3.6.29 \(\int \frac {(a+b \text {arccosh}(c x))^2}{d+e x^2} \, dx\) [529]

3.6.29.1 Optimal result

Integrand size = 20, antiderivative size = 763 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{d+e x^2} \, dx=\frac {(a+b \text {arccosh}(c x))^2 \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \text {arccosh}(c x))^2 \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {(a+b \text {arccosh}(c x))^2 \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \text {arccosh}(c x))^2 \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{\sqrt {-d} \sqrt {e}} \]

output

1/2*(a+b*arccosh(c*x))^2*ln(1-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c 
*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/(-d)^(1/2)/e^(1/2)-1/2*(a+b*arccosh(c*x))^2 
*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d-e)^( 
1/2)))/(-d)^(1/2)/e^(1/2)+1/2*(a+b*arccosh(c*x))^2*ln(1-(c*x+(c*x-1)^(1/2) 
*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/(-d)^(1/2)/e^(1/2 
)-1/2*(a+b*arccosh(c*x))^2*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/ 
(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/(-d)^(1/2)/e^(1/2)-b*(a+b*arccosh(c*x))*p 
olylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d- 
e)^(1/2)))/(-d)^(1/2)/e^(1/2)+b*(a+b*arccosh(c*x))*polylog(2,(c*x+(c*x-1)^ 
(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/(-d)^(1/2)/e 
^(1/2)-b*(a+b*arccosh(c*x))*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e 
^(1/2)/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/(-d)^(1/2)/e^(1/2)+b*(a+b*arccosh( 
c*x))*polylog(2,(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(- 
c^2*d-e)^(1/2)))/(-d)^(1/2)/e^(1/2)+b^2*polylog(3,-(c*x+(c*x-1)^(1/2)*(c*x 
+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/(-d)^(1/2)/e^(1/2)-b^2 
*polylog(3,(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d 
-e)^(1/2)))/(-d)^(1/2)/e^(1/2)+b^2*polylog(3,-(c*x+(c*x-1)^(1/2)*(c*x+1)^( 
1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/(-d)^(1/2)/e^(1/2)-b^2*poly 
log(3,(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d-e)^( 
1/2)))/(-d)^(1/2)/e^(1/2)
 

3.6.29.2 Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 623, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{d+e x^2} \, dx=\frac {-(a+b \text {arccosh}(c x))^2 \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )+(a+b \text {arccosh}(c x))^2 \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )+(a+b \text {arccosh}(c x))^2 \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )-(a+b \text {arccosh}(c x))^2 \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )+2 b (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )-2 b (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )-2 b (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )+2 b (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )-2 b^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )+2 b^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )+2 b^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )-2 b^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 \sqrt {-d} \sqrt {e}} \]

input

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

output

(-((a + b*ArcCosh[c*x])^2*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - S 
qrt[-(c^2*d) - e])]) + (a + b*ArcCosh[c*x])^2*Log[1 + (Sqrt[e]*E^ArcCosh[c 
*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) - e])] + (a + b*ArcCosh[c*x])^2*Log[1 
- (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])] - (a + b*Arc 
Cosh[c*x])^2*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) 
- e])] + 2*b*(a + b*ArcCosh[c*x])*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*S 
qrt[-d] - Sqrt[-(c^2*d) - e])] - 2*b*(a + b*ArcCosh[c*x])*PolyLog[2, (Sqrt 
[e]*E^ArcCosh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) - e])] - 2*b*(a + b*Arc 
Cosh[c*x])*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2* 
d) - e]))] + 2*b*(a + b*ArcCosh[c*x])*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/ 
(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])] - 2*b^2*PolyLog[3, (Sqrt[e]*E^ArcCosh[c 
*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])] + 2*b^2*PolyLog[3, (Sqrt[e]*E^ArcC 
osh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) - e])] + 2*b^2*PolyLog[3, -((Sqrt 
[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))] - 2*b^2*PolyLog[3, 
 (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*Sqrt[-d]* 
Sqrt[e])
 

3.6.29.3 Rubi [A] (verified)

Time = 1.76 (sec) , antiderivative size = 763, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6324, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(a+b \text {arccosh}(c x))^2}{d+e x^2} \, dx\)

\(\Big \downarrow \) 6324

\(\displaystyle \int \left (\frac {\sqrt {-d} (a+b \text {arccosh}(c x))^2}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} (a+b \text {arccosh}(c x))^2}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {(a+b \text {arccosh}(c x))^2 \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \text {arccosh}(c x))^2 \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {(a+b \text {arccosh}(c x))^2 \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \text {arccosh}(c x))^2 \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{\sqrt {-d} \sqrt {e}}\)

input

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

output

((a + b*ArcCosh[c*x])^2*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqr 
t[-(c^2*d) - e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcCosh[c*x])^2*Log[1 + 
(Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*Sqrt[-d]*S 
qrt[e]) + ((a + b*ArcCosh[c*x])^2*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt 
[-d] + Sqrt[-(c^2*d) - e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcCosh[c*x])^ 
2*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2* 
Sqrt[-d]*Sqrt[e]) - (b*(a + b*ArcCosh[c*x])*PolyLog[2, -((Sqrt[e]*E^ArcCos 
h[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])/(Sqrt[-d]*Sqrt[e]) + (b*(a + 
b*ArcCosh[c*x])*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c 
^2*d) - e])])/(Sqrt[-d]*Sqrt[e]) - (b*(a + b*ArcCosh[c*x])*PolyLog[2, -((S 
qrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(Sqrt[-d]*Sqrt 
[e]) + (b*(a + b*ArcCosh[c*x])*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt 
[-d] + Sqrt[-(c^2*d) - e])])/(Sqrt[-d]*Sqrt[e]) + (b^2*PolyLog[3, -((Sqrt[ 
e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])/(Sqrt[-d]*Sqrt[e]) 
 - (b^2*PolyLog[3, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - 
e])])/(Sqrt[-d]*Sqrt[e]) + (b^2*PolyLog[3, -((Sqrt[e]*E^ArcCosh[c*x])/(c*S 
qrt[-d] + Sqrt[-(c^2*d) - e]))])/(Sqrt[-d]*Sqrt[e]) - (b^2*PolyLog[3, (Sqr 
t[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(Sqrt[-d]*Sqrt[e] 
)
 

3.6.29.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6324

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && 
(p > 0 || IGtQ[n, 0])
 

3.6.29.4 Maple [F]

\[\int \frac {\left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{e \,x^{2}+d}d x\]

input

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

output

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

3.6.29.5 Fricas [F]

\[ \int \frac {(a+b \text {arccosh}(c x))^2}{d+e x^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{e x^{2} + d} \,d x } \]

input

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

output

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

3.6.29.6 Sympy [F]

\[ \int \frac {(a+b \text {arccosh}(c x))^2}{d+e x^2} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{d + e x^{2}}\, dx \]

input

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

output

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

3.6.29.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{d+e x^2} \, dx=\text {Exception raised: ValueError} \]

input

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

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

3.6.29.8 Giac [F]

\[ \int \frac {(a+b \text {arccosh}(c x))^2}{d+e x^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{e x^{2} + d} \,d x } \]

input

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

output

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

3.6.29.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{d+e x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{e\,x^2+d} \,d x \]

input

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

output

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