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

Optimal. Leaf size=191 \[ \frac{e^2 \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )}{4 b^2 d}+\frac{3 e^2 \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{4 b^2 d}-\frac{e^2 \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )}{4 b^2 d}-\frac{3 e^2 \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{4 b^2 d}-\frac{e^2 \sqrt{c+d x-1} (c+d x)^2 \sqrt{c+d x+1}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )} \]

[Out]

-((e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(b*d*(a + b*ArcCosh[c + d*x]))) + (e^2*Cosh[a/b]*Cosh
Integral[(a + b*ArcCosh[c + d*x])/b])/(4*b^2*d) + (3*e^2*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c + d*x]
))/b])/(4*b^2*d) - (e^2*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/(4*b^2*d) - (3*e^2*Sinh[(3*a)/b]*S
inhIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/(4*b^2*d)

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Rubi [A]  time = 0.275911, antiderivative size = 187, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {5866, 12, 5666, 3303, 3298, 3301} \[ \frac{e^2 \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )}{4 b^2 d}+\frac{3 e^2 \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{4 b^2 d}-\frac{e^2 \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )}{4 b^2 d}-\frac{3 e^2 \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{4 b^2 d}-\frac{e^2 \sqrt{c+d x-1} (c+d x)^2 \sqrt{c+d x+1}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(b*d*(a + b*ArcCosh[c + d*x]))) + (e^2*Cosh[a/b]*Cosh
Integral[a/b + ArcCosh[c + d*x]])/(4*b^2*d) + (3*e^2*Cosh[(3*a)/b]*CoshIntegral[(3*a)/b + 3*ArcCosh[c + d*x]])
/(4*b^2*d) - (e^2*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c + d*x]])/(4*b^2*d) - (3*e^2*Sinh[(3*a)/b]*SinhIntegra
l[(3*a)/b + 3*ArcCosh[c + d*x]])/(4*b^2*d)

Rule 5866

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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]

Rubi steps

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

Mathematica [A]  time = 1.70244, size = 150, normalized size = 0.79 \[ \frac{e^2 \left (\cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )+3 \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )-\sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )-3 \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )-\frac{4 b \sqrt{\frac{c+d x-1}{c+d x+1}} (c+d x+1) (c+d x)^2}{a+b \cosh ^{-1}(c+d x)}\right )}{4 b^2 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(e^2*((-4*b*(c + d*x)^2*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x))/(a + b*ArcCosh[c + d*x]) + Cosh[a/b]
*CoshIntegral[a/b + ArcCosh[c + d*x]] + 3*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcCosh[c + d*x])] - Sinh[a/b]*S
inhIntegral[a/b + ArcCosh[c + d*x]] - 3*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c + d*x])]))/(4*b^2*d)

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Maple [B]  time = 0.099, size = 374, normalized size = 2. \begin{align*}{\frac{1}{d} \left ({\frac{{e}^{2}}{8\,b \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) } \left ( -4\, \left ( dx+c \right ) ^{2}\sqrt{dx+c-1}\sqrt{dx+c+1}+\sqrt{dx+c-1}\sqrt{dx+c+1}+4\, \left ( dx+c \right ) ^{3}-3\,dx-3\,c \right ) }-{\frac{3\,{e}^{2}}{8\,{b}^{2}}{{\rm e}^{3\,{\frac{a}{b}}}}{\it Ei} \left ( 1,3\,{\rm arccosh} \left (dx+c\right )+3\,{\frac{a}{b}} \right ) }+{\frac{{e}^{2}}{8\,b \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) } \left ( -\sqrt{dx+c-1}\sqrt{dx+c+1}+dx+c \right ) }-{\frac{{e}^{2}}{8\,{b}^{2}}{{\rm e}^{{\frac{a}{b}}}}{\it Ei} \left ( 1,{\rm arccosh} \left (dx+c\right )+{\frac{a}{b}} \right ) }-{\frac{{e}^{2}}{8\,b \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) } \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) }-{\frac{{e}^{2}}{8\,{b}^{2}}{{\rm e}^{-{\frac{a}{b}}}}{\it Ei} \left ( 1,-{\rm arccosh} \left (dx+c\right )-{\frac{a}{b}} \right ) }-{\frac{{e}^{2}}{8\,b \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) } \left ( 4\, \left ( dx+c \right ) ^{3}-3\,dx-3\,c+4\, \left ( dx+c \right ) ^{2}\sqrt{dx+c-1}\sqrt{dx+c+1}-\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) }-{\frac{3\,{e}^{2}}{8\,{b}^{2}}{{\rm e}^{-3\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-3\,{\rm arccosh} \left (dx+c\right )-3\,{\frac{a}{b}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/8*(-4*(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+4*(d*x+c)^3-3*d*x-3*c)*
e^2/b/(a+b*arccosh(d*x+c))-3/8*e^2/b^2*exp(3*a/b)*Ei(1,3*arccosh(d*x+c)+3*a/b)+1/8*(-(d*x+c-1)^(1/2)*(d*x+c+1)
^(1/2)+d*x+c)*e^2/b/(a+b*arccosh(d*x+c))-1/8*e^2/b^2*exp(a/b)*Ei(1,arccosh(d*x+c)+a/b)-1/8*e^2/b*(d*x+c+(d*x+c
-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))-1/8*e^2/b^2*exp(-a/b)*Ei(1,-arccosh(d*x+c)-a/b)-1/8*e^2/b*(4*(
d*x+c)^3-3*d*x-3*c+4*(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d
*x+c))-3/8*e^2/b^2*exp(-3*a/b)*Ei(1,-3*arccosh(d*x+c)-3*a/b))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d^5*e^2*x^5 + 5*c*d^4*e^2*x^4 + c^5*e^2 - c^3*e^2 + (10*c^2*d^3*e^2 - d^3*e^2)*x^3 + (10*c^3*d^2*e^2 - 3*c*d
^2*e^2)*x^2 + (d^4*e^2*x^4 + 4*c*d^3*e^2*x^3 + c^4*e^2 - c^2*e^2 + (6*c^2*d^2*e^2 - d^2*e^2)*x^2 + 2*(2*c^3*d*
e^2 - c*d*e^2)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (5*c^4*d*e^2 - 3*c^2*d*e^2)*x)/(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((3*d^6*e^2*x^6 + 18*c*d^5*e^2*x^5 + 3*c^6*e^2 - 6*c^4*e^2 + 3*(15*c^2*d^
4*e^2 - 2*d^4*e^2)*x^4 + 3*c^2*e^2 + 12*(5*c^3*d^3*e^2 - 2*c*d^3*e^2)*x^3 + (3*d^4*e^2*x^4 + 12*c*d^3*e^2*x^3
+ 3*c^4*e^2 - c^2*e^2 + (18*c^2*d^2*e^2 - d^2*e^2)*x^2 + 2*(6*c^3*d*e^2 - c*d*e^2)*x)*(d*x + c + 1)*(d*x + c -
1) + 3*(15*c^4*d^2*e^2 - 12*c^2*d^2*e^2 + d^2*e^2)*x^2 + (6*d^5*e^2*x^5 + 30*c*d^4*e^2*x^4 + 6*c^5*e^2 - 7*c^
3*e^2 + (60*c^2*d^3*e^2 - 7*d^3*e^2)*x^3 + 2*c*e^2 + 3*(20*c^3*d^2*e^2 - 7*c*d^2*e^2)*x^2 + (30*c^4*d*e^2 - 21
*c^2*d*e^2 + 2*d*e^2)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + 6*(3*c^5*d*e^2 - 4*c^3*d*e^2 + c*d*e^2)*x)/(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., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}{b^{2} \operatorname{arcosh}\left (d x + c\right )^{2} + 2 \, a b \operatorname{arcosh}\left (d x + c\right ) + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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