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

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Rubi [A]  time = 0.47, 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 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 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]

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 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]

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*}

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Mathematica [A]  time = 0.79, size = 268, normalized size = 1.41 \[ -\frac {-c d \cosh \left (\frac {a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )-e \cosh \left (\frac {2 a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+a c d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )+b c d \sinh \left (\frac {a}{b}\right ) \cosh ^{-1}(c x) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )+a e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+b e \sinh \left (\frac {2 a}{b}\right ) \cosh ^{-1}(c x) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+b c^2 d x \sqrt {\frac {c x-1}{c x+1}}+b c^2 e x^2 \sqrt {\frac {c x-1}{c x+1}}+b c d \sqrt {\frac {c x-1}{c x+1}}+b c e x \sqrt {\frac {c x-1}{c x+1}}}{b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F]  time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e x + d}{b^{2} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname {arcosh}\left (c x\right ) + a^{2}}, x\right ) \]

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

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)

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maple [A]  time = 0.33, size = 285, normalized size = 1.50 \[ \frac {\frac {\left (-\sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) d}{2 b \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \Ei \left (1, \mathrm {arccosh}\left (c x \right )+\frac {a}{b}\right ) d}{2 b^{2}}-\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) d}{2 b \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\mathrm {arccosh}\left (c x \right )-\frac {a}{b}\right ) d}{2 b^{2}}+\frac {\left (-2 \sqrt {c x +1}\, \sqrt {c x -1}\, x c +2 c^{2} x^{2}-1\right ) e}{4 c \left (a +b \,\mathrm {arccosh}\left (c x \right )\right ) b}-\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \Ei \left (1, 2 \,\mathrm {arccosh}\left (c x \right )+\frac {2 a}{b}\right )}{2 c \,b^{2}}-\frac {e \left (2 c^{2} x^{2}-1+2 \sqrt {c x +1}\, \sqrt {c x -1}\, x c \right )}{4 c b \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \Ei \left (1, -2 \,\mathrm {arccosh}\left (c x \right )-\frac {2 a}{b}\right )}{2 c \,b^{2}}}{c} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ -\frac {c^{3} e x^{4} + c^{3} d x^{3} - c e x^{2} - c d x + {\left (c^{2} e x^{3} + c^{2} d x^{2} - e x - d\right )} \sqrt {c x + 1} \sqrt {c x - 1}}{a b c^{3} x^{2} + \sqrt {c x + 1} \sqrt {c x - 1} a b c^{2} x - a b c + {\left (b^{2} c^{3} x^{2} + \sqrt {c x + 1} \sqrt {c x - 1} b^{2} c^{2} x - b^{2} c\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )} + \int \frac {2 \, c^{5} e x^{5} + c^{5} d x^{4} - 4 \, c^{3} e x^{3} - 2 \, c^{3} d x^{2} + {\left (2 \, c^{3} e x^{3} + c^{3} d x^{2} + c d\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} + 2 \, c e x + {\left (4 \, c^{4} e x^{4} + 2 \, c^{4} d x^{3} - 4 \, c^{2} e x^{2} - c^{2} d x + e\right )} \sqrt {c x + 1} \sqrt {c x - 1} + c d}{a b c^{5} x^{4} + {\left (c x + 1\right )} {\left (c x - 1\right )} a b c^{3} x^{2} - 2 \, a b c^{3} x^{2} + a b c + 2 \, {\left (a b c^{4} x^{3} - a b c^{2} x\right )} \sqrt {c x + 1} \sqrt {c x - 1} + {\left (b^{2} c^{5} x^{4} + {\left (c x + 1\right )} {\left (c x - 1\right )} b^{2} c^{3} x^{2} - 2 \, b^{2} c^{3} x^{2} + b^{2} c + 2 \, {\left (b^{2} c^{4} x^{3} - b^{2} c^{2} x\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}\,{d x} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(a + b*acosh(c*x))^2,x)

[Out]

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

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

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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