\(\int \frac {d+e x^2}{\sqrt {a+b \text {arccosh}(c x)}} \, dx\) [170]

Optimal result

Integrand size = 20, antiderivative size = 287 \[ \int \frac {d+e x^2}{\sqrt {a+b \text {arccosh}(c x)}} \, dx=-\frac {d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}-\frac {e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}-\frac {e e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}+\frac {d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}+\frac {e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}+\frac {e e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3} \] Output:

-1/2*d*exp(a/b)*Pi^(1/2)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(1/2)/c-1 
/8*e*exp(a/b)*Pi^(1/2)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(1/2)/c^3-1 
/24*e*exp(3*a/b)*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^( 
1/2))/b^(1/2)/c^3+1/2*d*Pi^(1/2)*erfi((a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^ 
(1/2)/c/exp(a/b)+1/8*e*Pi^(1/2)*erfi((a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^( 
1/2)/c^3/exp(a/b)+1/24*e*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)*(a+b*arccosh(c*x))^ 
(1/2)/b^(1/2))/b^(1/2)/c^3/exp(3*a/b)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.74 \[ \int \frac {d+e x^2}{\sqrt {a+b \text {arccosh}(c x)}} \, dx=\frac {e^{-\frac {3 a}{b}} \left (3 \left (4 c^2 d+e\right ) e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c x)\right )+\sqrt {3} e \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+3 \left (4 c^2 d+e\right ) e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )+\sqrt {3} e e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arccosh}(c x))}{b}\right )\right )}{24 c^3 \sqrt {a+b \text {arccosh}(c x)}} \] Input:

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

Output:

(3*(4*c^2*d + e)*E^((4*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, a/b + Arc 
Cosh[c*x]] + Sqrt[3]*e*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-3*(a + 
 b*ArcCosh[c*x]))/b] + 3*(4*c^2*d + e)*E^((2*a)/b)*Sqrt[-((a + b*ArcCosh[c 
*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c*x])/b)] + Sqrt[3]*e*E^((6*a)/b)*Sqr 
t[a/b + ArcCosh[c*x]]*Gamma[1/2, (3*(a + b*ArcCosh[c*x]))/b])/(24*c^3*E^(( 
3*a)/b)*Sqrt[a + b*ArcCosh[c*x]])
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 287, 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.

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

\(\Big \downarrow \) 6324

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {\pi } e e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}-\frac {\sqrt {\frac {\pi }{3}} e e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}+\frac {\sqrt {\pi } e e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}+\frac {\sqrt {\frac {\pi }{3}} e e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 \sqrt {b} c^3}-\frac {\sqrt {\pi } d e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}+\frac {\sqrt {\pi } d e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}\)

Input:

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

Output:

-1/2*(d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(Sqrt[b]*c 
) - (e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*Sqrt[b]* 
c^3) - (e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sq 
rt[b]])/(8*Sqrt[b]*c^3) + (d*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b 
]])/(2*Sqrt[b]*c*E^(a/b)) + (e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt 
[b]])/(8*Sqrt[b]*c^3*E^(a/b)) + (e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*Arc 
Cosh[c*x]])/Sqrt[b]])/(8*Sqrt[b]*c^3*E^((3*a)/b))
 

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])
 

Maple [F]

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

Input:

int((e*x^2+d)/(a+b*arccosh(c*x))^(1/2),x)
 

Output:

int((e*x^2+d)/(a+b*arccosh(c*x))^(1/2),x)
 

Fricas [F(-2)]

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

integrate((e*x^2+d)/(a+b*arccosh(c*x))^(1/2),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

integrate((e*x**2+d)/(a+b*acosh(c*x))**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate((e*x^2+d)/(a+b*arccosh(c*x))^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((e*x^2+d)/(a+b*arccosh(c*x))^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((d + e*x^2)/(a + b*acosh(c*x))^(1/2),x)
 

Output:

int((d + e*x^2)/(a + b*acosh(c*x))^(1/2), x)
 

Reduce [F]

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

int((e*x^2+d)/(a+b*acosh(c*x))^(1/2),x)
 

Output:

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