3.4.46 \(\int \frac {x}{\sqrt {1-c^2 x^2} (a+b \cosh ^{-1}(c x))^2} \, dx\) [346]

Optimal. Leaf size=130 \[ -\frac {x \sqrt {-1+c x}}{b c \sqrt {1-c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\sqrt {-1+c x} \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b^2 c^2 \sqrt {1-c x}}+\frac {\sqrt {-1+c x} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{b^2 c^2 \sqrt {1-c x}} \]

[Out]

-x*(c*x-1)^(1/2)/b/c/(a+b*arccosh(c*x))/(-c*x+1)^(1/2)+cosh(a/b)*Shi((a+b*arccosh(c*x))/b)*(c*x-1)^(1/2)/b^2/c
^2/(-c*x+1)^(1/2)-Chi((a+b*arccosh(c*x))/b)*sinh(a/b)*(c*x-1)^(1/2)/b^2/c^2/(-c*x+1)^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {5950, 5881, 3384, 3379, 3382} \begin {gather*} -\frac {\sqrt {c x-1} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{b^2 c^2 \sqrt {1-c x}}+\frac {\sqrt {c x-1} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{b^2 c^2 \sqrt {1-c x}}-\frac {x \sqrt {c x-1}}{b c \sqrt {1-c x} \left (a+b \cosh ^{-1}(c x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3379

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 3382

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]

Rule 3384

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 5881

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Sinh[-a/b + x/b], x], x
, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rule 5950

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

Rubi steps

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

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Mathematica [A]
time = 0.16, size = 107, normalized size = 0.82 \begin {gather*} \frac {\sqrt {1-c^2 x^2} \left (b c x+\left (a+b \cosh ^{-1}(c x)\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )-\left (a+b \cosh ^{-1}(c x)\right ) \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )}{b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(280\) vs. \(2(118)=236\).
time = 2.95, size = 281, normalized size = 2.16

method	result	size
default	\(-\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right )}{2 c^{2} \left (c^{2} x^{2}-1\right ) b \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}-\frac {\left (\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}\, \expIntegral \left (1, \mathrm {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{\frac {-b \,\mathrm {arccosh}\left (c x \right )+a}{b}}}{2 b^{2} \left (c^{2} x^{2}-1\right ) c^{2}}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (\mathrm {arccosh}\left (c x \right ) {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\mathrm {arccosh}\left (c x \right )-\frac {a}{b}\right ) b +\sqrt {c x +1}\, \sqrt {c x -1}\, b +{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\mathrm {arccosh}\left (c x \right )-\frac {a}{b}\right ) a +b c x \right )}{2 c^{2} \left (c^{2} x^{2}-1\right ) b^{2} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}\)	\(281\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(-c^2*x^2+1)^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)/c^2/(c^2*x^2-1)/b/(a+b*arccosh(c*x))-1/2*
((c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*Ei(1,arccosh(c*x)+a/b)*exp((-b*arccosh(c*x)+a)/
b)/b^2/(c^2*x^2-1)/c^2+1/2*(-c^2*x^2+1)^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^2/(c^2*x^2-1)*(arccosh(c*x)*exp(-a
/b)*Ei(1,-arccosh(c*x)-a/b)*b+(c*x+1)^(1/2)*(c*x-1)^(1/2)*b+exp(-a/b)*Ei(1,-arccosh(c*x)-a/b)*a+b*c*x)/b^2/(a+
b*arccosh(c*x))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c^3*x^4 - c*x^2 + (c^2*x^3 - x)*sqrt(c*x + 1)*sqrt(c*x - 1))/(((c*x + 1)*sqrt(c*x - 1)*b^2*c^2*x + (b^2*c^3*
x^2 - b^2*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + ((c*x + 1)*sqrt(c*x - 1)*a
*b*c^2*x + (a*b*c^3*x^2 - a*b*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)) + integrate((c^5*x^5 + (c*x + 1)*(c*x - 1)*c^3
*x^3 - 3*c^3*x^3 + (2*c^4*x^4 - 3*c^2*x^2 + 1)*sqrt(c*x + 1)*sqrt(c*x - 1) + 2*c*x)/(((c*x + 1)^(3/2)*(c*x - 1
)*b^2*c^3*x^2 + 2*(b^2*c^4*x^3 - b^2*c^2*x)*(c*x + 1)*sqrt(c*x - 1) + (b^2*c^5*x^4 - 2*b^2*c^3*x^2 + b^2*c)*sq
rt(c*x + 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + ((c*x + 1)^(3/2)*(c*x - 1)*a*b*c^3*x^2 +
2*(a*b*c^4*x^3 - a*b*c^2*x)*(c*x + 1)*sqrt(c*x - 1) + (a*b*c^5*x^4 - 2*a*b*c^3*x^2 + a*b*c)*sqrt(c*x + 1))*sqr
t(-c*x + 1)), x)

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {1-c^2\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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