\(\int (d+e x^2) \sqrt {a+b \text {arccosh}(c x)} \, dx\) [161]

Optimal result

Integrand size = 20, antiderivative size = 322 \[ \int \left (d+e x^2\right ) \sqrt {a+b \text {arccosh}(c x)} \, dx=d x \sqrt {a+b \text {arccosh}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \text {arccosh}(c x)}-\frac {\sqrt {b} d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 c^3}-\frac {\sqrt {b} e e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{48 c^3}-\frac {\sqrt {b} d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 c^3}-\frac {\sqrt {b} e e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{48 c^3} \] Output:

d*x*(a+b*arccosh(c*x))^(1/2)+1/3*e*x^3*(a+b*arccosh(c*x))^(1/2)-1/4*b^(1/2 
)*d*exp(a/b)*Pi^(1/2)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))/c-1/16*b^(1/2) 
*e*exp(a/b)*Pi^(1/2)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))/c^3-1/144*b^(1/ 
2)*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))/c^3-1/4*b^(1/2)*d*Pi^(1/2)*erfi((a+b*arccosh(c*x))^(1/2)/b^(1/2))/c/e 
xp(a/b)-1/16*b^(1/2)*e*Pi^(1/2)*erfi((a+b*arccosh(c*x))^(1/2)/b^(1/2))/c^3 
/exp(a/b)-1/144*b^(1/2)*e*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)*(a+b*arccosh(c*x)) 
^(1/2)/b^(1/2))/c^3/exp(3*a/b)
 

Mathematica [A] (verified)

Time = 1.78 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.98 \[ \int \left (d+e x^2\right ) \sqrt {a+b \text {arccosh}(c x)} \, dx=\frac {d e^{-\frac {a}{b}} \sqrt {a+b \text {arccosh}(c x)} \left (\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arccosh}(c x)\right )}{\sqrt {\frac {a}{b}+\text {arccosh}(c x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {-\frac {a+b \text {arccosh}(c x)}{b}}}\right )}{2 c}+\frac {e e^{-\frac {3 a}{b}} \sqrt {a+b \text {arccosh}(c x)} \left (9 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arccosh}(c x)\right )+\sqrt {3} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {3}{2},-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+9 e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {3}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )+\sqrt {3} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {3}{2},\frac {3 (a+b \text {arccosh}(c x))}{b}\right )\right )}{72 c^3 \sqrt {-\frac {(a+b \text {arccosh}(c x))^2}{b^2}}} \] Input:

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

Output:

(d*Sqrt[a + b*ArcCosh[c*x]]*((E^((2*a)/b)*Gamma[3/2, a/b + ArcCosh[c*x]])/ 
Sqrt[a/b + ArcCosh[c*x]] + Gamma[3/2, -((a + b*ArcCosh[c*x])/b)]/Sqrt[-((a 
 + b*ArcCosh[c*x])/b)]))/(2*c*E^(a/b)) + (e*Sqrt[a + b*ArcCosh[c*x]]*(9*E^ 
((4*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[3/2, a/b + ArcCosh[c*x]] + 
 Sqrt[3]*Sqrt[a/b + ArcCosh[c*x]]*Gamma[3/2, (-3*(a + b*ArcCosh[c*x]))/b] 
+ 9*E^((2*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[3/2, -((a + b*ArcCosh[c*x]) 
/b)] + Sqrt[3]*E^((6*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[3/2, (3*( 
a + b*ArcCosh[c*x]))/b]))/(72*c^3*E^((3*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])^ 
2/b^2)])
 

Rubi [A] (verified)

Time = 1.68 (sec) , antiderivative size = 322, 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.

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

\(\Big \downarrow \) 6324

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {\pi } \sqrt {b} e e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 c^3}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{48 c^3}-\frac {\sqrt {\pi } \sqrt {b} e e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 c^3}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{48 c^3}-\frac {\sqrt {\pi } \sqrt {b} d e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {\pi } \sqrt {b} d e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 c}+d x \sqrt {a+b \text {arccosh}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \text {arccosh}(c x)}\)

Input:

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

Output:

d*x*Sqrt[a + b*ArcCosh[c*x]] + (e*x^3*Sqrt[a + b*ArcCosh[c*x]])/3 - (Sqrt[ 
b]*d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*c) - (Sqrt 
[b]*e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(16*c^3) - ( 
Sqrt[b]*e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sq 
rt[b]])/(48*c^3) - (Sqrt[b]*d*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[ 
b]])/(4*c*E^(a/b)) - (Sqrt[b]*e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqr 
t[b]])/(16*c^3*E^(a/b)) - (Sqrt[b]*e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*A 
rcCosh[c*x]])/Sqrt[b]])/(48*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]

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

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

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

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

Output:

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

Maxima [F]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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