3.151 \(\int \frac{c e+d e x}{(a+b \cosh ^{-1}(c+d x))^4} \, dx\)

Optimal. Leaf size=218 \[ \frac{2 e \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}-\frac{2 e \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}-\frac{e (c+d x)^2}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac{2 e \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{e}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac{e \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3} \]

[Out]

-(e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(3*b*d*(a + b*ArcCosh[c + d*x])^3) + e/(6*b^2*d*(a + b*Arc
Cosh[c + d*x])^2) - (e*(c + d*x)^2)/(3*b^2*d*(a + b*ArcCosh[c + d*x])^2) - (2*e*Sqrt[-1 + c + d*x]*(c + d*x)*S
qrt[1 + c + d*x])/(3*b^3*d*(a + b*ArcCosh[c + d*x])) + (2*e*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c + d
*x]))/b])/(3*b^4*d) - (2*e*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c + d*x]))/b])/(3*b^4*d)

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Rubi [A]  time = 0.510687, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5866, 12, 5668, 5775, 5666, 3303, 3298, 3301, 5676} \[ \frac{2 e \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c+d x)\right )}{3 b^4 d}-\frac{2 e \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c+d x)\right )}{3 b^4 d}-\frac{e (c+d x)^2}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac{2 e \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{e}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac{e \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)/(a + b*ArcCosh[c + d*x])^4,x]

[Out]

-(e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(3*b*d*(a + b*ArcCosh[c + d*x])^3) + e/(6*b^2*d*(a + b*Arc
Cosh[c + d*x])^2) - (e*(c + d*x)^2)/(3*b^2*d*(a + b*ArcCosh[c + d*x])^2) - (2*e*Sqrt[-1 + c + d*x]*(c + d*x)*S
qrt[1 + c + d*x])/(3*b^3*d*(a + b*ArcCosh[c + d*x])) + (2*e*Cosh[(2*a)/b]*CoshIntegral[(2*a)/b + 2*ArcCosh[c +
d*x]])/(3*b^4*d) - (2*e*Sinh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcCosh[c + d*x]])/(3*b^4*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 5668

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[(c*(m + 1))/(b*(n + 1)), Int[(x^(m + 1)*(a + b*ArcCosh
[c*x])^(n + 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] + Dist[m/(b*c*(n + 1)), Int[(x^(m - 1)*(a + b*ArcCosh[c
*x])^(n + 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 5775

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[((f*x)^m*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] - Dist[(f
*m)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1
, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && LtQ[n, -1] && GtQ[d1, 0] && LtQ[d2, 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]

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 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]

Rubi steps

\begin{align*} \int \frac{c e+d e x}{\left (a+b \cosh ^{-1}(c+d x)\right )^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e x}{\left (a+b \cosh ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=\frac{e \operatorname{Subst}\left (\int \frac{x}{\left (a+b \cosh ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=-\frac{e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}-\frac{e \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} \sqrt{1+x} \left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{3 b d}+\frac{(2 e) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-1+x} \sqrt{1+x} \left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{3 b d}\\ &=-\frac{e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac{e}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac{e (c+d x)^2}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac{(2 e) \operatorname{Subst}\left (\int \frac{x}{\left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{3 b^2 d}\\ &=-\frac{e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac{e}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac{e (c+d x)^2}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac{2 e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{(2 e) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 b^3 d}\\ &=-\frac{e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac{e}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac{e (c+d x)^2}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac{2 e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{\left (2 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+d x)\right )}{3 b^3 d}-\frac{\left (2 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+d x)\right )}{3 b^3 d}\\ &=-\frac{e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac{e}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac{e (c+d x)^2}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac{2 e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{2 e \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c+d x)\right )}{3 b^4 d}-\frac{2 e \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c+d x)\right )}{3 b^4 d}\\ \end{align*}

Mathematica [A]  time = 0.993619, size = 195, normalized size = 0.89 \[ \frac{e \left (-\frac{2 b^3 \sqrt{c+d x-1} (c+d x) \sqrt{c+d x+1}}{\left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac{b^2 \left (1-2 (c+d x)^2\right )}{\left (a+b \cosh ^{-1}(c+d x)\right )^2}+4 \left (\cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )-\sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )+\log \left (a+b \cosh ^{-1}(c+d x)\right )\right )-\frac{4 b \sqrt{c+d x-1} (c+d x) \sqrt{c+d x+1}}{a+b \cosh ^{-1}(c+d x)}-4 \log \left (a+b \cosh ^{-1}(c+d x)\right )\right )}{6 b^4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)/(a + b*ArcCosh[c + d*x])^4,x]

[Out]

(e*((-2*b^3*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(a + b*ArcCosh[c + d*x])^3 + (b^2*(1 - 2*(c + d*x)
^2))/(a + b*ArcCosh[c + d*x])^2 - (4*b*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(a + b*ArcCosh[c + d*x]
) - 4*Log[a + b*ArcCosh[c + d*x]] + 4*(Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCosh[c + d*x])] + Log[a + b*ArcC
osh[c + d*x]] - Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c + d*x])])))/(6*b^4*d)

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Maple [A]  time = 0.066, size = 353, normalized size = 1.6 \begin{align*}{\frac{1}{d} \left ({\frac{e \left ( 2\,{b}^{2} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{2}+4\,ab{\rm arccosh} \left (dx+c\right )-{\rm arccosh} \left (dx+c\right ){b}^{2}+2\,{a}^{2}-ab+{b}^{2} \right ) }{12\,{b}^{3} \left ({b}^{3} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{3}+3\,a{b}^{2} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{2}+3\,{a}^{2}b{\rm arccosh} \left (dx+c\right )+{a}^{3} \right ) } \left ( -2\,\sqrt{dx+c+1}\sqrt{dx+c-1} \left ( dx+c \right ) +2\, \left ( dx+c \right ) ^{2}-1 \right ) }-{\frac{e}{3\,{b}^{4}}{{\rm e}^{2\,{\frac{a}{b}}}}{\it Ei} \left ( 1,2\,{\rm arccosh} \left (dx+c\right )+2\,{\frac{a}{b}} \right ) }-{\frac{e}{12\,b \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) ^{3}} \left ( 2\, \left ( dx+c \right ) ^{2}-1+2\,\sqrt{dx+c+1}\sqrt{dx+c-1} \left ( dx+c \right ) \right ) }-{\frac{e}{12\,{b}^{2} \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) ^{2}} \left ( 2\, \left ( dx+c \right ) ^{2}-1+2\,\sqrt{dx+c+1}\sqrt{dx+c-1} \left ( dx+c \right ) \right ) }-{\frac{e}{6\,{b}^{3} \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) } \left ( 2\, \left ( dx+c \right ) ^{2}-1+2\,\sqrt{dx+c+1}\sqrt{dx+c-1} \left ( dx+c \right ) \right ) }-{\frac{e}{3\,{b}^{4}}{{\rm e}^{-2\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-2\,{\rm arccosh} \left (dx+c\right )-2\,{\frac{a}{b}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)/(a+b*arccosh(d*x+c))^4,x)

[Out]

1/d*(1/12*(-2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c)+2*(d*x+c)^2-1)*e*(2*b^2*arccosh(d*x+c)^2+4*a*b*arccosh(d
*x+c)-arccosh(d*x+c)*b^2+2*a^2-a*b+b^2)/b^3/(b^3*arccosh(d*x+c)^3+3*a*b^2*arccosh(d*x+c)^2+3*a^2*b*arccosh(d*x
+c)+a^3)-1/3*e/b^4*exp(2*a/b)*Ei(1,2*arccosh(d*x+c)+2*a/b)-1/12*e/b*(2*(d*x+c)^2-1+2*(d*x+c+1)^(1/2)*(d*x+c-1)
^(1/2)*(d*x+c))/(a+b*arccosh(d*x+c))^3-1/12*e/b^2*(2*(d*x+c)^2-1+2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c))/(a
+b*arccosh(d*x+c))^2-1/6*e/b^3*(2*(d*x+c)^2-1+2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c))/(a+b*arccosh(d*x+c))-
1/3*e/b^4*exp(-2*a/b)*Ei(1,-2*arccosh(d*x+c)-2*a/b))

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)/(a+b*arccosh(d*x+c))^4,x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{d e x + c e}{b^{4} \operatorname{arcosh}\left (d x + c\right )^{4} + 4 \, a b^{3} \operatorname{arcosh}\left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \operatorname{arcosh}\left (d x + c\right )^{2} + 4 \, a^{3} b \operatorname{arcosh}\left (d x + c\right ) + a^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)/(a+b*arccosh(d*x+c))^4,x, algorithm="fricas")

[Out]

integral((d*e*x + c*e)/(b^4*arccosh(d*x + c)^4 + 4*a*b^3*arccosh(d*x + c)^3 + 6*a^2*b^2*arccosh(d*x + c)^2 + 4
*a^3*b*arccosh(d*x + c) + a^4), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)/(a+b*acosh(d*x+c))**4,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d e x + c e}{{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)/(a+b*arccosh(d*x+c))^4,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)/(b*arccosh(d*x + c) + a)^4, x)