\(\int \frac {(d-c^2 d x^2)^{3/2}}{a+b \text {arccosh}(c x)} \, dx\) [71]

Optimal result

Integrand size = 26, antiderivative size = 314 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx=\frac {d \sqrt {d-c^2 d x^2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d \sqrt {d-c^2 d x^2} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 d \sqrt {d-c^2 d x^2} \log (a+b \text {arccosh}(c x))}{8 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d \sqrt {d-c^2 d x^2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d \sqrt {d-c^2 d x^2} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 b c \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

1/2*d*(-c^2*d*x^2+d)^(1/2)*cosh(2*a/b)*Chi(2*(a+b*arccosh(c*x))/b)/b/c/(c* 
x-1)^(1/2)/(c*x+1)^(1/2)-1/8*d*(-c^2*d*x^2+d)^(1/2)*cosh(4*a/b)*Chi(4*(a+b 
*arccosh(c*x))/b)/b/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3/8*d*(-c^2*d*x^2+d)^(1/ 
2)*ln(a+b*arccosh(c*x))/b/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/2*d*(-c^2*d*x^2+ 
d)^(1/2)*sinh(2*a/b)*Shi(2*(a+b*arccosh(c*x))/b)/b/c/(c*x-1)^(1/2)/(c*x+1) 
^(1/2)+1/8*d*(-c^2*d*x^2+d)^(1/2)*sinh(4*a/b)*Shi(4*(a+b*arccosh(c*x))/b)/ 
b/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.62 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.47 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx=\frac {d \sqrt {d-c^2 d x^2} \left (4 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-3 \log (a+b \text {arccosh}(c x))-4 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{8 b c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \] Input:

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

Output:

(d*Sqrt[d - c^2*d*x^2]*(4*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCosh[c*x] 
)] - Cosh[(4*a)/b]*CoshIntegral[4*(a/b + ArcCosh[c*x])] - 3*Log[a + b*ArcC 
osh[c*x]] - 4*Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] + Sinh[(4 
*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])]))/(8*b*c*Sqrt[(-1 + c*x)/(1 + 
c*x)]*(1 + c*x))
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.50, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6321, 3042, 3793, 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.

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

\(\Big \downarrow \) 6321

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {d \sqrt {d-c^2 d x^2} \left (-\frac {1}{2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )+\frac {3}{8} \log (a+b \text {arccosh}(c x))\right )}{b c \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

-((d*Sqrt[d - c^2*d*x^2]*(-1/2*(Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCo 
sh[c*x]))/b]) + (Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b])/8 
 + (3*Log[a + b*ArcCosh[c*x]])/8 + (Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*A 
rcCosh[c*x]))/b])/2 - (Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x])) 
/b])/8))/(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))
 

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 6321

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/((1 + c*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, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p, 0]
 

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.75

method	result	size
default	\(-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-6 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )-6 \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c x +\operatorname {expIntegral}_{1}\left (4 \,\operatorname {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}+\operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}-4 \,\operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}-4 \,\operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}\right ) d}{16 \left (c x -1\right ) \left (c x +1\right ) c b}\)	\(234\)

Input:

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

Output:

-1/16*(-d*(c^2*x^2-1))^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)* 
(-6*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln(a+b*arccosh(c*x))-6*ln(a+b*arccosh(c*x) 
)*c*x+Ei(1,4*arccosh(c*x)+4*a/b)*exp((b*arccosh(c*x)+4*a)/b)+Ei(1,-4*arcco 
sh(c*x)-4*a/b)*exp(-(-b*arccosh(c*x)+4*a)/b)-4*Ei(1,2*arccosh(c*x)+2*a/b)* 
exp((b*arccosh(c*x)+2*a)/b)-4*Ei(1,-2*arccosh(c*x)-2*a/b)*exp(-(-b*arccosh 
(c*x)+2*a)/b))*d/(c*x-1)/(c*x+1)/c/b
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((-d*(c*x - 1)*(c*x + 1))**(3/2)/(a + b*acosh(c*x)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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