\(\int \frac {x (d-c^2 d x^2)^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx\) [314]

Optimal result

Integrand size = 27, antiderivative size = 363 \[ \int \frac {x \left (d-c^2 d x^2\right )^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=-\frac {2 d^2 x (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}-\frac {d^2 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {5 d^2 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}+\frac {d^2 e^{\frac {6 a}{b}} \sqrt {\frac {3 \pi }{2}} \text {erf}\left (\frac {\sqrt {6} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}-\frac {d^2 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {5 d^2 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}+\frac {d^2 e^{-\frac {6 a}{b}} \sqrt {\frac {3 \pi }{2}} \text {erfi}\left (\frac {\sqrt {6} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2} \] Output:

-2*d^2*x*(c*x-1)^(5/2)*(c*x+1)^(5/2)/b/c/(a+b*arccosh(c*x))^(1/2)-1/4*d^2* 
exp(4*a/b)*Pi^(1/2)*erf(2*(a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^2+5/ 
32*d^2*exp(2*a/b)*2^(1/2)*Pi^(1/2)*erf(2^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^ 
(1/2))/b^(3/2)/c^2+1/32*d^2*exp(6*a/b)*6^(1/2)*Pi^(1/2)*erf(6^(1/2)*(a+b*a 
rccosh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^2-1/4*d^2*Pi^(1/2)*erfi(2*(a+b*arcco 
sh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^2/exp(4*a/b)+5/32*d^2*2^(1/2)*Pi^(1/2)*e 
rfi(2^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^2/exp(2*a/b)+1/32* 
d^2*6^(1/2)*Pi^(1/2)*erfi(6^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(3/2 
)/c^2/exp(6*a/b)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 4.06 (sec) , antiderivative size = 567, normalized size of antiderivative = 1.56, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6357, 6322, 3042, 3793, 2009, 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.

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

\(\Big \downarrow \) 6357

\(\displaystyle \frac {12 c d^2 \int \frac {x^2 (c x-1)^{3/2} (c x+1)^{3/2}}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b}-\frac {2 d^2 \int \frac {(c x-1)^{3/2} (c x+1)^{3/2}}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b c}-\frac {2 d^2 x (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}\)

\(\Big \downarrow \) 6322

\(\displaystyle -\frac {2 d^2 \int \frac {\sinh ^4\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{b^2 c^2}+\frac {12 c d^2 \int \frac {x^2 (c x-1)^{3/2} (c x+1)^{3/2}}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b}-\frac {2 d^2 x (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 d^2 \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )^4}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{b^2 c^2}+\frac {12 c d^2 \int \frac {x^2 (c x-1)^{3/2} (c x+1)^{3/2}}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b}-\frac {2 d^2 x (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}\)

\(\Big \downarrow \) 3793

\(\displaystyle -\frac {2 d^2 \int \left (\frac {\cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 \sqrt {a+b \text {arccosh}(c x)}}-\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 \sqrt {a+b \text {arccosh}(c x)}}+\frac {3}{8 \sqrt {a+b \text {arccosh}(c x)}}\right )d(a+b \text {arccosh}(c x))}{b^2 c^2}+\frac {12 c d^2 \int \frac {x^2 (c x-1)^{3/2} (c x+1)^{3/2}}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b}-\frac {2 d^2 x (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {12 c d^2 \int \frac {x^2 (c x-1)^{3/2} (c x+1)^{3/2}}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b}-\frac {2 d^2 \left (\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {3}{4} \sqrt {a+b \text {arccosh}(c x)}\right )}{b^2 c^2}-\frac {2 d^2 x (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}\)

\(\Big \downarrow \) 6368

\(\displaystyle \frac {12 d^2 \int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^4\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{b^2 c^2}-\frac {2 d^2 \left (\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {3}{4} \sqrt {a+b \text {arccosh}(c x)}\right )}{b^2 c^2}-\frac {2 d^2 x (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {12 d^2 \int \left (\frac {\cosh \left (\frac {6 a}{b}-\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{32 \sqrt {a+b \text {arccosh}(c x)}}-\frac {\cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{16 \sqrt {a+b \text {arccosh}(c x)}}-\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{32 \sqrt {a+b \text {arccosh}(c x)}}+\frac {1}{16 \sqrt {a+b \text {arccosh}(c x)}}\right )d(a+b \text {arccosh}(c x))}{b^2 c^2}-\frac {2 d^2 \left (\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {3}{4} \sqrt {a+b \text {arccosh}(c x)}\right )}{b^2 c^2}-\frac {2 d^2 x (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 d^2 \left (\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\pi } \sqrt {b} e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {3}{4} \sqrt {a+b \text {arccosh}(c x)}\right )}{b^2 c^2}+\frac {12 d^2 \left (-\frac {1}{64} \sqrt {\pi } \sqrt {b} e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{64} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{64} \sqrt {\frac {\pi }{6}} \sqrt {b} e^{\frac {6 a}{b}} \text {erf}\left (\frac {\sqrt {6} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{64} \sqrt {\pi } \sqrt {b} e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{64} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{64} \sqrt {\frac {\pi }{6}} \sqrt {b} e^{-\frac {6 a}{b}} \text {erfi}\left (\frac {\sqrt {6} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {a+b \text {arccosh}(c x)}\right )}{b^2 c^2}-\frac {2 d^2 x (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}\)

Input:

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

Output:

(-2*d^2*x*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(b*c*Sqrt[a + b*ArcCosh[c*x]]) 
 - (2*d^2*((3*Sqrt[a + b*ArcCosh[c*x]])/4 + (Sqrt[b]*E^((4*a)/b)*Sqrt[Pi]* 
Erf[(2*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/32 - (Sqrt[b]*E^((2*a)/b)*Sqrt[ 
Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/4 + (Sqrt[b]*Sqrt[P 
i]*Erfi[(2*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(32*E^((4*a)/b)) - (Sqrt[b] 
*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*E^((2*a)/ 
b))))/(b^2*c^2) + (12*d^2*(Sqrt[a + b*ArcCosh[c*x]]/8 - (Sqrt[b]*E^((4*a)/ 
b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/64 - (Sqrt[b]*E^((2 
*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/64 + (S 
qrt[b]*E^((6*a)/b)*Sqrt[Pi/6]*Erf[(Sqrt[6]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[ 
b]])/64 - (Sqrt[b]*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(6 
4*E^((4*a)/b)) - (Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x] 
])/Sqrt[b]])/(64*E^((2*a)/b)) + (Sqrt[b]*Sqrt[Pi/6]*Erfi[(Sqrt[6]*Sqrt[a + 
 b*ArcCosh[c*x]])/Sqrt[b]])/(64*E^((6*a)/b))))/(b^2*c^2)
 

Defintions of rubi rules used

rule 2009

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

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 6322

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*( 
(d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[(1/(b*c))*Simp[(d1 + e1*x)^p/ 
(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p]   Subst[Int[x^n*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[2*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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

\[ \int \frac {x \left (d-c^2 d x^2\right )^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=d^{2} \left (\int \frac {x}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\, dx + \int \left (- \frac {2 c^{2} x^{3}}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\right )\, dx + \int \frac {c^{4} x^{5}}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\, dx\right ) \] Input:

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

Output:

d**2*(Integral(x/(a*sqrt(a + b*acosh(c*x)) + b*sqrt(a + b*acosh(c*x))*acos 
h(c*x)), x) + Integral(-2*c**2*x**3/(a*sqrt(a + b*acosh(c*x)) + b*sqrt(a + 
 b*acosh(c*x))*acosh(c*x)), x) + Integral(c**4*x**5/(a*sqrt(a + b*acosh(c* 
x)) + b*sqrt(a + b*acosh(c*x))*acosh(c*x)), x))
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x \left (d-c^2 d x^2\right )^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=d^{2} \left (\left (\int \frac {\sqrt {\mathit {acosh} \left (c x \right ) b +a}\, x^{5}}{\mathit {acosh} \left (c x \right )^{2} b^{2}+2 \mathit {acosh} \left (c x \right ) a b +a^{2}}d x \right ) c^{4}-2 \left (\int \frac {\sqrt {\mathit {acosh} \left (c x \right ) b +a}\, x^{3}}{\mathit {acosh} \left (c x \right )^{2} b^{2}+2 \mathit {acosh} \left (c x \right ) a b +a^{2}}d x \right ) c^{2}+\int \frac {\sqrt {\mathit {acosh} \left (c x \right ) b +a}\, x}{\mathit {acosh} \left (c x \right )^{2} b^{2}+2 \mathit {acosh} \left (c x \right ) a b +a^{2}}d x \right ) \] Input:

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

Output:

d**2*(int((sqrt(acosh(c*x)*b + a)*x**5)/(acosh(c*x)**2*b**2 + 2*acosh(c*x) 
*a*b + a**2),x)*c**4 - 2*int((sqrt(acosh(c*x)*b + a)*x**3)/(acosh(c*x)**2* 
b**2 + 2*acosh(c*x)*a*b + a**2),x)*c**2 + int((sqrt(acosh(c*x)*b + a)*x)/( 
acosh(c*x)**2*b**2 + 2*acosh(c*x)*a*b + a**2),x))