3.3.79 \(\int \frac {(1-c^2 x^2)^{3/2}}{a+b \cosh ^{-1}(c x)} \, dx\) [279]

Optimal. Leaf size=239 \[ \frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{2 b c \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{8 b c \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \log \left (a+b \cosh ^{-1}(c x)\right )}{8 b c \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{2 b c \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{8 b c \sqrt {-1+c x}} \]

[Out]

1/2*Chi(2*(a+b*arccosh(c*x))/b)*cosh(2*a/b)*(-c*x+1)^(1/2)/b/c/(c*x-1)^(1/2)-1/8*Chi(4*(a+b*arccosh(c*x))/b)*c
osh(4*a/b)*(-c*x+1)^(1/2)/b/c/(c*x-1)^(1/2)-3/8*ln(a+b*arccosh(c*x))*(-c*x+1)^(1/2)/b/c/(c*x-1)^(1/2)-1/2*Shi(
2*(a+b*arccosh(c*x))/b)*sinh(2*a/b)*(-c*x+1)^(1/2)/b/c/(c*x-1)^(1/2)+1/8*Shi(4*(a+b*arccosh(c*x))/b)*sinh(4*a/
b)*(-c*x+1)^(1/2)/b/c/(c*x-1)^(1/2)

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Rubi [A]
time = 0.17, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5906, 3393, 3384, 3379, 3382} \begin {gather*} \frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{2 b c \sqrt {c x-1}}-\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{8 b c \sqrt {c x-1}}-\frac {\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{2 b c \sqrt {c x-1}}+\frac {\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{8 b c \sqrt {c x-1}}-\frac {3 \sqrt {1-c x} \log \left (a+b \cosh ^{-1}(c x)\right )}{8 b c \sqrt {c x-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - c*x]*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(2*b*c*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]
*Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(8*b*c*Sqrt[-1 + c*x]) - (3*Sqrt[1 - c*x]*Log[a + b*A
rcCosh[c*x]])/(8*b*c*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/
(2*b*c*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(8*b*c*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 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 5906

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

Rubi steps

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

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Mathematica [A]
time = 0.34, size = 147, normalized size = 0.62 \begin {gather*} -\frac {\sqrt {1-c^2 x^2} \left (-4 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+3 \log \left (a+b \cosh ^{-1}(c x)\right )+4 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )\right )}{8 b c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/8*(Sqrt[1 - c^2*x^2]*(-4*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCosh[c*x])] + Cosh[(4*a)/b]*CoshIntegral[4*
(a/b + ArcCosh[c*x])] + 3*Log[a + b*ArcCosh[c*x]] + 4*Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] - Sin
h[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])]))/(b*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))

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Maple [A]
time = 3.24, size = 413, normalized size = 1.73

method	result	size
default	\(-\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \expIntegral \left (1, 4 \,\mathrm {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\mathrm {arccosh}\left (c x \right )+4 a}{b}}}{16 \left (c x +1\right ) \left (c x -1\right ) c b}-\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \expIntegral \left (1, -4 \,\mathrm {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\mathrm {arccosh}\left (c x \right )+4 a}{b}}}{16 \left (c x +1\right ) \left (c x -1\right ) c b}-\frac {3 \sqrt {-c^{2} x^{2}+1}\, \ln \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}{8 \sqrt {c x -1}\, \sqrt {c x +1}\, c b}+\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \expIntegral \left (1, 2 \,\mathrm {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\mathrm {arccosh}\left (c x \right )+2 a}{b}}}{4 \left (c x +1\right ) \left (c x -1\right ) c b}+\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \expIntegral \left (1, -2 \,\mathrm {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\mathrm {arccosh}\left (c x \right )+2 a}{b}}}{4 \left (c x +1\right ) \left (c x -1\right ) c b}\)	\(413\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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(1,4*arccosh(c*x)+4*a/b)*exp((b*arccos
h(c*x)+4*a)/b)/(c*x+1)/(c*x-1)/c/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,-
4*arccosh(c*x)-4*a/b)*exp(-(-b*arccosh(c*x)+4*a)/b)/(c*x+1)/(c*x-1)/c/b-3/8*(-c^2*x^2+1)^(1/2)/(c*x-1)^(1/2)/(
c*x+1)^(1/2)/c*ln(a+b*arccosh(c*x))/b+1/4*(-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
,2*arccosh(c*x)+2*a/b)*exp((b*arccosh(c*x)+2*a)/b)/(c*x+1)/(c*x-1)/c/b+1/4*(-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,-2*arccosh(c*x)-2*a/b)*exp(-(-b*arccosh(c*x)+2*a)/b)/(c*x+1)/(c*x-1)/c/b

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

[Out]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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