\(\int \frac {x (a+b \text {arccosh}(c x))}{(d+e x^2)^3} \, dx\) [406]

Optimal result

Integrand size = 19, antiderivative size = 167 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=-\frac {b c x \sqrt {-1+c x} \sqrt {1+c x}}{8 d \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \text {arccosh}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {b c \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{8 d^{3/2} e \left (c^2 d+e\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-1/8*b*c*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(c^2*d+e)/(e*x^2+d)-1/4*(a+b*arcc 
osh(c*x))/e/(e*x^2+d)^2+1/8*b*c*(2*c^2*d+e)*(c^2*x^2-1)^(1/2)*arctanh((c^2 
*d+e)^(1/2)*x/d^(1/2)/(c^2*x^2-1)^(1/2))/d^(3/2)/e/(c^2*d+e)^(3/2)/(c*x-1) 
^(1/2)/(c*x+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.10 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\frac {1}{8} \left (-\frac {\frac {2 a}{e}+\frac {b c x \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}{d \left (c^2 d+e\right )}}{\left (d+e x^2\right )^2}-\frac {2 b \text {arccosh}(c x)}{e \left (d+e x^2\right )^2}-\frac {b c \left (2 c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \arctan \left (\frac {\sqrt {-c^2 d-e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{d^{3/2} \left (-c^2 d-e\right )^{3/2} e \sqrt {-1+c^2 x^2}}\right ) \] Input:

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

Output:

(-(((2*a)/e + (b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2))/(d*(c^2*d + 
 e)))/(d + e*x^2)^2) - (2*b*ArcCosh[c*x])/(e*(d + e*x^2)^2) - (b*c*(2*c^2* 
d + e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTan[(Sqrt[-(c^2*d) - e]*x)/(Sqrt[d] 
*Sqrt[-1 + c^2*x^2])])/(d^(3/2)*(-(c^2*d) - e)^(3/2)*e*Sqrt[-1 + c^2*x^2]) 
)/8
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6372, 648, 296, 291, 221}

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.

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

\(\Big \downarrow \) 6372

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

\(\Big \downarrow \) 648

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

\(\Big \downarrow \) 296

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-1/4*(a + b*ArcCosh[c*x])/(e*(d + e*x^2)^2) + (b*c*Sqrt[-1 + c^2*x^2]*(-1/ 
2*(e*x*Sqrt[-1 + c^2*x^2])/(d*(c^2*d + e)*(d + e*x^2)) + ((2*c^2*d + e)*Ar 
cTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(2*d^(3/2)*(c^2*d 
 + e)^(3/2))))/(4*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

Defintions of rubi rules used

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 296

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) 
), x] + Simp[(b*c + 2*(p + 1)*(b*c - a*d))/(2*a*(p + 1)*(b*c - a*d))   Int[ 
(a + b*x^2)^(p + 1)*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, q}, x] && N 
eQ[b*c - a*d, 0] && EqQ[2*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1 
]) && NeQ[p, -1]
 

rule 648

Int[((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) 
^2)^(p_), x_Symbol] :> Simp[(c + d*x)^FracPart[m]*((e + f*x)^FracPart[m]/(c 
*e + d*f*x^2)^FracPart[m])   Int[(c*e + d*f*x^2)^m*(a + b*x^2)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[m, n] && EqQ[d*e + c*f, 0] && 
  !(EqQ[p, 2] && LtQ[m, -1])
 

rule 6372

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])/(2*e*(p + 1))), 
x] - Simp[b*(c/(2*e*(p + 1)))   Int[(d + e*x^2)^(p + 1)/(Sqrt[1 + c*x]*Sqrt 
[-1 + c*x]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[c^2*d + e, 0] && 
 NeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1125\) vs. \(2(141)=282\).

Time = 0.22 (sec) , antiderivative size = 1126, normalized size of antiderivative = 6.74

method	result	size
parts	\(\text {Expression too large to display}\)	\(1126\)
derivativedivides	\(\text {Expression too large to display}\)	\(1149\)
default	\(\text {Expression too large to display}\)	\(1149\)

Input:

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

Output:

-1/4*a/e/(e*x^2+d)^2+b/c^2*(-1/4*c^6/e/(c^2*e*x^2+c^2*d)^2*arccosh(c*x)-1/ 
16*c^4*e^2*(2*ln(2*((-(c^2*d+e)/e)^(1/2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1 
/2)*c*x-e)/(e*c*x-(-c^2*d*e)^(1/2)))*c^6*d^2*e*x^2+2*ln(2*((-(c^2*d+e)/e)^ 
(1/2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(e*c*x-(-c^2*d*e)^(1/2)) 
)*c^6*d^3-2*ln(-2*(-(-(c^2*d+e)/e)^(1/2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1 
/2)*c*x+e)/(e*c*x+(-c^2*d*e)^(1/2)))*c^6*x^2*d^2*e-2*ln(-2*(-(-(c^2*d+e)/e 
)^(1/2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/(e*c*x+(-c^2*d*e)^(1/2 
)))*c^6*d^3+3*ln(2*((-(c^2*d+e)/e)^(1/2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1 
/2)*c*x-e)/(e*c*x-(-c^2*d*e)^(1/2)))*c^4*d*e^2*x^2+3*ln(2*((-(c^2*d+e)/e)^ 
(1/2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(e*c*x-(-c^2*d*e)^(1/2)) 
)*c^4*d^2*e-3*ln(-2*(-(-(c^2*d+e)/e)^(1/2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^ 
(1/2)*c*x+e)/(e*c*x+(-c^2*d*e)^(1/2)))*c^4*x^2*d*e^2-3*ln(-2*(-(-(c^2*d+e) 
/e)^(1/2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/(e*c*x+(-c^2*d*e)^(1 
/2)))*c^4*d^2*e+2*c^3*d*e*(-(c^2*d+e)/e)^(1/2)*(c^2*x^2-1)^(1/2)*(-c^2*d*e 
)^(1/2)*x+ln(2*((-(c^2*d+e)/e)^(1/2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2)* 
c*x-e)/(e*c*x-(-c^2*d*e)^(1/2)))*e^3*c^2*x^2+ln(2*((-(c^2*d+e)/e)^(1/2)*(c 
^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(e*c*x-(-c^2*d*e)^(1/2)))*c^2*d* 
e^2-ln(-2*(-(-(c^2*d+e)/e)^(1/2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x+ 
e)/(e*c*x+(-c^2*d*e)^(1/2)))*c^2*x^2*e^3-ln(-2*(-(-(c^2*d+e)/e)^(1/2)*(c^2 
*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/(e*c*x+(-c^2*d*e)^(1/2)))*c^2*d...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 584 vs. \(2 (141) = 282\).

Time = 0.20 (sec) , antiderivative size = 1233, normalized size of antiderivative = 7.38 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/16*(2*(2*a + b)*c^4*d^4 + 2*(4*a + b)*c^2*d^3*e + 4*a*d^2*e^2 + 2*(b*c 
^4*d^2*e^2 + b*c^2*d*e^3)*x^4 + 4*(b*c^4*d^3*e + b*c^2*d^2*e^2)*x^2 - (2*b 
*c^3*d^3 + b*c*d^2*e + (2*b*c^3*d*e^2 + b*c*e^3)*x^4 + 2*(2*b*c^3*d^2*e + 
b*c*d*e^2)*x^2)*sqrt(c^2*d^2 + d*e)*log(-(2*c^2*d^2 - (4*c^4*d^2 + 4*c^2*d 
*e + e^2)*x^2 + d*e - 2*sqrt(c^2*d^2 + d*e)*((2*c^3*d + c*e)*x^2 - c*d) - 
2*sqrt(c^2*x^2 - 1)*(sqrt(c^2*d^2 + d*e)*(2*c^2*d + e)*x + 2*(c^3*d^2 + c* 
d*e)*x))/(e*x^2 + d)) - 4*((b*c^4*d^2*e^2 + 2*b*c^2*d*e^3 + b*e^4)*x^4 + 2 
*(b*c^4*d^3*e + 2*b*c^2*d^2*e^2 + b*d*e^3)*x^2)*log(c*x + sqrt(c^2*x^2 - 1 
)) - 4*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2 + (b*c^4*d^2*e^2 + 2*b*c^2*d 
*e^3 + b*e^4)*x^4 + 2*(b*c^4*d^3*e + 2*b*c^2*d^2*e^2 + b*d*e^3)*x^2)*log(- 
c*x + sqrt(c^2*x^2 - 1)) + 2*sqrt(c^2*x^2 - 1)*((b*c^3*d^2*e^2 + b*c*d*e^3 
)*x^3 + (b*c^3*d^3*e + b*c*d^2*e^2)*x))/(c^4*d^6*e + 2*c^2*d^5*e^2 + d^4*e 
^3 + (c^4*d^4*e^3 + 2*c^2*d^3*e^4 + d^2*e^5)*x^4 + 2*(c^4*d^5*e^2 + 2*c^2* 
d^4*e^3 + d^3*e^4)*x^2), -1/8*((2*a + b)*c^4*d^4 + (4*a + b)*c^2*d^3*e + 2 
*a*d^2*e^2 + (b*c^4*d^2*e^2 + b*c^2*d*e^3)*x^4 + 2*(b*c^4*d^3*e + b*c^2*d^ 
2*e^2)*x^2 - (2*b*c^3*d^3 + b*c*d^2*e + (2*b*c^3*d*e^2 + b*c*e^3)*x^4 + 2* 
(2*b*c^3*d^2*e + b*c*d*e^2)*x^2)*sqrt(-c^2*d^2 - d*e)*arctan((sqrt(-c^2*d^ 
2 - d*e)*sqrt(c^2*x^2 - 1)*e*x - sqrt(-c^2*d^2 - d*e)*(c*e*x^2 + c*d))/(c^ 
2*d^2 + d*e)) - 2*((b*c^4*d^2*e^2 + 2*b*c^2*d*e^3 + b*e^4)*x^4 + 2*(b*c^4* 
d^3*e + 2*b*c^2*d^2*e^2 + b*d*e^3)*x^2)*log(c*x + sqrt(c^2*x^2 - 1)) - ...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/8*(c^4*log(e*x^2 + d)/(c^4*d^2*e + 2*c^2*d*e^2 + e^3) + 8*c*integrate(1 
/4/(c^3*e^3*x^7 + (2*c^3*d*e^2 - c*e^3)*x^5 - c*d^2*e*x + (c^3*d^2*e - 2*c 
*d*e^2)*x^3 + (c^2*e^3*x^6 + (2*c^2*d*e^2 - e^3)*x^4 - d^2*e + (c^2*d^2*e 
- 2*d*e^2)*x^2)*e^(1/2*log(c*x + 1) + 1/2*log(c*x - 1))), x) - (c^4*d^2 + 
c^2*d*e + (c^4*d*e + c^2*e^2)*x^2 - 2*(c^4*d^2 + 2*c^2*d*e + e^2)*log(c*x 
+ sqrt(c*x + 1)*sqrt(c*x - 1)) + (c^4*e^2*x^4 + 2*c^4*d*e*x^2 + c^4*d^2)*l 
og(c*x + 1) + (c^4*e^2*x^4 + 2*c^4*d*e*x^2 + c^4*d^2)*log(c*x - 1))/(c^4*d 
^4*e + 2*c^2*d^3*e^2 + d^2*e^3 + (c^4*d^2*e^3 + 2*c^2*d*e^4 + e^5)*x^4 + 2 
*(c^4*d^3*e^2 + 2*c^2*d^2*e^3 + d*e^4)*x^2))*b - 1/4*a/(e^3*x^4 + 2*d*e^2* 
x^2 + d^2*e)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\frac {4 \left (\int \frac {\mathit {acosh} \left (c x \right ) x}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{2} e +8 \left (\int \frac {\mathit {acosh} \left (c x \right ) x}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b d \,e^{2} x^{2}+4 \left (\int \frac {\mathit {acosh} \left (c x \right ) x}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,e^{3} x^{4}-a}{4 e \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

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

Output:

(4*int((acosh(c*x)*x)/(d**3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e**3*x**6),x 
)*b*d**2*e + 8*int((acosh(c*x)*x)/(d**3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + 
e**3*x**6),x)*b*d*e**2*x**2 + 4*int((acosh(c*x)*x)/(d**3 + 3*d**2*e*x**2 + 
 3*d*e**2*x**4 + e**3*x**6),x)*b*e**3*x**4 - a)/(4*e*(d**2 + 2*d*e*x**2 + 
e**2*x**4))