 5914, 5879, 75, 5883, 102, 12} \begin {gather*} -\frac {40 \sqrt {1-a x} \sqrt {a x+1}}{27 a^4}-\frac {4 x \sqrt {a x-1} \cosh ^{-1}(a x)}{3 a^3 \sqrt {1-a x}}-\frac {2 x^2 \sqrt {1-a x} \sqrt {a x+1}}{27 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^2}{3 a^2}-\frac {2 \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^2}{3 a^4}-\frac {2 x^3 \sqrt {a x-1} \cosh ^{-1}(a x)}{9 a \sqrt {1-a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-40*Sqrt[1 - a*x]*Sqrt[1 + a*x])/(27*a^4) - (2*x^2*Sqrt[1 - a*x]*Sqrt[1 + a*x])/(27*a^2) - (4*x*Sqrt[-1 + a*x
]*ArcCosh[a*x])/(3*a^3*Sqrt[1 - a*x]) - (2*x^3*Sqrt[-1 + a*x]*ArcCosh[a*x])/(9*a*Sqrt[1 - a*x]) - (2*Sqrt[1 -
a^2*x^2]*ArcCosh[a*x]^2)/(3*a^4) - (x^2*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^2)/(3*a^2)

Rule 12

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

Rule 75

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

Rule 102

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

Rule 5879

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

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 5914

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

Rule 5938

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

Rubi steps

\begin {align*} \int \frac {x^3 \cosh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x^3 \cosh ^{-1}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{3 a^2 \sqrt {1-a^2 x^2}}+\frac {\left (2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x \cosh ^{-1}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{3 a^2 \sqrt {1-a^2 x^2}}-\frac {\left (2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int x^2 \cosh ^{-1}(a x) \, dx}{3 a \sqrt {1-a^2 x^2}}\\ &=-\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{9 a \sqrt {1-a^2 x^2}}-\frac {2 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{3 a^4 \sqrt {1-a^2 x^2}}-\frac {x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{3 a^2 \sqrt {1-a^2 x^2}}+\frac {\left (2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{9 \sqrt {1-a^2 x^2}}-\frac {\left (4 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \cosh ^{-1}(a x) \, dx}{3 a^3 \sqrt {1-a^2 x^2}}\\ &=-\frac {2 x^2 (1-a x) (1+a x)}{27 a^2 \sqrt {1-a^2 x^2}}-\frac {4 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{3 a^3 \sqrt {1-a^2 x^2}}-\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{9 a \sqrt {1-a^2 x^2}}-\frac {2 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{3 a^4 \sqrt {1-a^2 x^2}}-\frac {x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{3 a^2 \sqrt {1-a^2 x^2}}+\frac {\left (2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {2 x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{27 a^2 \sqrt {1-a^2 x^2}}+\frac {\left (4 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{3 a^2 \sqrt {1-a^2 x^2}}\\ &=-\frac {4 (1-a x) (1+a x)}{3 a^4 \sqrt {1-a^2 x^2}}-\frac {2 x^2 (1-a x) (1+a x)}{27 a^2 \sqrt {1-a^2 x^2}}-\frac {4 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{3 a^3 \sqrt {1-a^2 x^2}}-\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{9 a \sqrt {1-a^2 x^2}}-\frac {2 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{3 a^4 \sqrt {1-a^2 x^2}}-\frac {x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{3 a^2 \sqrt {1-a^2 x^2}}+\frac {\left (4 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{27 a^2 \sqrt {1-a^2 x^2}}\\ &=-\frac {40 (1-a x) (1+a x)}{27 a^4 \sqrt {1-a^2 x^2}}-\frac {2 x^2 (1-a x) (1+a x)}{27 a^2 \sqrt {1-a^2 x^2}}-\frac {4 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{3 a^3 \sqrt {1-a^2 x^2}}-\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{9 a \sqrt {1-a^2 x^2}}-\frac {2 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{3 a^4 \sqrt {1-a^2 x^2}}-\frac {x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{3 a^2 \sqrt {1-a^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 123, normalized size = 0.69 \begin {gather*} \left (-\frac {40}{27 a^4}-\frac {2 x^2}{27 a^2}\right ) \sqrt {1-a^2 x^2}+\frac {2 x \sqrt {1-a^2 x^2} \left (6+a^2 x^2\right ) \cosh ^{-1}(a x)}{9 a^3 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\sqrt {1-a^2 x^2} \left (2+a^2 x^2\right ) \cosh ^{-1}(a x)^2}{3 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-40/(27*a^4) - (2*x^2)/(27*a^2))*Sqrt[1 - a^2*x^2] + (2*x*Sqrt[1 - a^2*x^2]*(6 + a^2*x^2)*ArcCosh[a*x])/(9*a^
3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (Sqrt[1 - a^2*x^2]*(2 + a^2*x^2)*ArcCosh[a*x]^2)/(3*a^4)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(342\) vs. \(2(145)=290\).
time = 3.26, size = 343, normalized size = 1.94

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (4 a^{4} x^{4}-5 a^{2} x^{2}+4 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}-3 \sqrt {a x +1}\, \sqrt {a x -1}\, a x +1\right ) \left (9 \mathrm {arccosh}\left (a x \right )^{2}-6 \,\mathrm {arccosh}\left (a x \right )+2\right )}{216 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \left (\sqrt {a x +1}\, \sqrt {a x -1}\, a x +a^{2} x^{2}-1\right ) \left (\mathrm {arccosh}\left (a x \right )^{2}-2 \,\mathrm {arccosh}\left (a x \right )+2\right )}{8 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x +1}\, \sqrt {a x -1}\, a x -1\right ) \left (\mathrm {arccosh}\left (a x \right )^{2}+2 \,\mathrm {arccosh}\left (a x \right )+2\right )}{8 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (4 a^{4} x^{4}-5 a^{2} x^{2}-4 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}+3 \sqrt {a x +1}\, \sqrt {a x -1}\, a x +1\right ) \left (9 \mathrm {arccosh}\left (a x \right )^{2}+6 \,\mathrm {arccosh}\left (a x \right )+2\right )}{216 a^{4} \left (a^{2} x^{2}-1\right )}\)	\(343\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/216*(-a^2*x^2+1)^(1/2)*(4*a^4*x^4-5*a^2*x^2+4*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a^3*x^3-3*(a*x+1)^(1/2)*(a*x-1)^(
1/2)*a*x+1)*(9*arccosh(a*x)^2-6*arccosh(a*x)+2)/a^4/(a^2*x^2-1)-3/8*(-a^2*x^2+1)^(1/2)*((a*x+1)^(1/2)*(a*x-1)^
(1/2)*a*x+a^2*x^2-1)*(arccosh(a*x)^2-2*arccosh(a*x)+2)/a^4/(a^2*x^2-1)-3/8*(-a^2*x^2+1)^(1/2)*(a^2*x^2-(a*x+1)
^(1/2)*(a*x-1)^(1/2)*a*x-1)*(arccosh(a*x)^2+2*arccosh(a*x)+2)/a^4/(a^2*x^2-1)-1/216*(-a^2*x^2+1)^(1/2)*(4*a^4*
x^4-5*a^2*x^2-4*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a^3*x^3+3*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a*x+1)*(9*arccosh(a*x)^2+6*a
rccosh(a*x)+2)/a^4/(a^2*x^2-1)

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Maxima [C] Result contains complex when optimal does not.
time = 0.47, size = 105, normalized size = 0.59 \begin {gather*} -\frac {1}{3} \, {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname {arcosh}\left (a x\right )^{2} + \frac {2 \, {\left (-i \, \sqrt {a^{2} x^{2} - 1} x^{2} - \frac {20 i \, \sqrt {a^{2} x^{2} - 1}}{a^{2}}\right )}}{27 \, a^{2}} + \frac {2 \, {\left (i \, a^{2} x^{3} + 6 i \, x\right )} \operatorname {arcosh}\left (a x\right )}{9 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(sqrt(-a^2*x^2 + 1)*x^2/a^2 + 2*sqrt(-a^2*x^2 + 1)/a^4)*arccosh(a*x)^2 + 2/27*(-I*sqrt(a^2*x^2 - 1)*x^2 -
20*I*sqrt(a^2*x^2 - 1)/a^2)/a^2 + 2/9*(I*a^2*x^3 + 6*I*x)*arccosh(a*x)/a^3

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Fricas [A]
time = 0.35, size = 150, normalized size = 0.85 \begin {gather*} -\frac {9 \, {\left (a^{4} x^{4} + a^{2} x^{2} - 2\right )} \sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} - 6 \, {\left (a^{3} x^{3} + 6 \, a x\right )} \sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) + 2 \, {\left (a^{4} x^{4} + 19 \, a^{2} x^{2} - 20\right )} \sqrt {-a^{2} x^{2} + 1}}{27 \, {\left (a^{6} x^{2} - a^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*(9*(a^4*x^4 + a^2*x^2 - 2)*sqrt(-a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 - 1))^2 - 6*(a^3*x^3 + 6*a*x)*sqrt(
a^2*x^2 - 1)*sqrt(-a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 - 1)) + 2*(a^4*x^4 + 19*a^2*x^2 - 20)*sqrt(-a^2*x^2 + 1
))/(a^6*x^2 - a^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \operatorname {acosh}^{2}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*acosh(a*x)**2/sqrt(-(a*x - 1)*(a*x + 1)), 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(x^3*arccosh(a*x)^2/(-a^2*x^2+1)^(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 = -0.01 \begin {gather*} \int \frac {x^3\,{\mathrm {acosh}\left (a\,x\right )}^2}{\sqrt {1-a^2\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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