3.4.75 \(\int \frac {x^2 (d-c^2 d x^2)^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx\) [375]

3.4.75.1 Optimal result

Integrand size = 29, antiderivative size = 462 \[ \int \frac {x^2 \left (d-c^2 d x^2\right )^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=-\frac {2 d^2 x^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {5 d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {d^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {3 d^2 e^{\frac {5 a}{b}} \sqrt {5 \pi } \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {d^2 e^{\frac {7 a}{b}} \sqrt {7 \pi } \text {erf}\left (\frac {\sqrt {7} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {5 d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {d^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {3 d^2 e^{-\frac {5 a}{b}} \sqrt {5 \pi } \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {d^2 e^{-\frac {7 a}{b}} \sqrt {7 \pi } \text {erfi}\left (\frac {\sqrt {7} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3} \]

output

5/64*d^2*exp(a/b)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)/c 
^3+5/64*d^2*erfi((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)/c^3/ex 
p(a/b)+1/64*d^2*exp(3*a/b)*erf(3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*3 
^(1/2)*Pi^(1/2)/b^(3/2)/c^3+1/64*d^2*erfi(3^(1/2)*(a+b*arccosh(c*x))^(1/2) 
/b^(1/2))*3^(1/2)*Pi^(1/2)/b^(3/2)/c^3/exp(3*a/b)-3/64*d^2*exp(5*a/b)*erf( 
5^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*5^(1/2)*Pi^(1/2)/b^(3/2)/c^3-3/6 
4*d^2*erfi(5^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*5^(1/2)*Pi^(1/2)/b^(3 
/2)/c^3/exp(5*a/b)+1/64*d^2*exp(7*a/b)*erf(7^(1/2)*(a+b*arccosh(c*x))^(1/2 
)/b^(1/2))*7^(1/2)*Pi^(1/2)/b^(3/2)/c^3+1/64*d^2*erfi(7^(1/2)*(a+b*arccosh 
(c*x))^(1/2)/b^(1/2))*7^(1/2)*Pi^(1/2)/b^(3/2)/c^3/exp(7*a/b)-2*d^2*x^2*(c 
*x-1)^(5/2)*(c*x+1)^(5/2)/b/c/(a+b*arccosh(c*x))^(1/2)
 

3.4.75.2 Mathematica [A] (warning: unable to verify)

Time = 2.48 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.08 \[ \int \frac {x^2 \left (d-c^2 d x^2\right )^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=-\frac {d^2 e^{-\frac {7 a}{b}} \left (10 e^{\frac {7 a}{b}} \sqrt {\frac {-1+c x}{1+c x}}+10 c e^{\frac {7 a}{b}} x \sqrt {\frac {-1+c x}{1+c x}}+5 e^{\frac {8 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c x)\right )-\sqrt {7} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {7 (a+b \text {arccosh}(c x))}{b}\right )+3 \sqrt {5} e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )-\sqrt {3} e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-5 e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )+\sqrt {3} e^{\frac {10 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-3 \sqrt {5} e^{\frac {12 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {5 (a+b \text {arccosh}(c x))}{b}\right )+\sqrt {7} e^{\frac {14 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {7 (a+b \text {arccosh}(c x))}{b}\right )+2 e^{\frac {7 a}{b}} \sinh (3 \text {arccosh}(c x))-6 e^{\frac {7 a}{b}} \sinh (5 \text {arccosh}(c x))+2 e^{\frac {7 a}{b}} \sinh (7 \text {arccosh}(c x))\right )}{64 b c^3 \sqrt {a+b \text {arccosh}(c x)}} \]

input

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

output

-1/64*(d^2*(10*E^((7*a)/b)*Sqrt[(-1 + c*x)/(1 + c*x)] + 10*c*E^((7*a)/b)*x 
*Sqrt[(-1 + c*x)/(1 + c*x)] + 5*E^((8*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma 
[1/2, a/b + ArcCosh[c*x]] - Sqrt[7]*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[ 
1/2, (-7*(a + b*ArcCosh[c*x]))/b] + 3*Sqrt[5]*E^((2*a)/b)*Sqrt[-((a + b*Ar 
cCosh[c*x])/b)]*Gamma[1/2, (-5*(a + b*ArcCosh[c*x]))/b] - Sqrt[3]*E^((4*a) 
/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcCosh[c*x]))/b 
] - 5*E^((6*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, -((a + b*ArcC 
osh[c*x])/b)] + Sqrt[3]*E^((10*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, ( 
3*(a + b*ArcCosh[c*x]))/b] - 3*Sqrt[5]*E^((12*a)/b)*Sqrt[a/b + ArcCosh[c*x 
]]*Gamma[1/2, (5*(a + b*ArcCosh[c*x]))/b] + Sqrt[7]*E^((14*a)/b)*Sqrt[a/b 
+ ArcCosh[c*x]]*Gamma[1/2, (7*(a + b*ArcCosh[c*x]))/b] + 2*E^((7*a)/b)*Sin 
h[3*ArcCosh[c*x]] - 6*E^((7*a)/b)*Sinh[5*ArcCosh[c*x]] + 2*E^((7*a)/b)*Sin 
h[7*ArcCosh[c*x]]))/(b*c^3*E^((7*a)/b)*Sqrt[a + b*ArcCosh[c*x]])
 

3.4.75.3 Rubi [A] (verified)

Time = 2.40 (sec) , antiderivative size = 723, normalized size of antiderivative = 1.56, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {6357, 6368, 5971, 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.

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

\(\Big \downarrow \) 6357

\(\displaystyle \frac {14 c d^2 \int \frac {x^3 (c x-1)^{3/2} (c x+1)^{3/2}}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b}-\frac {4 d^2 \int \frac {x (c x-1)^{3/2} (c x+1)^{3/2}}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b c}-\frac {2 d^2 x^2 (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}\)

\(\Big \downarrow \) 6368

\(\displaystyle \frac {14 d^2 \int \frac {\cosh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^4\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{b^2 c^3}-\frac {4 d^2 \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^4\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{b^2 c^3}-\frac {2 d^2 x^2 (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {14 d^2 \int \left (\frac {\cosh \left (\frac {7 a}{b}-\frac {7 (a+b \text {arccosh}(c x))}{b}\right )}{64 \sqrt {a+b \text {arccosh}(c x)}}-\frac {\cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{64 \sqrt {a+b \text {arccosh}(c x)}}-\frac {3 \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{64 \sqrt {a+b \text {arccosh}(c x)}}+\frac {3 \cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{64 \sqrt {a+b \text {arccosh}(c x)}}\right )d(a+b \text {arccosh}(c x))}{b^2 c^3}-\frac {4 d^2 \int \left (\frac {\cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 \sqrt {a+b \text {arccosh}(c x)}}-\frac {3 \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 \sqrt {a+b \text {arccosh}(c x)}}+\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 \sqrt {a+b \text {arccosh}(c x)}}\right )d(a+b \text {arccosh}(c x))}{b^2 c^3}-\frac {2 d^2 x^2 (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {4 d^2 \left (\frac {1}{16} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \sqrt {b} e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{16} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \sqrt {b} e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}+\frac {14 d^2 \left (\frac {3}{128} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{128} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{128} \sqrt {\frac {\pi }{5}} \sqrt {b} e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{128} \sqrt {\frac {\pi }{7}} \sqrt {b} e^{\frac {7 a}{b}} \text {erf}\left (\frac {\sqrt {7} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {3}{128} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{128} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{128} \sqrt {\frac {\pi }{5}} \sqrt {b} e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{128} \sqrt {\frac {\pi }{7}} \sqrt {b} e^{-\frac {7 a}{b}} \text {erfi}\left (\frac {\sqrt {7} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}-\frac {2 d^2 x^2 (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}\)

input

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

output

(-2*d^2*x^2*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(b*c*Sqrt[a + b*ArcCosh[c*x] 
]) - (4*d^2*((Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b 
]])/16 - (Sqrt[b]*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c 
*x]])/Sqrt[b]])/32 + (Sqrt[b]*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + 
 b*ArcCosh[c*x]])/Sqrt[b]])/32 + (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh 
[c*x]]/Sqrt[b]])/(16*E^(a/b)) - (Sqrt[b]*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + 
 b*ArcCosh[c*x]])/Sqrt[b]])/(32*E^((3*a)/b)) + (Sqrt[b]*Sqrt[Pi/5]*Erfi[(S 
qrt[5]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(32*E^((5*a)/b))))/(b^2*c^3) + 
(14*d^2*((3*Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]] 
)/128 - (Sqrt[b]*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c* 
x]])/Sqrt[b]])/128 - (Sqrt[b]*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + 
 b*ArcCosh[c*x]])/Sqrt[b]])/128 + (Sqrt[b]*E^((7*a)/b)*Sqrt[Pi/7]*Erf[(Sqr 
t[7]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/128 + (3*Sqrt[b]*Sqrt[Pi]*Erfi[Sq 
rt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(128*E^(a/b)) - (Sqrt[b]*Sqrt[3*Pi]*Erfi[ 
(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(128*E^((3*a)/b)) - (Sqrt[b]* 
Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(128*E^((5*a) 
/b)) + (Sqrt[b]*Sqrt[Pi/7]*Erfi[(Sqrt[7]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b] 
])/(128*E^((7*a)/b))))/(b^2*c^3)
 

3.4.75.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6357

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^m*Simp[Sqrt[1 + c*x]*Sqrt[-1 + c* 
x]*(d + e*x^2)^p]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (Simp[ 
f*(m/(b*c*(n + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]   Int[(f 
*x)^(m - 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^( 
n + 1), x], x] - Simp[c*((m + 2*p + 1)/(b*f*(n + 1)))*Simp[(d + e*x^2)^p/(( 
1 + c*x)^p*(-1 + c*x)^p)]   Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x 
)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d, e, 
f, m, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 
2*p + 1, 0] && IGtQ[m, -3]
 

rule 6368

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x 
_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))* 
Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p]   Subst[In 
t[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c 
*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[ 
e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]
 

3.4.75.4 Maple [F]

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

input

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

output

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

3.4.75.5 Fricas [F(-2)]

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

input

integrate(x^2*(-c^2*d*x^2+d)^2/(a+b*arccosh(c*x))^(3/2),x, algorithm="fric 
as")
 

output

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

3.4.75.6 Sympy [F]

\[ \int \frac {x^2 \left (d-c^2 d x^2\right )^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=d^{2} \left (\int \frac {x^{2}}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\, dx + \int \left (- \frac {2 c^{2} x^{4}}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\right )\, dx + \int \frac {c^{4} x^{6}}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\, dx\right ) \]

input

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

output

d**2*(Integral(x**2/(a*sqrt(a + b*acosh(c*x)) + b*sqrt(a + b*acosh(c*x))*a 
cosh(c*x)), x) + Integral(-2*c**2*x**4/(a*sqrt(a + b*acosh(c*x)) + b*sqrt( 
a + b*acosh(c*x))*acosh(c*x)), x) + Integral(c**4*x**6/(a*sqrt(a + b*acosh 
(c*x)) + b*sqrt(a + b*acosh(c*x))*acosh(c*x)), x))
 

3.4.75.7 Maxima [F]

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

input

integrate(x^2*(-c^2*d*x^2+d)^2/(a+b*arccosh(c*x))^(3/2),x, algorithm="maxi 
ma")
 

output

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

3.4.75.8 Giac [F]

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

input

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

output

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

3.4.75.9 Mupad [F(-1)]

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

input

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

output

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