\(\int \frac {a+b \text {arccosh}(c x)}{(d+e x^2)^{3/2}} \, dx\) [145]

Optimal result

Integrand size = 20, antiderivative size = 101 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {x (a+b \text {arccosh}(c x))}{d \sqrt {d+e x^2}}-\frac {b \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{d \sqrt {e} \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

x*(a+b*arccosh(c*x))/d/(e*x^2+d)^(1/2)-b*(c^2*x^2-1)^(1/2)*arctanh(e^(1/2) 
*(c^2*x^2-1)^(1/2)/c/(e*x^2+d)^(1/2))/d/e^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2 
)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 13.23 (sec) , antiderivative size = 556, normalized size of antiderivative = 5.50 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {a x+b x \text {arccosh}(c x)+\frac {2 b (-1+c x)^{3/2} \sqrt {\frac {\left (c \sqrt {d}-i \sqrt {e}\right ) (1+c x)}{\left (c \sqrt {d}+i \sqrt {e}\right ) (-1+c x)}} \left (\frac {c \left (-i c \sqrt {d}+\sqrt {e}\right ) \left (i \sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {1+\frac {i c \sqrt {d}}{\sqrt {e}}-c x+\frac {i \sqrt {e} x}{\sqrt {d}}}{1-c x}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{2-2 c x}}\right ),\frac {4 i c \sqrt {d} \sqrt {e}}{\left (c \sqrt {d}+i \sqrt {e}\right )^2}\right )}{-1+c x}+c \sqrt {d} \left (-c \sqrt {d}+i \sqrt {e}\right ) \sqrt {\frac {\left (c^2 d+e\right ) \left (d+e x^2\right )}{d e (-1+c x)^2}} \sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{1-c x}} \operatorname {EllipticPi}\left (\frac {2 c \sqrt {d}}{c \sqrt {d}+i \sqrt {e}},\arcsin \left (\sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{2-2 c x}}\right ),\frac {4 i c \sqrt {d} \sqrt {e}}{\left (c \sqrt {d}+i \sqrt {e}\right )^2}\right )\right )}{c \left (c^2 d+e\right ) \sqrt {1+c x} \sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{1-c x}}}}{d \sqrt {d+e x^2}} \] Input:

Integrate[(a + b*ArcCosh[c*x])/(d + e*x^2)^(3/2),x]
 

Output:

(a*x + b*x*ArcCosh[c*x] + (2*b*(-1 + c*x)^(3/2)*Sqrt[((c*Sqrt[d] - I*Sqrt[ 
e])*(1 + c*x))/((c*Sqrt[d] + I*Sqrt[e])*(-1 + c*x))]*((c*((-I)*c*Sqrt[d] + 
 Sqrt[e])*(I*Sqrt[d] + Sqrt[e]*x)*Sqrt[(1 + (I*c*Sqrt[d])/Sqrt[e] - c*x + 
(I*Sqrt[e]*x)/Sqrt[d])/(1 - c*x)]*EllipticF[ArcSin[Sqrt[-((-1 + (I*Sqrt[e] 
*x)/Sqrt[d] + c*((I*Sqrt[d])/Sqrt[e] + x))/(2 - 2*c*x))]], ((4*I)*c*Sqrt[d 
]*Sqrt[e])/(c*Sqrt[d] + I*Sqrt[e])^2])/(-1 + c*x) + c*Sqrt[d]*(-(c*Sqrt[d] 
) + I*Sqrt[e])*Sqrt[((c^2*d + e)*(d + e*x^2))/(d*e*(-1 + c*x)^2)]*Sqrt[-(( 
-1 + (I*Sqrt[e]*x)/Sqrt[d] + c*((I*Sqrt[d])/Sqrt[e] + x))/(1 - c*x))]*Elli 
pticPi[(2*c*Sqrt[d])/(c*Sqrt[d] + I*Sqrt[e]), ArcSin[Sqrt[-((-1 + (I*Sqrt[ 
e]*x)/Sqrt[d] + c*((I*Sqrt[d])/Sqrt[e] + x))/(2 - 2*c*x))]], ((4*I)*c*Sqrt 
[d]*Sqrt[e])/(c*Sqrt[d] + I*Sqrt[e])^2]))/(c*(c^2*d + e)*Sqrt[1 + c*x]*Sqr 
t[-((-1 + (I*Sqrt[e]*x)/Sqrt[d] + c*((I*Sqrt[d])/Sqrt[e] + x))/(1 - c*x))] 
))/(d*Sqrt[d + e*x^2])
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6323, 27, 2038, 353, 66, 221}

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.

\(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 6323

\(\displaystyle \frac {x (a+b \text {arccosh}(c x))}{d \sqrt {d+e x^2}}-b c \int \frac {x}{d \sqrt {c x-1} \sqrt {c x+1} \sqrt {e x^2+d}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x (a+b \text {arccosh}(c x))}{d \sqrt {d+e x^2}}-\frac {b c \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1} \sqrt {e x^2+d}}dx}{d}\)

\(\Big \downarrow \) 2038

\(\displaystyle \frac {x (a+b \text {arccosh}(c x))}{d \sqrt {d+e x^2}}-\frac {b c \sqrt {c^2 x^2-1} \int \frac {x}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx}{d \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 353

\(\displaystyle \frac {x (a+b \text {arccosh}(c x))}{d \sqrt {d+e x^2}}-\frac {b c \sqrt {c^2 x^2-1} \int \frac {1}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2}{2 d \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {x (a+b \text {arccosh}(c x))}{d \sqrt {d+e x^2}}-\frac {b c \sqrt {c^2 x^2-1} \int \frac {1}{c^2-e x^4}d\frac {\sqrt {c^2 x^2-1}}{\sqrt {e x^2+d}}}{d \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {x (a+b \text {arccosh}(c x))}{d \sqrt {d+e x^2}}-\frac {b \sqrt {c^2 x^2-1} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{d \sqrt {e} \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

Int[(a + b*ArcCosh[c*x])/(d + e*x^2)^(3/2),x]
 

Output:

(x*(a + b*ArcCosh[c*x]))/(d*Sqrt[d + e*x^2]) - (b*Sqrt[-1 + c^2*x^2]*ArcTa 
nh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(d*Sqrt[e]*Sqrt[-1 + 
 c*x]*Sqrt[1 + c*x])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 353

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] 
 :> Simp[1/2   Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ 
{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
 

rule 2038

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p 
_)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_), x_Symbol] :> Simp[(a1 + b1*x^(n/2)) 
^FracPart[p]*((a2 + b2*x^(n/2))^FracPart[p]/(a1*a2 + b1*b2*x^n)^FracPart[p] 
)   Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a1, b1, a2, 
 b2, c, d, n, p, q}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] &&  !(Eq 
Q[n, 2] && IGtQ[q, 0])
 

rule 6323

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symb 
ol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcCosh[c*x])   u, 
 x] - Simp[b*c   Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x] 
, x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] 
|| ILtQ[p + 1/2, 0])
 

Maple [F]

\[\int \frac {a +b \,\operatorname {arccosh}\left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]

Input:

int((a+b*arccosh(c*x))/(e*x^2+d)^(3/2),x)
 

Output:

int((a+b*arccosh(c*x))/(e*x^2+d)^(3/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 332, normalized size of antiderivative = 3.29 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx=\left [\frac {4 \, \sqrt {e x^{2} + d} b e x \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 4 \, \sqrt {e x^{2} + d} a e x + {\left (b e x^{2} + b d\right )} \sqrt {e} \log \left (8 \, c^{4} e^{2} x^{4} + c^{4} d^{2} - 6 \, c^{2} d e + 8 \, {\left (c^{4} d e - c^{2} e^{2}\right )} x^{2} - 4 \, {\left (2 \, c^{3} e x^{2} + c^{3} d - c e\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {e x^{2} + d} \sqrt {e} + e^{2}\right )}{4 \, {\left (d e^{2} x^{2} + d^{2} e\right )}}, \frac {2 \, \sqrt {e x^{2} + d} b e x \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, \sqrt {e x^{2} + d} a e x + {\left (b e x^{2} + b d\right )} \sqrt {-e} \arctan \left (\frac {{\left (2 \, c^{2} e x^{2} + c^{2} d - e\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {e x^{2} + d} \sqrt {-e}}{2 \, {\left (c^{3} e^{2} x^{4} - c d e + {\left (c^{3} d e - c e^{2}\right )} x^{2}\right )}}\right )}{2 \, {\left (d e^{2} x^{2} + d^{2} e\right )}}\right ] \] Input:

integrate((a+b*arccosh(c*x))/(e*x^2+d)^(3/2),x, algorithm="fricas")
 

Output:

[1/4*(4*sqrt(e*x^2 + d)*b*e*x*log(c*x + sqrt(c^2*x^2 - 1)) + 4*sqrt(e*x^2 
+ d)*a*e*x + (b*e*x^2 + b*d)*sqrt(e)*log(8*c^4*e^2*x^4 + c^4*d^2 - 6*c^2*d 
*e + 8*(c^4*d*e - c^2*e^2)*x^2 - 4*(2*c^3*e*x^2 + c^3*d - c*e)*sqrt(c^2*x^ 
2 - 1)*sqrt(e*x^2 + d)*sqrt(e) + e^2))/(d*e^2*x^2 + d^2*e), 1/2*(2*sqrt(e* 
x^2 + d)*b*e*x*log(c*x + sqrt(c^2*x^2 - 1)) + 2*sqrt(e*x^2 + d)*a*e*x + (b 
*e*x^2 + b*d)*sqrt(-e)*arctan(1/2*(2*c^2*e*x^2 + c^2*d - e)*sqrt(c^2*x^2 - 
 1)*sqrt(e*x^2 + d)*sqrt(-e)/(c^3*e^2*x^4 - c*d*e + (c^3*d*e - c*e^2)*x^2) 
))/(d*e^2*x^2 + d^2*e)]
 

Sympy [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((a+b*acosh(c*x))/(e*x**2+d)**(3/2),x)
 

Output:

Integral((a + b*acosh(c*x))/(d + e*x**2)**(3/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+b*arccosh(c*x))/(e*x^2+d)^(3/2),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+c^2*d>0)', see `assume?` for m 
ore detail
 

Giac [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((a+b*arccosh(c*x))/(e*x^2+d)^(3/2),x, algorithm="giac")
 

Output:

integrate((b*arccosh(c*x) + a)/(e*x^2 + d)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \] Input:

int((a + b*acosh(c*x))/(d + e*x^2)^(3/2),x)
 

Output:

int((a + b*acosh(c*x))/(d + e*x^2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {\mathit {acosh} \left (c x \right ) b +a}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x \] Input:

int((a+b*acosh(c*x))/(e*x^2+d)^(3/2),x)
                                                                                    
                                                                                    
 

Output:

int((a+b*acosh(c*x))/(e*x^2+d)^(3/2),x)