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

Optimal. Leaf size=110 \[ -\frac{2 i b^2 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac{2 i b^2 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^2 (c+d x)}+\frac{4 b \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^2} \]

[Out]

-((a + b*ArcCosh[c + d*x])^2/(d*e^2*(c + d*x))) + (4*b*(a + b*ArcCosh[c + d*x])*ArcTan[E^ArcCosh[c + d*x]])/(d
*e^2) - ((2*I)*b^2*PolyLog[2, (-I)*E^ArcCosh[c + d*x]])/(d*e^2) + ((2*I)*b^2*PolyLog[2, I*E^ArcCosh[c + d*x]])
/(d*e^2)

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Rubi [A]  time = 0.242946, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {5866, 12, 5662, 5761, 4180, 2279, 2391} \[ -\frac{2 i b^2 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac{2 i b^2 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^2 (c+d x)}+\frac{4 b \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + b*ArcCosh[c + d*x])^2/(d*e^2*(c + d*x))) + (4*b*(a + b*ArcCosh[c + d*x])*ArcTan[E^ArcCosh[c + d*x]])/(d
*e^2) - ((2*I)*b^2*PolyLog[2, (-I)*E^ArcCosh[c + d*x]])/(d*e^2) + ((2*I)*b^2*PolyLog[2, I*E^ArcCosh[c + d*x]])
/(d*e^2)

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 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5761

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

Rule 4180

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c
+ d*x)^m*ArcTanh[E^(-(I*e) + f*fz*x)/E^(I*k*Pi)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*
Log[1 - E^(-(I*e) + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e)
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{\left (a+b \cosh ^{-1}(c+d x)\right )^2}{(c e+d e x)^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^2}{e^2 x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^2}{x^2} \, dx,x,c+d x\right )}{d e^2}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^2 (c+d x)}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}(x)}{\sqrt{-1+x} x \sqrt{1+x}} \, dx,x,c+d x\right )}{d e^2}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^2 (c+d x)}+\frac{(2 b) \operatorname{Subst}\left (\int (a+b x) \text{sech}(x) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^2}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^2 (c+d x)}+\frac{4 b \left (a+b \cosh ^{-1}(c+d x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{\left (2 i b^2\right ) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^2}+\frac{\left (2 i b^2\right ) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^2}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^2 (c+d x)}+\frac{4 b \left (a+b \cosh ^{-1}(c+d x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{\left (2 i b^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac{\left (2 i b^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}\\ &=-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^2 (c+d x)}+\frac{4 b \left (a+b \cosh ^{-1}(c+d x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac{2 i b^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac{2 i b^2 \text{Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}\\ \end{align*}

Mathematica [A]  time = 0.721384, size = 161, normalized size = 1.46 \[ \frac{-i b^2 \left (2 \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c+d x)}\right )-2 \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c+d x)}\right )+\cosh ^{-1}(c+d x) \left (-\frac{i \cosh ^{-1}(c+d x)}{c+d x}+2 \log \left (1-i e^{-\cosh ^{-1}(c+d x)}\right )-2 \log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right )\right )\right )-\frac{a^2}{c+d x}+2 a b \left (2 \tan ^{-1}\left (\tanh \left (\frac{1}{2} \cosh ^{-1}(c+d x)\right )\right )-\frac{\cosh ^{-1}(c+d x)}{c+d x}\right )}{d e^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-(a^2/(c + d*x)) + 2*a*b*(-(ArcCosh[c + d*x]/(c + d*x)) + 2*ArcTan[Tanh[ArcCosh[c + d*x]/2]]) - I*b^2*(ArcCos
h[c + d*x]*(((-I)*ArcCosh[c + d*x])/(c + d*x) + 2*Log[1 - I/E^ArcCosh[c + d*x]] - 2*Log[1 + I/E^ArcCosh[c + d*
x]]) + 2*PolyLog[2, (-I)/E^ArcCosh[c + d*x]] - 2*PolyLog[2, I/E^ArcCosh[c + d*x]]))/(d*e^2)

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Maple [A]  time = 0.049, size = 290, normalized size = 2.6 \begin{align*} -{\frac{{a}^{2}}{d{e}^{2} \left ( dx+c \right ) }}-{\frac{{b}^{2} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{2}}{d{e}^{2} \left ( dx+c \right ) }}-{\frac{2\,i{b}^{2}{\rm arccosh} \left (dx+c\right )}{d{e}^{2}}\ln \left ( 1+i \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) \right ) }+{\frac{2\,i{b}^{2}{\rm arccosh} \left (dx+c\right )}{d{e}^{2}}\ln \left ( 1-i \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) \right ) }-{\frac{2\,i{b}^{2}}{d{e}^{2}}{\it dilog} \left ( 1+i \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) \right ) }+{\frac{2\,i{b}^{2}}{d{e}^{2}}{\it dilog} \left ( 1-i \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) \right ) }-2\,{\frac{ab{\rm arccosh} \left (dx+c\right )}{d{e}^{2} \left ( dx+c \right ) }}-2\,{\frac{ab\sqrt{dx+c-1}\sqrt{dx+c+1}\arctan \left ({\frac{1}{\sqrt{ \left ( dx+c \right ) ^{2}-1}}} \right ) }{d{e}^{2}\sqrt{ \left ( dx+c \right ) ^{2}-1}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d*a^2/e^2/(d*x+c)-1/d*b^2/e^2*arccosh(d*x+c)^2/(d*x+c)-2*I/d*b^2/e^2*arccosh(d*x+c)*ln(1+I*(d*x+c+(d*x+c-1)
^(1/2)*(d*x+c+1)^(1/2)))+2*I/d*b^2/e^2*arccosh(d*x+c)*ln(1-I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))-2*I/d*b^
2/e^2*dilog(1+I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))+2*I/d*b^2/e^2*dilog(1-I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c
+1)^(1/2)))-2/d*a*b/e^2/(d*x+c)*arccosh(d*x+c)-2/d*a*b/e^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/((d*x+c)^2-1)^(1/2)
*arctan(1/((d*x+c)^2-1)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError