\(\int \frac {(d+e x^2)^2}{\sqrt {a+b \text {arccosh}(c x)}} \, dx\) [169]

Optimal result

Integrand size = 22, antiderivative size = 608 \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+b \text {arccosh}(c x)}} \, dx=-\frac {d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}-\frac {d e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}-\frac {e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 \sqrt {b} c^5}-\frac {d e e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}-\frac {e^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5}-\frac {e^2 e^{\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5}+\frac {d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}+\frac {d e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}+\frac {e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 \sqrt {b} c^5}+\frac {d e e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}+\frac {e^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5}+\frac {e^2 e^{-\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5} \] Output:

-1/2*d^2*exp(a/b)*Pi^(1/2)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(1/2)/c 
-1/4*d*e*exp(a/b)*Pi^(1/2)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(1/2)/c 
^3-1/16*e^2*exp(a/b)*Pi^(1/2)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(1/2 
)/c^5-1/12*d*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^3-1/32*e^2*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^5-1/160*e^2*exp(5*a/b)*5^(1/ 
2)*Pi^(1/2)*erf(5^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(1/2)/c^5+1/2* 
d^2*Pi^(1/2)*erfi((a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(1/2)/c/exp(a/b)+1/4 
*d*e*Pi^(1/2)*erfi((a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(1/2)/c^3/exp(a/b)+ 
1/16*e^2*Pi^(1/2)*erfi((a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(1/2)/c^5/exp(a 
/b)+1/12*d*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)+1/32*e^2*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)*(a+b*arcc 
osh(c*x))^(1/2)/b^(1/2))/b^(1/2)/c^5/exp(3*a/b)+1/160*e^2*5^(1/2)*Pi^(1/2) 
*erfi(5^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(1/2)/c^5/exp(5*a/b)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 530, normalized size of antiderivative = 0.87 \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+b \text {arccosh}(c x)}} \, dx=\frac {e^{-\frac {5 a}{b}} \left (30 \left (8 c^4 d^2+4 c^2 d e+e^2\right ) e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c x)\right )+3 \sqrt {5} e^2 \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )+40 \sqrt {3} c^2 d e e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+15 \sqrt {3} e^2 e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+240 c^4 d^2 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )+120 c^2 d e e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )+30 e^2 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )+40 \sqrt {3} c^2 d e e^{\frac {8 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+15 \sqrt {3} e^2 e^{\frac {8 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+3 \sqrt {5} e^2 e^{\frac {10 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {5 (a+b \text {arccosh}(c x))}{b}\right )\right )}{480 c^5 \sqrt {a+b \text {arccosh}(c x)}} \] Input:

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

Output:

(30*(8*c^4*d^2 + 4*c^2*d*e + e^2)*E^((6*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gam 
ma[1/2, a/b + ArcCosh[c*x]] + 3*Sqrt[5]*e^2*Sqrt[-((a + b*ArcCosh[c*x])/b) 
]*Gamma[1/2, (-5*(a + b*ArcCosh[c*x]))/b] + 40*Sqrt[3]*c^2*d*e*E^((2*a)/b) 
*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcCosh[c*x]))/b] + 
 15*Sqrt[3]*e^2*E^((2*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-3 
*(a + b*ArcCosh[c*x]))/b] + 240*c^4*d^2*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[ 
c*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c*x])/b)] + 120*c^2*d*e*E^((4*a)/b)* 
Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c*x])/b)] + 30 
*e^2*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, -((a + b*ArcCo 
sh[c*x])/b)] + 40*Sqrt[3]*c^2*d*e*E^((8*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gam 
ma[1/2, (3*(a + b*ArcCosh[c*x]))/b] + 15*Sqrt[3]*e^2*E^((8*a)/b)*Sqrt[a/b 
+ ArcCosh[c*x]]*Gamma[1/2, (3*(a + b*ArcCosh[c*x]))/b] + 3*Sqrt[5]*e^2*E^( 
(10*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (5*(a + b*ArcCosh[c*x]))/b]) 
/(480*c^5*E^((5*a)/b)*Sqrt[a + b*ArcCosh[c*x]])
 

Rubi [A] (verified)

Time = 1.43 (sec) , antiderivative size = 608, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.

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

\(\Big \downarrow \) 6324

\(\displaystyle \int \left (\frac {d^2}{\sqrt {a+b \text {arccosh}(c x)}}+\frac {2 d e x^2}{\sqrt {a+b \text {arccosh}(c x)}}+\frac {e^2 x^4}{\sqrt {a+b \text {arccosh}(c x)}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {\pi } e^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 \sqrt {b} c^5}-\frac {\sqrt {3 \pi } e^2 e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5}-\frac {\sqrt {\frac {\pi }{5}} e^2 e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5}+\frac {\sqrt {\pi } e^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 \sqrt {b} c^5}+\frac {\sqrt {3 \pi } e^2 e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5}+\frac {\sqrt {\frac {\pi }{5}} e^2 e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 \sqrt {b} c^5}-\frac {\sqrt {\pi } d e e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}-\frac {\sqrt {\frac {\pi }{3}} d e e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}+\frac {\sqrt {\pi } d e e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}+\frac {\sqrt {\frac {\pi }{3}} d e e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^3}-\frac {\sqrt {\pi } d^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}+\frac {\sqrt {\pi } d^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}\)

Input:

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

Output:

-1/2*(d^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(Sqrt[b] 
*c) - (d*e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*Sqrt 
[b]*c^3) - (e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(1 
6*Sqrt[b]*c^5) - (d*e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCo 
sh[c*x]])/Sqrt[b]])/(4*Sqrt[b]*c^3) - (e^2*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqr 
t[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(32*Sqrt[b]*c^5) - (e^2*E^((5*a)/ 
b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(32*Sqrt[b] 
*c^5) + (d^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c 
*E^(a/b)) + (d*e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*Sqrt[ 
b]*c^3*E^(a/b)) + (e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(1 
6*Sqrt[b]*c^5*E^(a/b)) + (d*e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[ 
c*x]])/Sqrt[b]])/(4*Sqrt[b]*c^3*E^((3*a)/b)) + (e^2*Sqrt[3*Pi]*Erfi[(Sqrt[ 
3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(32*Sqrt[b]*c^5*E^((3*a)/b)) + (e^2 
*Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(32*Sqrt[b]* 
c^5*E^((5*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]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

integrate((e*x^2+d)^2/(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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+b \text {arccosh}(c x)}} \, dx=\left (\int \frac {\sqrt {\mathit {acosh} \left (c x \right ) b +a}}{\mathit {acosh} \left (c x \right ) b +a}d x \right ) d^{2}+\left (\int \frac {\sqrt {\mathit {acosh} \left (c x \right ) b +a}\, x^{4}}{\mathit {acosh} \left (c x \right ) b +a}d x \right ) e^{2}+2 \left (\int \frac {\sqrt {\mathit {acosh} \left (c x \right ) b +a}\, x^{2}}{\mathit {acosh} \left (c x \right ) b +a}d x \right ) d e \] Input:

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

Output:

int(sqrt(acosh(c*x)*b + a)/(acosh(c*x)*b + a),x)*d**2 + int((sqrt(acosh(c* 
x)*b + a)*x**4)/(acosh(c*x)*b + a),x)*e**2 + 2*int((sqrt(acosh(c*x)*b + a) 
*x**2)/(acosh(c*x)*b + a),x)*d*e