\(\int \frac {d-c^2 d x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx\) [310]

Optimal result

Integrand size = 24, antiderivative size = 233 \[ \int \frac {d-c^2 d x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\frac {2 d (-1+c x)^{3/2} (1+c x)^{3/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {3 d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}-\frac {d e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}+\frac {3 d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}-\frac {d e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c} \] Output:

2*d*(c*x-1)^(3/2)*(c*x+1)^(3/2)/b/c/(a+b*arccosh(c*x))^(1/2)+3/4*d*exp(a/b 
)*Pi^(1/2)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c-1/4*d*exp(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 
+3/4*d*Pi^(1/2)*erfi((a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c/exp(a/b)- 
1/4*d*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(3 
/2)/c/exp(3*a/b)
 

Mathematica [A] (warning: unable to verify)

Time = 1.06 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.06 \[ \int \frac {d-c^2 d x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\frac {e^{-\frac {3 a}{b}} \left (-3 d e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c x)\right )-\sqrt {3} d \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+d e^{\frac {2 a}{b}} \left (3 \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )+e^{a/b} \left (-6 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)+\sqrt {3} e^{\frac {3 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+2 \sinh (3 \text {arccosh}(c x))\right )\right )\right )}{4 b c \sqrt {a+b \text {arccosh}(c x)}} \] Input:

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

Output:

(-3*d*E^((4*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, a/b + ArcCosh[c*x]] 
- Sqrt[3]*d*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcCosh[ 
c*x]))/b] + d*E^((2*a)/b)*(3*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, -( 
(a + b*ArcCosh[c*x])/b)] + E^(a/b)*(-6*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x 
) + Sqrt[3]*E^((3*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (3*(a + b*ArcC 
osh[c*x]))/b] + 2*Sinh[3*ArcCosh[c*x]])))/(4*b*c*E^((3*a)/b)*Sqrt[a + b*Ar 
cCosh[c*x]])
 

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6319, 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.

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

\(\Big \downarrow \) 6319

\(\displaystyle \frac {2 d (c x-1)^{3/2} (c x+1)^{3/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}-\frac {6 c d \int \frac {x \sqrt {c x-1} \sqrt {c x+1}}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b}\)

\(\Big \downarrow \) 6368

\(\displaystyle \frac {2 d (c x-1)^{3/2} (c x+1)^{3/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}-\frac {6 d \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{b^2 c}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {2 d (c x-1)^{3/2} (c x+1)^{3/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}-\frac {6 d \int \left (\frac {\cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c x)}}-\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c x)}}\right )d(a+b \text {arccosh}(c x))}{b^2 c}\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(2*d*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(b*c*Sqrt[a + b*ArcCosh[c*x]]) - (6 
*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]])/S 
qrt[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]])/Sqr 
t[b]])/(8*E^((3*a)/b))))/(b^2*c)
 

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 6319

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[Simp[Sqrt[1 + c*x]*Sqrt[-1 + c*x]*(d + e*x^2)^p]*((a + b*A 
rcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c*((2*p + 1)/(b*(n + 1)))*Si 
mp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]   Int[x*(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, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IntegerQ[2*p]
 

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]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

integrate((-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]

\[ \int \frac {d-c^2 d x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=- d \left (\int \frac {c^{2} x^{2}}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\, dx + \int \left (- \frac {1}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\right )\, dx\right ) \] Input:

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

Output:

-d*(Integral(c**2*x**2/(a*sqrt(a + b*acosh(c*x)) + b*sqrt(a + b*acosh(c*x) 
)*acosh(c*x)), x) + Integral(-1/(a*sqrt(a + b*acosh(c*x)) + b*sqrt(a + b*a 
cosh(c*x))*acosh(c*x)), x))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(d*( - 3*acosh(c*x)*int((sqrt(acosh(c*x)*b + a)*x**4)/(acosh(c*x)**2*b**2* 
c**2*x**2 - acosh(c*x)**2*b**2 + 2*acosh(c*x)*a*b*c**2*x**2 - 2*acosh(c*x) 
*a*b + a**2*c**2*x**2 - a**2),x)*b**2*c**5 + 3*acosh(c*x)*int((sqrt(acosh( 
c*x)*b + a)*x**2)/(acosh(c*x)**2*b**2*c**2*x**2 - acosh(c*x)**2*b**2 + 2*a 
cosh(c*x)*a*b*c**2*x**2 - 2*acosh(c*x)*a*b + a**2*c**2*x**2 - a**2),x)*b** 
2*c**3 + 6*acosh(c*x)*int((sqrt(c*x + 1)*sqrt(c*x - 1)*sqrt(acosh(c*x)*b + 
 a)*acosh(c*x)*x**3)/(acosh(c*x)**2*b**2*c**2*x**2 - acosh(c*x)**2*b**2 + 
2*acosh(c*x)*a*b*c**2*x**2 - 2*acosh(c*x)*a*b + a**2*c**2*x**2 - a**2),x)* 
b**2*c**4 + 6*acosh(c*x)*int((sqrt(c*x + 1)*sqrt(c*x - 1)*sqrt(acosh(c*x)* 
b + a)*x**3)/(acosh(c*x)**2*b**2*c**2*x**2 - acosh(c*x)**2*b**2 + 2*acosh( 
c*x)*a*b*c**2*x**2 - 2*acosh(c*x)*a*b + a**2*c**2*x**2 - a**2),x)*a*b*c**4 
 - 2*sqrt(c*x + 1)*sqrt(c*x - 1)*sqrt(acosh(c*x)*b + a)*c**2*x**2 - 4*sqrt 
(c*x + 1)*sqrt(c*x - 1)*sqrt(acosh(c*x)*b + a) - 3*int((sqrt(acosh(c*x)*b 
+ a)*x**4)/(acosh(c*x)**2*b**2*c**2*x**2 - acosh(c*x)**2*b**2 + 2*acosh(c* 
x)*a*b*c**2*x**2 - 2*acosh(c*x)*a*b + a**2*c**2*x**2 - a**2),x)*a*b*c**5 + 
 3*int((sqrt(acosh(c*x)*b + a)*x**2)/(acosh(c*x)**2*b**2*c**2*x**2 - acosh 
(c*x)**2*b**2 + 2*acosh(c*x)*a*b*c**2*x**2 - 2*acosh(c*x)*a*b + a**2*c**2* 
x**2 - a**2),x)*a*b*c**3 + 6*int((sqrt(c*x + 1)*sqrt(c*x - 1)*sqrt(acosh(c 
*x)*b + a)*acosh(c*x)*x**3)/(acosh(c*x)**2*b**2*c**2*x**2 - acosh(c*x)**2* 
b**2 + 2*acosh(c*x)*a*b*c**2*x**2 - 2*acosh(c*x)*a*b + a**2*c**2*x**2 -...