\(\int \frac {x^2 (d-c^2 d x^2)}{(a+b \text {arccosh}(c x))^{3/2}} \, dx\) [308]

Optimal result

Integrand size = 27, antiderivative size = 340 \[ \int \frac {x^2 \left (d-c^2 d x^2\right )}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\frac {2 d x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c^3}+\frac {d e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}-\frac {d e^{\frac {5 a}{b}} \sqrt {5 \pi } \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}+\frac {d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c^3}+\frac {d e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}-\frac {d e^{-\frac {5 a}{b}} \sqrt {5 \pi } \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3} \] Output:

2*d*x^2*(c*x-1)^(3/2)*(c*x+1)^(3/2)/b/c/(a+b*arccosh(c*x))^(1/2)+1/8*d*exp 
(a/b)*Pi^(1/2)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^3+1/16*d*ex 
p(3*a/b)*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^ 
(3/2)/c^3-1/16*d*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^(3/2)/c^3+1/8*d*Pi^(1/2)*erfi((a+b*arccosh(c*x))^(1/2)/ 
b^(1/2))/b^(3/2)/c^3/exp(a/b)+1/16*d*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)*(a+b*ar 
ccosh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^3/exp(3*a/b)-1/16*d*5^(1/2)*Pi^(1/2)* 
erfi(5^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^3/exp(5*a/b)
 

Mathematica [A] (warning: unable to verify)

Time = 1.15 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.13 \[ \int \frac {x^2 \left (d-c^2 d x^2\right )}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\frac {d e^{-\frac {5 a}{b}} \left (-4 e^{\frac {5 a}{b}} \sqrt {\frac {-1+c x}{1+c x}}-4 c e^{\frac {5 a}{b}} x \sqrt {\frac {-1+c x}{1+c x}}-2 e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c x)\right )-\sqrt {5} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )+\sqrt {3} e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+2 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )-\sqrt {3} e^{\frac {8 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\sqrt {5} e^{\frac {10 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {5 (a+b \text {arccosh}(c x))}{b}\right )-2 e^{\frac {5 a}{b}} \sinh (3 \text {arccosh}(c x))+2 e^{\frac {5 a}{b}} \sinh (5 \text {arccosh}(c x))\right )}{16 b c^3 \sqrt {a+b \text {arccosh}(c x)}} \] Input:

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

Output:

(d*(-4*E^((5*a)/b)*Sqrt[(-1 + c*x)/(1 + c*x)] - 4*c*E^((5*a)/b)*x*Sqrt[(-1 
 + c*x)/(1 + c*x)] - 2*E^((6*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, a/b 
 + ArcCosh[c*x]] - Sqrt[5]*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-5* 
(a + b*ArcCosh[c*x]))/b] + Sqrt[3]*E^((2*a)/b)*Sqrt[-((a + b*ArcCosh[c*x]) 
/b)]*Gamma[1/2, (-3*(a + b*ArcCosh[c*x]))/b] + 2*E^((4*a)/b)*Sqrt[-((a + b 
*ArcCosh[c*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c*x])/b)] - Sqrt[3]*E^((8*a 
)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (3*(a + b*ArcCosh[c*x]))/b] + Sqr 
t[5]*E^((10*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (5*(a + b*ArcCosh[c* 
x]))/b] - 2*E^((5*a)/b)*Sinh[3*ArcCosh[c*x]] + 2*E^((5*a)/b)*Sinh[5*ArcCos 
h[c*x]]))/(16*b*c^3*E^((5*a)/b)*Sqrt[a + b*ArcCosh[c*x]])
 

Rubi [A] (verified)

Time = 1.99 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.54, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, 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.

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

\(\Big \downarrow \) 6357

\(\displaystyle -\frac {10 c d \int \frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b}+\frac {4 d \int \frac {x \sqrt {c x-1} \sqrt {c x+1}}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b c}+\frac {2 d x^2 (c x-1)^{3/2} (c x+1)^{3/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}\)

\(\Big \downarrow \) 6368

\(\displaystyle -\frac {10 d \int \frac {\cosh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{b^2 c^3}+\frac {4 d \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{b^2 c^3}+\frac {2 d x^2 (c x-1)^{3/2} (c x+1)^{3/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {4 d \int \left (\frac {\cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c x)}}-\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c x)}}\right )d(a+b \text {arccosh}(c x))}{b^2 c^3}-\frac {10 d \int \left (\frac {\cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 \sqrt {a+b \text {arccosh}(c x)}}+\frac {\cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 \sqrt {a+b \text {arccosh}(c x)}}-\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 \sqrt {a+b \text {arccosh}(c x)}}\right )d(a+b \text {arccosh}(c x))}{b^2 c^3}+\frac {2 d x^2 (c x-1)^{3/2} (c x+1)^{3/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {4 d \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}-\frac {10 d \left (-\frac {1}{16} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \sqrt {b} e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{16} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \sqrt {b} e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}+\frac {2 d x^2 (c x-1)^{3/2} (c x+1)^{3/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}\)

Input:

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

Output:

(2*d*x^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(b*c*Sqrt[a + b*ArcCosh[c*x]]) 
+ (4*d*(-1/8*(Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b 
]]) + (Sqrt[b]*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x] 
])/Sqrt[b]])/8 - (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]) 
/(8*E^(a/b)) + (Sqrt[b]*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]]) 
/Sqrt[b]])/(8*E^((3*a)/b))))/(b^2*c^3) - (10*d*(-1/16*(Sqrt[b]*E^(a/b)*Sqr 
t[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]) + (Sqrt[b]*E^((3*a)/b)*Sqrt[P 
i/3]*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 - (Sq 
rt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(16*E^(a/b)) + (Sqr 
t[b]*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(32*E^(( 
3*a)/b)) + (Sqrt[b]*Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcCosh[c*x]])/Sqr 
t[b]])/(32*E^((5*a)/b))))/(b^2*c^3)
 

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

integrate(x^2*(-c^2*d*x^2+d)/(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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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