\(\int \frac {a+b \text {arccosh}(c x)}{(d+e x^2)^{7/2}} \, dx\) [147]

Optimal result

Integrand size = 20, antiderivative size = 264 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{7/2}} \, dx=-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{15 d \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}-\frac {2 b c \left (3 c^2 d+2 e\right ) \sqrt {-1+c x} \sqrt {1+c x}}{15 d^2 \left (c^2 d+e\right )^2 \sqrt {d+e x^2}}+\frac {x (a+b \text {arccosh}(c x))}{5 d \left (d+e x^2\right )^{5/2}}+\frac {4 x (a+b \text {arccosh}(c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {8 x (a+b \text {arccosh}(c x))}{15 d^3 \sqrt {d+e x^2}}-\frac {8 b \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{15 d^3 \sqrt {e} \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-1/15*b*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(c^2*d+e)/(e*x^2+d)^(3/2)-2/15*b*c 
*(3*c^2*d+2*e)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/(c^2*d+e)^2/(e*x^2+d)^(1/2) 
+1/5*x*(a+b*arccosh(c*x))/d/(e*x^2+d)^(5/2)+4/15*x*(a+b*arccosh(c*x))/d^2/ 
(e*x^2+d)^(3/2)+8/15*x*(a+b*arccosh(c*x))/d^3/(e*x^2+d)^(1/2)-8/15*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^3/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 = 2.83 (sec) , antiderivative size = 685, normalized size of antiderivative = 2.59 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{7/2}} \, dx=\frac {\frac {a x \left (15 d^2+20 d e x^2+8 e^2 x^4\right )}{d^3}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right ) \left (e \left (5 d+4 e x^2\right )+c^2 d \left (7 d+6 e x^2\right )\right )}{d^2 \left (c^2 d+e\right )^2}+\frac {b x \left (15 d^2+20 d e x^2+8 e^2 x^4\right ) \text {arccosh}(c x)}{d^3}+\frac {16 b (-1+c x)^{3/2} \sqrt {\frac {\left (c \sqrt {d}-i \sqrt {e}\right ) (1+c x)}{\left (c \sqrt {d}+i \sqrt {e}\right ) (-1+c x)}} \left (d+e x^2\right )^2 \left (\frac {c \left (-i c \sqrt {d}+\sqrt {e}\right ) \left (i \sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {1+\frac {i c \sqrt {d}}{\sqrt {e}}-c x+\frac {i \sqrt {e} x}{\sqrt {d}}}{1-c x}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{2-2 c x}}\right ),\frac {4 i c \sqrt {d} \sqrt {e}}{\left (c \sqrt {d}+i \sqrt {e}\right )^2}\right )}{-1+c x}+c \sqrt {d} \left (-c \sqrt {d}+i \sqrt {e}\right ) \sqrt {\frac {\left (c^2 d+e\right ) \left (d+e x^2\right )}{d e (-1+c x)^2}} \sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{1-c x}} \operatorname {EllipticPi}\left (\frac {2 c \sqrt {d}}{c \sqrt {d}+i \sqrt {e}},\arcsin \left (\sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{2-2 c x}}\right ),\frac {4 i c \sqrt {d} \sqrt {e}}{\left (c \sqrt {d}+i \sqrt {e}\right )^2}\right )\right )}{c d^3 \left (c^2 d+e\right ) \sqrt {1+c x} \sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{1-c x}}}}{15 \left (d+e x^2\right )^{5/2}} \] Input:

Integrate[(a + b*ArcCosh[c*x])/(d + e*x^2)^(7/2),x]
 

Output:

((a*x*(15*d^2 + 20*d*e*x^2 + 8*e^2*x^4))/d^3 - (b*c*Sqrt[-1 + c*x]*Sqrt[1 
+ c*x]*(d + e*x^2)*(e*(5*d + 4*e*x^2) + c^2*d*(7*d + 6*e*x^2)))/(d^2*(c^2* 
d + e)^2) + (b*x*(15*d^2 + 20*d*e*x^2 + 8*e^2*x^4)*ArcCosh[c*x])/d^3 + (16 
*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))]*(d + e*x^2)^2*((c*((-I)*c*Sqrt[d] + Sqrt[e])*(I*S 
qrt[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))]*EllipticPi[(2*c*Sq 
rt[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*d^3*(c^2*d + e)*Sqrt[1 + c*x]*Sqrt[-((-1 + 
(I*Sqrt[e]*x)/Sqrt[d] + c*((I*Sqrt[d])/Sqrt[e] + x))/(1 - c*x))]))/(15*(d 
+ e*x^2)^(5/2))
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6323, 27, 2038, 7266, 1193, 27, 87, 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.

\(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 6323

\(\displaystyle -b c \int \frac {x \left (8 e^2 x^4+20 d e x^2+15 d^2\right )}{15 d^3 \sqrt {c x-1} \sqrt {c x+1} \left (e x^2+d\right )^{5/2}}dx+\frac {8 x (a+b \text {arccosh}(c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \text {arccosh}(c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \text {arccosh}(c x))}{5 d \left (d+e x^2\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c \int \frac {x \left (8 e^2 x^4+20 d e x^2+15 d^2\right )}{\sqrt {c x-1} \sqrt {c x+1} \left (e x^2+d\right )^{5/2}}dx}{15 d^3}+\frac {8 x (a+b \text {arccosh}(c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \text {arccosh}(c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \text {arccosh}(c x))}{5 d \left (d+e x^2\right )^{5/2}}\)

\(\Big \downarrow \) 2038

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \frac {x \left (8 e^2 x^4+20 d e x^2+15 d^2\right )}{\sqrt {c^2 x^2-1} \left (e x^2+d\right )^{5/2}}dx}{15 d^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 x (a+b \text {arccosh}(c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \text {arccosh}(c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \text {arccosh}(c x))}{5 d \left (d+e x^2\right )^{5/2}}\)

\(\Big \downarrow \) 7266

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \frac {8 e^2 x^4+20 d e x^2+15 d^2}{\sqrt {c^2 x^2-1} \left (e x^2+d\right )^{5/2}}dx^2}{30 d^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 x (a+b \text {arccosh}(c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \text {arccosh}(c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \text {arccosh}(c x))}{5 d \left (d+e x^2\right )^{5/2}}\)

\(\Big \downarrow \) 1193

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \left (\frac {2 \int \frac {3 \left (4 e \left (d c^2+e\right ) x^2+d \left (7 d c^2+6 e\right )\right )}{\sqrt {c^2 x^2-1} \left (e x^2+d\right )^{3/2}}dx^2}{3 \left (c^2 d+e\right )}+\frac {2 d^2 \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}\right )}{30 d^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 x (a+b \text {arccosh}(c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \text {arccosh}(c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \text {arccosh}(c x))}{5 d \left (d+e x^2\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \left (\frac {2 \int \frac {4 e \left (d c^2+e\right ) x^2+d \left (7 d c^2+6 e\right )}{\sqrt {c^2 x^2-1} \left (e x^2+d\right )^{3/2}}dx^2}{c^2 d+e}+\frac {2 d^2 \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}\right )}{30 d^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 x (a+b \text {arccosh}(c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \text {arccosh}(c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \text {arccosh}(c x))}{5 d \left (d+e x^2\right )^{5/2}}\)

\(\Big \downarrow \) 87

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \left (\frac {2 \left (4 \left (c^2 d+e\right ) \int \frac {1}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2+\frac {2 d \sqrt {c^2 x^2-1} \left (3 c^2 d+2 e\right )}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{c^2 d+e}+\frac {2 d^2 \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}\right )}{30 d^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 x (a+b \text {arccosh}(c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \text {arccosh}(c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \text {arccosh}(c x))}{5 d \left (d+e x^2\right )^{5/2}}\)

\(\Big \downarrow \) 66

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \left (\frac {2 \left (8 \left (c^2 d+e\right ) \int \frac {1}{c^2-e x^4}d\frac {\sqrt {c^2 x^2-1}}{\sqrt {e x^2+d}}+\frac {2 d \sqrt {c^2 x^2-1} \left (3 c^2 d+2 e\right )}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{c^2 d+e}+\frac {2 d^2 \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}\right )}{30 d^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 x (a+b \text {arccosh}(c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \text {arccosh}(c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \text {arccosh}(c x))}{5 d \left (d+e x^2\right )^{5/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {8 x (a+b \text {arccosh}(c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \text {arccosh}(c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \text {arccosh}(c x))}{5 d \left (d+e x^2\right )^{5/2}}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {2 \left (\frac {8 \left (c^2 d+e\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{c \sqrt {e}}+\frac {2 d \sqrt {c^2 x^2-1} \left (3 c^2 d+2 e\right )}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{c^2 d+e}+\frac {2 d^2 \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}\right )}{30 d^3 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

Int[(a + b*ArcCosh[c*x])/(d + e*x^2)^(7/2),x]
 

Output:

(x*(a + b*ArcCosh[c*x]))/(5*d*(d + e*x^2)^(5/2)) + (4*x*(a + b*ArcCosh[c*x 
]))/(15*d^2*(d + e*x^2)^(3/2)) + (8*x*(a + b*ArcCosh[c*x]))/(15*d^3*Sqrt[d 
 + e*x^2]) - (b*c*Sqrt[-1 + c^2*x^2]*((2*d^2*Sqrt[-1 + c^2*x^2])/((c^2*d + 
 e)*(d + e*x^2)^(3/2)) + (2*((2*d*(3*c^2*d + 2*e)*Sqrt[-1 + c^2*x^2])/((c^ 
2*d + e)*Sqrt[d + e*x^2]) + (8*(c^2*d + e)*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2* 
x^2])/(c*Sqrt[d + e*x^2])])/(c*Sqrt[e])))/(c^2*d + e)))/(30*d^3*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 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

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 1193

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x 
 + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + b*x + c*x^2)^p, d + 
e*x, x]}, Simp[R*(d + e*x)^(m + 1)*((f + g*x)^(n + 1)/((m + 1)*(e*f - d*g)) 
), x] + Simp[1/((m + 1)*(e*f - d*g))   Int[(d + e*x)^(m + 1)*(f + g*x)^n*Ex 
pandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x], x]] /; FreeQ[{a 
, b, c, d, e, f, g, n}, x] && IGtQ[p, 0] && ILtQ[2*m, -2] &&  !IntegerQ[n] 
&&  !(EqQ[m, -2] && EqQ[p, 1] && EqQ[2*c*d - b*e, 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])
 

rule 7266

Int[(u_)*(x_)^(m_.), x_Symbol] :> Simp[1/(m + 1)   Subst[Int[SubstFor[x^(m 
+ 1), u, x], x], x, x^(m + 1)], x] /; FreeQ[m, x] && NeQ[m, -1] && Function 
OfQ[x^(m + 1), u, x]
 

Maple [F]

\[\int \frac {a +b \,\operatorname {arccosh}\left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {7}{2}}}d x\]

Input:

int((a+b*arccosh(c*x))/(e*x^2+d)^(7/2),x)
 

Output:

int((a+b*arccosh(c*x))/(e*x^2+d)^(7/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 674 vs. \(2 (220) = 440\).

Time = 0.21 (sec) , antiderivative size = 1360, normalized size of antiderivative = 5.15 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((a+b*arccosh(c*x))/(e*x^2+d)^(7/2),x, algorithm="fricas")
 

Output:

[1/15*(2*(b*c^4*d^5 + 2*b*c^2*d^4*e + (b*c^4*d^2*e^3 + 2*b*c^2*d*e^4 + b*e 
^5)*x^6 + b*d^3*e^2 + 3*(b*c^4*d^3*e^2 + 2*b*c^2*d^2*e^3 + b*d*e^4)*x^4 + 
3*(b*c^4*d^4*e + 2*b*c^2*d^3*e^2 + b*d^2*e^3)*x^2)*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) + (8*(b*c^4*d^ 
2*e^3 + 2*b*c^2*d*e^4 + b*e^5)*x^5 + 20*(b*c^4*d^3*e^2 + 2*b*c^2*d^2*e^3 + 
 b*d*e^4)*x^3 + 15*(b*c^4*d^4*e + 2*b*c^2*d^3*e^2 + b*d^2*e^3)*x)*sqrt(e*x 
^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) + (8*(a*c^4*d^2*e^3 + 2*a*c^2*d*e^4 + 
 a*e^5)*x^5 + 20*(a*c^4*d^3*e^2 + 2*a*c^2*d^2*e^3 + a*d*e^4)*x^3 + 15*(a*c 
^4*d^4*e + 2*a*c^2*d^3*e^2 + a*d^2*e^3)*x - (7*b*c^3*d^4*e + 5*b*c*d^3*e^2 
 + 2*(3*b*c^3*d^2*e^3 + 2*b*c*d*e^4)*x^4 + (13*b*c^3*d^3*e^2 + 9*b*c*d^2*e 
^3)*x^2)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d))/(c^4*d^8*e + 2*c^2*d^7*e^2 + 
d^6*e^3 + (c^4*d^5*e^4 + 2*c^2*d^4*e^5 + d^3*e^6)*x^6 + 3*(c^4*d^6*e^3 + 2 
*c^2*d^5*e^4 + d^4*e^5)*x^4 + 3*(c^4*d^7*e^2 + 2*c^2*d^6*e^3 + d^5*e^4)*x^ 
2), 1/15*(4*(b*c^4*d^5 + 2*b*c^2*d^4*e + (b*c^4*d^2*e^3 + 2*b*c^2*d*e^4 + 
b*e^5)*x^6 + b*d^3*e^2 + 3*(b*c^4*d^3*e^2 + 2*b*c^2*d^2*e^3 + b*d*e^4)*x^4 
 + 3*(b*c^4*d^4*e + 2*b*c^2*d^3*e^2 + b*d^2*e^3)*x^2)*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)) + (8*(b*c^4*d^2*e^3 + 2*b*c^2*d* 
e^4 + b*e^5)*x^5 + 20*(b*c^4*d^3*e^2 + 2*b*c^2*d^2*e^3 + b*d*e^4)*x^3 +...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{7/2}} \, dx=\text {Timed out} \] Input:

integrate((a+b*acosh(c*x))/(e*x**2+d)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((a+b*arccosh(c*x))/(e*x^2+d)^(7/2),x, algorithm="maxima")
 

Output:

1/15*a*(8*x/(sqrt(e*x^2 + d)*d^3) + 4*x/((e*x^2 + d)^(3/2)*d^2) + 3*x/((e* 
x^2 + d)^(5/2)*d)) + b*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(e 
*x^2 + d)^(7/2), x)
 

Giac [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((a+b*arccosh(c*x))/(e*x^2+d)^(7/2),x, algorithm="giac")
 

Output:

integrate((b*arccosh(c*x) + a)/(e*x^2 + d)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^{7/2}} \,d x \] Input:

int((a + b*acosh(c*x))/(d + e*x^2)^(7/2),x)
 

Output:

int((a + b*acosh(c*x))/(d + e*x^2)^(7/2), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {\mathit {acosh} \left (c x \right ) b +a}{\left (e \,x^{2}+d \right )^{\frac {7}{2}}}d x \] Input:

int((a+b*acosh(c*x))/(e*x^2+d)^(7/2),x)
 

Output:

int((a+b*acosh(c*x))/(e*x^2+d)^(7/2),x)