3.140 \(\int \frac {1}{(a+b \cosh ^{-1}(c+d x))^2} \, dx\)

Optimal. Leaf size=98 \[ \frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{b^2 d}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{b^2 d}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )} \]

[Out]

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

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Rubi [A]  time = 0.25, antiderivative size = 94, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5864, 5656, 5781, 3303, 3298, 3301} \[ \frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )}{b^2 d}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )}{b^2 d}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c + d*x])^(-2),x]

[Out]

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

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]

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 5656

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

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 5864

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

Rubi steps

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

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Mathematica [A]  time = 1.03, size = 143, normalized size = 1.46 \[ \frac {\sqrt {\frac {c+d x-1}{c+d x+1}} \coth \left (\frac {1}{2} \cosh ^{-1}(c+d x)\right ) \left (\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )-\log \left (a+b \cosh ^{-1}(c+d x)\right )\right )-\frac {b \sqrt {c+d x-1} \sqrt {c+d x+1}}{a+b \cosh ^{-1}(c+d x)}+\log \left (\frac {b \cosh ^{-1}(c+d x)}{a}+1\right )}{b^2 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F]  time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b^{2} \operatorname {arcosh}\left (d x + c\right )^{2} + 2 \, a b \operatorname {arcosh}\left (d x + c\right ) + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arccosh(d*x + c) + a)^(-2), x)

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maple [A]  time = 0.04, size = 139, normalized size = 1.42 \[ \frac {-\frac {d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\mathrm {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{2 b^{2}}+\frac {-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c}{2 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \Ei \left (1, \mathrm {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{2 b^{2}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {d^{3} x^{3} + 3 \, c d^{2} x^{2} + c^{3} + {\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left (3 \, c^{2} d - d\right )} x - c}{a b d^{3} x^{2} + 2 \, a b c d^{2} x + {\left (c^{2} d - d\right )} a b + {\left (a b d^{2} x + a b c d\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + {\left (c^{2} d - d\right )} b^{2} + {\left (b^{2} d^{2} x + b^{2} c d\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1}\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )} + \int \frac {d^{4} x^{4} + 4 \, c d^{3} x^{3} + c^{4} + {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )} {\left (d x + c + 1\right )} {\left (d x + c - 1\right )} + 2 \, {\left (3 \, c^{2} d^{2} - d^{2}\right )} x^{2} + {\left (2 \, d^{3} x^{3} + 6 \, c d^{2} x^{2} + 2 \, c^{3} + {\left (6 \, c^{2} d - d\right )} x - c\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} - 2 \, c^{2} + 4 \, {\left (c^{3} d - c d\right )} x + 1}{a b d^{4} x^{4} + 4 \, a b c d^{3} x^{3} + 2 \, {\left (3 \, c^{2} d^{2} - d^{2}\right )} a b x^{2} + 4 \, {\left (c^{3} d - c d\right )} a b x + {\left (a b d^{2} x^{2} + 2 \, a b c d x + a b c^{2}\right )} {\left (d x + c + 1\right )} {\left (d x + c - 1\right )} + {\left (c^{4} - 2 \, c^{2} + 1\right )} a b + 2 \, {\left (a b d^{3} x^{3} + 3 \, a b c d^{2} x^{2} + {\left (3 \, c^{2} d - d\right )} a b x + {\left (c^{3} - c\right )} a b\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 2 \, {\left (3 \, c^{2} d^{2} - d^{2}\right )} b^{2} x^{2} + 4 \, {\left (c^{3} d - c d\right )} b^{2} x + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\left (d x + c + 1\right )} {\left (d x + c - 1\right )} + {\left (c^{4} - 2 \, c^{2} + 1\right )} b^{2} + 2 \, {\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + {\left (3 \, c^{2} d - d\right )} b^{2} x + {\left (c^{3} - c\right )} b^{2}\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1}\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d^3*x^3 + 3*c*d^2*x^2 + c^3 + (d^2*x^2 + 2*c*d*x + c^2 - 1)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (3*c^2*d -
d)*x - c)/(a*b*d^3*x^2 + 2*a*b*c*d^2*x + (c^2*d - d)*a*b + (a*b*d^2*x + a*b*c*d)*sqrt(d*x + c + 1)*sqrt(d*x +
c - 1) + (b^2*d^3*x^2 + 2*b^2*c*d^2*x + (c^2*d - d)*b^2 + (b^2*d^2*x + b^2*c*d)*sqrt(d*x + c + 1)*sqrt(d*x +
c - 1))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)) + integrate((d^4*x^4 + 4*c*d^3*x^3 + c^4 + (d^2*x^
2 + 2*c*d*x + c^2 + 1)*(d*x + c + 1)*(d*x + c - 1) + 2*(3*c^2*d^2 - d^2)*x^2 + (2*d^3*x^3 + 6*c*d^2*x^2 + 2*c^
3 + (6*c^2*d - d)*x - c)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) - 2*c^2 + 4*(c^3*d - c*d)*x + 1)/(a*b*d^4*x^4 + 4
*a*b*c*d^3*x^3 + 2*(3*c^2*d^2 - d^2)*a*b*x^2 + 4*(c^3*d - c*d)*a*b*x + (a*b*d^2*x^2 + 2*a*b*c*d*x + a*b*c^2)*(
d*x + c + 1)*(d*x + c - 1) + (c^4 - 2*c^2 + 1)*a*b + 2*(a*b*d^3*x^3 + 3*a*b*c*d^2*x^2 + (3*c^2*d - d)*a*b*x +
(c^3 - c)*a*b)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 2*(3*c^2*d^2 - d^2)*b^2*
x^2 + 4*(c^3*d - c*d)*b^2*x + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*(d*x + c + 1)*(d*x + c - 1) + (c^4 - 2*c^2
+ 1)*b^2 + 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + (3*c^2*d - d)*b^2*x + (c^3 - c)*b^2)*sqrt(d*x + c + 1)*sqrt(d*x
+ c - 1))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*acosh(c + d*x))^2,x)

[Out]

int(1/(a + b*acosh(c + d*x))^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \operatorname {acosh}{\left (c + d x \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*acosh(c + d*x))**(-2), x)

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