3.2.40 \(\int \frac {1}{(a+b \cosh ^{-1}(c+d x))^2} \, dx\) [140]

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

[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 = 0.18, antiderivative size = 98, normalized size of antiderivative = 1.00, 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 = {5995, 5880, 5953, 3384, 3379, 3382} \begin {gather*} \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 )} \end {gather*}

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*ArcC
osh[c + d*x])/b])/(b^2*d) - (Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/(b^2*d)

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 5880

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]*Sqrt[
-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && LtQ[n, -1]

Rule 5953

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(1/(b*c^(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Subs
t[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1,
e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]

Rule 5995

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 {\text {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 {\text {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 {\text {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 ) \text {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 ) \text {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 = 0.73, size = 143, normalized size = 1.46 \begin {gather*} \frac {-\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{a+b \cosh ^{-1}(c+d x)}+\log \left (1+\frac {b \cosh ^{-1}(c+d x)}{a}\right )+\sqrt {\frac {-1+c+d x}{1+c+d x}} \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 )-\log \left (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 )\right )}{b^2 d} \end {gather*}

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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Maple [A]
time = 41.24, size = 139, normalized size = 1.42

method	result	size
derivativedivides	\(\frac {\frac {-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c}{2 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \mathrm {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\mathrm {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{2 b^{2}}}{d}\)	\(139\)
default	\(\frac {\frac {-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c}{2 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \mathrm {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\mathrm {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{2 b^{2}}}{d}\)	\(139\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(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)-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))

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

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

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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