3.2.39 \(\int \frac {c e+d e x}{(a+b \cosh ^{-1}(c+d x))^2} \, dx\) [139]

Optimal. Leaf size=110 \[ -\frac {e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{b^2 d}-\frac {e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{b^2 d} \]

[Out]

e*Chi(2*(a+b*arccosh(d*x+c))/b)*cosh(2*a/b)/b^2/d-e*Shi(2*(a+b*arccosh(d*x+c))/b)*sinh(2*a/b)/b^2/d-e*(d*x+c)*
(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 = 0.10, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5996, 12, 5885, 3384, 3379, 3382} \begin {gather*} \frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{b^2 d}-\frac {e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{b^2 d}-\frac {e \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b d \left (a+b \cosh ^{-1}(c+d x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

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 5885

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^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^
(n + 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, a + b*ArcCosh[c*x]], x] /; Free
Q[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rule 5996

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
]

Rubi steps

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

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Mathematica [A]
time = 0.36, size = 108, normalized size = 0.98 \begin {gather*} \frac {e \left (-\frac {b \sqrt {\frac {-1+c+d x}{1+c+d x}} \left (c+c^2+2 c d x+d x (1+d x)\right )}{a+b \cosh ^{-1}(c+d x)}+\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 )\right )}{b^2 d} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]
time = 57.51, size = 170, normalized size = 1.55

method	result	size
derivativedivides	\(\frac {\frac {\left (-2 \left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+2 \left (d x +c \right )^{2}-1\right ) e}{4 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \expIntegral \left (1, 2 \,\mathrm {arccosh}\left (d x +c \right )+\frac {2 a}{b}\right )}{2 b^{2}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{4 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \,\mathrm {arccosh}\left (d x +c \right )-\frac {2 a}{b}\right )}{2 b^{2}}}{d}\)	\(170\)
default	\(\frac {\frac {\left (-2 \left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+2 \left (d x +c \right )^{2}-1\right ) e}{4 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \expIntegral \left (1, 2 \,\mathrm {arccosh}\left (d x +c \right )+\frac {2 a}{b}\right )}{2 b^{2}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{4 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \,\mathrm {arccosh}\left (d x +c \right )-\frac {2 a}{b}\right )}{2 b^{2}}}{d}\)	\(170\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/4*(-2*(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+2*(d*x+c)^2-1)*e/b/(a+b*arccosh(d*x+c))-1/2*e/b^2*exp(2*a
/b)*Ei(1,2*arccosh(d*x+c)+2*a/b)-1/4/b*e*(2*(d*x+c)^2-1+2*(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccos
h(d*x+c))-1/2/b^2*e*exp(-2*a/b)*Ei(1,-2*arccosh(d*x+c)-2*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((d*e*x+c*e)/(a+b*arccosh(d*x+c))^2,x, algorithm="maxima")

[Out]

-(d^4*x^4*e + 4*c*d^3*x^3*e + (6*c^2*d^2 - d^2)*x^2*e + 2*(2*c^3*d - c*d)*x*e + (d^3*x^3*e + 3*c*d^2*x^2*e + (
3*c^2*d - d)*x*e + (c^3 - c)*e)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (c^4 - c^2)*e)/(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((2*d^5*x^5*e + 10*c*d^4*x^4*e + 4*(5*c^2*d^3 - d^3)*x^3*e + 4*(5*c^3*d^2 -
3*c*d^2)*x^2*e + 2*(d^3*x^3*e + 3*c*d^2*x^2*e + 3*c^2*d*x*e + c^3*e)*(d*x + c + 1)*(d*x + c - 1) + 2*(5*c^4*d
- 6*c^2*d + d)*x*e + (4*d^4*x^4*e + 16*c*d^3*x^3*e + 4*(6*c^2*d^2 - d^2)*x^2*e + 8*(2*c^3*d - c*d)*x*e + (4*c
^4 - 4*c^2 + 1)*e)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + 2*(c^5 - 2*c^3 + c)*e)/(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((d*e*x+c*e)/(a+b*arccosh(d*x+c))^2,x, algorithm="fricas")

[Out]

integral((d*x + c)*e/(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*} e \left (\int \frac {c}{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 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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e*(Integral(c/(a**2 + 2*a*b*acosh(c + d*x) + b**2*acosh(c + d*x)**2), x) + Integral(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 = 0.00, 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((d*e*x+c*e)/(a+b*arccosh(d*x+c))^2,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)/(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 {c\,e+d\,e\,x}{{\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((c*e + d*e*x)/(a + b*acosh(c + d*x))^2,x)

[Out]

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

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