\(\int \frac {(d-c^2 d x^2)^{5/2}}{a+b \text {arccosh}(c x)} \, dx\) [70]

Optimal result

Integrand size = 26, antiderivative size = 458 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\frac {15 d^2 \sqrt {d-c^2 d x^2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 d^2 \sqrt {d-c^2 d x^2} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^2 \sqrt {d-c^2 d x^2} \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 \sqrt {d-c^2 d x^2} \log (a+b \text {arccosh}(c x))}{16 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {15 d^2 \sqrt {d-c^2 d x^2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 d^2 \sqrt {d-c^2 d x^2} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 \sqrt {d-c^2 d x^2} \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

15/32*d^2*(-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)-3/16*d^2*(-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)+1/32*d^2*(-c^2*d* 
x^2+d)^(1/2)*cosh(6*a/b)*Chi(6*(a+b*arccosh(c*x))/b)/b/c/(c*x-1)^(1/2)/(c* 
x+1)^(1/2)-5/16*d^2*(-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)-15/32*d^2*(-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)+3/16*d^2*(-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)-1/32*d^2*(-c^2*d*x^2+d)^(1/2)*sinh(6*a/b)*Shi(6*(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 = 0.92 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.43 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (15 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-6 \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+\cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (6 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-10 \log (a+b \text {arccosh}(c x))-15 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+6 \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-\sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{32 b c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \] Input:

Integrate[(d - c^2*d*x^2)^(5/2)/(a + b*ArcCosh[c*x]),x]
 

Output:

(d^2*Sqrt[d - c^2*d*x^2]*(15*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCosh[c 
*x])] - 6*Cosh[(4*a)/b]*CoshIntegral[4*(a/b + ArcCosh[c*x])] + Cosh[(6*a)/ 
b]*CoshIntegral[6*(a/b + ArcCosh[c*x])] - 10*Log[a + b*ArcCosh[c*x]] - 15* 
Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] + 6*Sinh[(4*a)/b]*SinhI 
ntegral[4*(a/b + ArcCosh[c*x])] - Sinh[(6*a)/b]*SinhIntegral[6*(a/b + ArcC 
osh[c*x])]))/(32*b*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.45, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {6321, 3042, 25, 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.

\(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx\)

\(\Big \downarrow \) 6321

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3793

\(\displaystyle -\frac {d^2 \sqrt {d-c^2 d x^2} \int \left (-\frac {\cosh \left (\frac {6 a}{b}-\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{32 (a+b \text {arccosh}(c x))}+\frac {3 \cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{16 (a+b \text {arccosh}(c x))}-\frac {15 \cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{32 (a+b \text {arccosh}(c x))}+\frac {5}{16 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b c \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \left (\frac {15}{32} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-\frac {3}{16} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{32} \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )-\frac {15}{32} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+\frac {3}{16} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{32} \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )-\frac {5}{16} \log (a+b \text {arccosh}(c x))\right )}{b c \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

Int[(d - c^2*d*x^2)^(5/2)/(a + b*ArcCosh[c*x]),x]
 

Output:

(d^2*Sqrt[d - c^2*d*x^2]*((15*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh 
[c*x]))/b])/32 - (3*Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b] 
)/16 + (Cosh[(6*a)/b]*CoshIntegral[(6*(a + b*ArcCosh[c*x]))/b])/32 - (5*Lo 
g[a + b*ArcCosh[c*x]])/16 - (15*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCo 
sh[c*x]))/b])/32 + (3*Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/ 
b])/16 - (Sinh[(6*a)/b]*SinhIntegral[(6*(a + b*ArcCosh[c*x]))/b])/32))/(b* 
c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 = 0.26 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.66

method	result	size
default	\(\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (20 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )+20 \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c x +\operatorname {expIntegral}_{1}\left (6 \,\operatorname {arccosh}\left (c x \right )+\frac {6 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+6 a}{b}}+\operatorname {expIntegral}_{1}\left (-6 \,\operatorname {arccosh}\left (c x \right )-\frac {6 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+6 a}{b}}-6 \,\operatorname {expIntegral}_{1}\left (4 \,\operatorname {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}+15 \,\operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}+15 \,\operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}-6 \,\operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}\right ) d^{2}}{64 \left (c x -1\right ) \left (c x +1\right ) c b}\)	\(302\)

Input:

int((-c^2*d*x^2+d)^(5/2)/(a+b*arccosh(c*x)),x,method=_RETURNVERBOSE)
 

Output:

1/64*(-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)*( 
20*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln(a+b*arccosh(c*x))+20*ln(a+b*arccosh(c*x) 
)*c*x+Ei(1,6*arccosh(c*x)+6*a/b)*exp((b*arccosh(c*x)+6*a)/b)+Ei(1,-6*arcco 
sh(c*x)-6*a/b)*exp(-(-b*arccosh(c*x)+6*a)/b)-6*Ei(1,4*arccosh(c*x)+4*a/b)* 
exp((b*arccosh(c*x)+4*a)/b)+15*Ei(1,2*arccosh(c*x)+2*a/b)*exp((b*arccosh(c 
*x)+2*a)/b)+15*Ei(1,-2*arccosh(c*x)-2*a/b)*exp(-(-b*arccosh(c*x)+2*a)/b)-6 
*Ei(1,-4*arccosh(c*x)-4*a/b)*exp(-(-b*arccosh(c*x)+4*a)/b))*d^2/(c*x-1)/(c 
*x+1)/c/b
 

Fricas [F]

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

integrate((-c^2*d*x^2+d)^(5/2)/(a+b*arccosh(c*x)),x, algorithm="fricas")
 

Output:

integral((c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2)*sqrt(-c^2*d*x^2 + d)/(b*arcco 
sh(c*x) + a), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\text {Timed out} \] Input:

integrate((-c**2*d*x**2+d)**(5/2)/(a+b*acosh(c*x)),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((-c^2*d*x^2+d)^(5/2)/(a+b*arccosh(c*x)),x, algorithm="maxima")
 

Output:

integrate((-c^2*d*x^2 + d)^(5/2)/(b*arccosh(c*x) + a), x)
 

Giac [F]

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

integrate((-c^2*d*x^2+d)^(5/2)/(a+b*arccosh(c*x)),x, algorithm="giac")
 

Output:

integrate((-c^2*d*x^2 + d)^(5/2)/(b*arccosh(c*x) + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {{\left (d-c^2\,d\,x^2\right )}^{5/2}}{a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2}}{a+b \text {arccosh}(c x)} \, dx=\sqrt {d}\, d^{2} \left (\int \frac {\sqrt {-c^{2} x^{2}+1}}{\mathit {acosh} \left (c x \right ) b +a}d x +\left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, x^{4}}{\mathit {acosh} \left (c x \right ) b +a}d x \right ) c^{4}-2 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, x^{2}}{\mathit {acosh} \left (c x \right ) b +a}d x \right ) c^{2}\right ) \] Input:

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

Output:

sqrt(d)*d**2*(int(sqrt( - c**2*x**2 + 1)/(acosh(c*x)*b + a),x) + int((sqrt 
( - c**2*x**2 + 1)*x**4)/(acosh(c*x)*b + a),x)*c**4 - 2*int((sqrt( - c**2* 
x**2 + 1)*x**2)/(acosh(c*x)*b + a),x)*c**2)