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

Optimal. Leaf size=190 \[ \frac{e \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{b^2 c^2}-\frac{e \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{b^2 c^2}+\frac{d \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{b^2 c}-\frac{d \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{b^2 c}-\frac{d \sqrt{c x-1} \sqrt{c x+1}}{b c \left (a+b \cosh ^{-1}(c x)\right )}-\frac{e x \sqrt{c x-1} \sqrt{c x+1}}{b c \left (a+b \cosh ^{-1}(c x)\right )} \]

[Out]

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

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Rubi [A]  time = 0.474596, antiderivative size = 186, normalized size of antiderivative = 0.98, number of steps used = 11, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {5804, 5656, 5781, 3303, 3298, 3301, 5666} \[ \frac{e \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{b^2 c^2}-\frac{e \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{b^2 c^2}+\frac{d \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{b^2 c}-\frac{d \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{b^2 c}-\frac{d \sqrt{c x-1} \sqrt{c x+1}}{b c \left (a+b \cosh ^{-1}(c x)\right )}-\frac{e x \sqrt{c x-1} \sqrt{c x+1}}{b c \left (a+b \cosh ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(a + b*ArcCosh[c*x])^2,x]

[Out]

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

Rule 5804

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -1]

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 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 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 5666

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(
a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] + Dist[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a +
b*x)^(n + 1)*Cosh[x]^(m - 1)*(m - (m + 1)*Cosh[x]^2), x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rubi steps

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

Mathematica [A]  time = 2.43025, size = 157, normalized size = 0.83 \[ \frac{c d \left (\cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )-\sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+e \left (\cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-\sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+\log \left (a+b \cosh ^{-1}(c x)\right )\right )-\frac{b c \sqrt{\frac{c x-1}{c x+1}} (c x+1) (d+e x)}{a+b \cosh ^{-1}(c x)}-e \log \left (a+b \cosh ^{-1}(c x)\right )}{b^2 c^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.106, size = 285, normalized size = 1.5 \begin{align*}{\frac{1}{c} \left ({\frac{d}{2\,b \left ( a+b{\rm arccosh} \left (cx\right ) \right ) } \left ( -\sqrt{cx-1}\sqrt{cx+1}+cx \right ) }-{\frac{d}{2\,{b}^{2}}{{\rm e}^{{\frac{a}{b}}}}{\it Ei} \left ( 1,{\rm arccosh} \left (cx\right )+{\frac{a}{b}} \right ) }-{\frac{d}{2\,b \left ( a+b{\rm arccosh} \left (cx\right ) \right ) } \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }-{\frac{d}{2\,{b}^{2}}{{\rm e}^{-{\frac{a}{b}}}}{\it Ei} \left ( 1,-{\rm arccosh} \left (cx\right )-{\frac{a}{b}} \right ) }+{\frac{e}{4\,c \left ( a+b{\rm arccosh} \left (cx\right ) \right ) b} \left ( -2\,\sqrt{cx+1}\sqrt{cx-1}xc+2\,{c}^{2}{x}^{2}-1 \right ) }-{\frac{e}{2\,c{b}^{2}}{{\rm e}^{2\,{\frac{a}{b}}}}{\it Ei} \left ( 1,2\,{\rm arccosh} \left (cx\right )+2\,{\frac{a}{b}} \right ) }-{\frac{e}{4\,c \left ( a+b{\rm arccosh} \left (cx\right ) \right ) b} \left ( 2\,{c}^{2}{x}^{2}-1+2\,\sqrt{cx+1}\sqrt{cx-1}xc \right ) }-{\frac{e}{2\,c{b}^{2}}{{\rm e}^{-2\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-2\,{\rm arccosh} \left (cx\right )-2\,{\frac{a}{b}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(1/2*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x)*d/b/(a+b*arccosh(c*x))-1/2/b^2*exp(a/b)*Ei(1,arccosh(c*x)+a/b)*d-1
/2/b*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(a+b*arccosh(c*x))*d-1/2/b^2*exp(-a/b)*Ei(1,-arccosh(c*x)-a/b)*d+1/4*(-
2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+2*c^2*x^2-1)*e/c/(a+b*arccosh(c*x))/b-1/2/c*e/b^2*exp(2*a/b)*Ei(1,2*arccosh(
c*x)+2*a/b)-1/4/c*e/b*(2*c^2*x^2-1+2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c)/(a+b*arccosh(c*x))-1/2/c*e/b^2*exp(-2*a/
b)*Ei(1,-2*arccosh(c*x)-2*a/b))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{c^{3} e x^{4} + c^{3} d x^{3} - c e x^{2} - c d x +{\left (c^{2} e x^{3} + c^{2} d x^{2} - e x - d\right )} \sqrt{c x + 1} \sqrt{c x - 1}}{a b c^{3} x^{2} + \sqrt{c x + 1} \sqrt{c x - 1} a b c^{2} x - a b c +{\left (b^{2} c^{3} x^{2} + \sqrt{c x + 1} \sqrt{c x - 1} b^{2} c^{2} x - b^{2} c\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )} + \int \frac{2 \, c^{5} e x^{5} + c^{5} d x^{4} - 4 \, c^{3} e x^{3} - 2 \, c^{3} d x^{2} +{\left (2 \, c^{3} e x^{3} + c^{3} d x^{2} + c d\right )}{\left (c x + 1\right )}{\left (c x - 1\right )} + 2 \, c e x +{\left (4 \, c^{4} e x^{4} + 2 \, c^{4} d x^{3} - 4 \, c^{2} e x^{2} - c^{2} d x + e\right )} \sqrt{c x + 1} \sqrt{c x - 1} + c d}{a b c^{5} x^{4} +{\left (c x + 1\right )}{\left (c x - 1\right )} a b c^{3} x^{2} - 2 \, a b c^{3} x^{2} + a b c + 2 \,{\left (a b c^{4} x^{3} - a b c^{2} x\right )} \sqrt{c x + 1} \sqrt{c x - 1} +{\left (b^{2} c^{5} x^{4} +{\left (c x + 1\right )}{\left (c x - 1\right )} b^{2} c^{3} x^{2} - 2 \, b^{2} c^{3} x^{2} + b^{2} c + 2 \,{\left (b^{2} c^{4} x^{3} - b^{2} c^{2} x\right )} \sqrt{c x + 1} \sqrt{c x - 1}\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c^3*e*x^4 + c^3*d*x^3 - c*e*x^2 - c*d*x + (c^2*e*x^3 + c^2*d*x^2 - e*x - d)*sqrt(c*x + 1)*sqrt(c*x - 1))/(a*
b*c^3*x^2 + sqrt(c*x + 1)*sqrt(c*x - 1)*a*b*c^2*x - a*b*c + (b^2*c^3*x^2 + sqrt(c*x + 1)*sqrt(c*x - 1)*b^2*c^2
*x - b^2*c)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))) + integrate((2*c^5*e*x^5 + c^5*d*x^4 - 4*c^3*e*x^3 - 2*c^3
*d*x^2 + (2*c^3*e*x^3 + c^3*d*x^2 + c*d)*(c*x + 1)*(c*x - 1) + 2*c*e*x + (4*c^4*e*x^4 + 2*c^4*d*x^3 - 4*c^2*e*
x^2 - c^2*d*x + e)*sqrt(c*x + 1)*sqrt(c*x - 1) + c*d)/(a*b*c^5*x^4 + (c*x + 1)*(c*x - 1)*a*b*c^3*x^2 - 2*a*b*c
^3*x^2 + a*b*c + 2*(a*b*c^4*x^3 - a*b*c^2*x)*sqrt(c*x + 1)*sqrt(c*x - 1) + (b^2*c^5*x^4 + (c*x + 1)*(c*x - 1)*
b^2*c^3*x^2 - 2*b^2*c^3*x^2 + b^2*c + 2*(b^2*c^4*x^3 - b^2*c^2*x)*sqrt(c*x + 1)*sqrt(c*x - 1))*log(c*x + sqrt(
c*x + 1)*sqrt(c*x - 1))), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{e x + d}{b^{2} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname{arcosh}\left (c x\right ) + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e x + d}{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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