\(\int \frac {x (a+b \text {csch}^{-1}(c x))}{(d+e x^2)^{5/2}} \, dx\) [159]

Optimal result

Integrand size = 21, antiderivative size = 144 \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {b c x \sqrt {-1-c^2 x^2}}{3 d \left (c^2 d-e\right ) \sqrt {-c^2 x^2} \sqrt {d+e x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}-\frac {b c x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{3 d^{3/2} e \sqrt {-c^2 x^2}} \] Output:

1/3*b*c*x*(-c^2*x^2-1)^(1/2)/d/(c^2*d-e)/(-c^2*x^2)^(1/2)/(e*x^2+d)^(1/2)- 
1/3*(a+b*arccsch(c*x))/e/(e*x^2+d)^(3/2)-1/3*b*c*x*arctan((e*x^2+d)^(1/2)/ 
d^(1/2)/(-c^2*x^2-1)^(1/2))/d^(3/2)/e/(-c^2*x^2)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.33 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.92 \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {-\frac {2 a}{e}+\frac {2 b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (d+e x^2\right )}{d \left (c^2 d-e\right )}+\frac {b \sqrt {1+\frac {d}{e x^2}} \left (d+e x^2\right ) \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,-\frac {1}{c^2 x^2},-\frac {d}{e x^2}\right )}{c d e x}-\frac {2 b \text {csch}^{-1}(c x)}{e}}{6 \left (d+e x^2\right )^{3/2}} \] Input:

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

Output:

((-2*a)/e + (2*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(d + e*x^2))/(d*(c^2*d - e)) + 
(b*Sqrt[1 + d/(e*x^2)]*(d + e*x^2)*AppellF1[1, 1/2, 1/2, 2, -(1/(c^2*x^2)) 
, -(d/(e*x^2))])/(c*d*e*x) - (2*b*ArcCsch[c*x])/e)/(6*(d + e*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.91, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6854, 354, 107, 104, 217}

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.

\(\displaystyle \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 6854

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 107

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 217

\(\displaystyle \frac {b c x \left (\frac {2 e \sqrt {-c^2 x^2-1}}{d \left (c^2 d-e\right ) \sqrt {d+e x^2}}-\frac {2 \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{d^{3/2}}\right )}{6 e \sqrt {-c^2 x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}\)

Input:

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

Output:

-1/3*(a + b*ArcCsch[c*x])/(e*(d + e*x^2)^(3/2)) + (b*c*x*((2*e*Sqrt[-1 - c 
^2*x^2])/(d*(c^2*d - e)*Sqrt[d + e*x^2]) - (2*ArcTan[Sqrt[d + e*x^2]/(Sqrt 
[d]*Sqrt[-1 - c^2*x^2])])/d^(3/2)))/(6*e*Sqrt[-(c^2*x^2)])
 

Defintions of rubi rules used

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 107

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 
 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 
 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x 
] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 6854

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

Maple [F]

\[\int \frac {x \left (a +b \,\operatorname {arccsch}\left (c x \right )\right )}{\left (x^{2} e +d \right )^{\frac {5}{2}}}d x\]

Input:

int(x*(a+b*arccsch(c*x))/(e*x^2+d)^(5/2),x)
 

Output:

int(x*(a+b*arccsch(c*x))/(e*x^2+d)^(5/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 341 vs. \(2 (120) = 240\).

Time = 0.19 (sec) , antiderivative size = 698, normalized size of antiderivative = 4.85 \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\left [-\frac {4 \, {\left (b c^{2} d^{3} - b d^{2} e\right )} \sqrt {e x^{2} + d} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (b c^{2} d^{3} + {\left (b c^{2} d e^{2} - b e^{3}\right )} x^{4} - b d^{2} e + 2 \, {\left (b c^{2} d^{2} e - b d e^{2}\right )} x^{2}\right )} \sqrt {d} \log \left (\frac {{\left (c^{4} d^{2} + 6 \, c^{2} d e + e^{2}\right )} x^{4} + 8 \, {\left (c^{2} d^{2} + d e\right )} x^{2} + 4 \, {\left ({\left (c^{3} d + c e\right )} x^{3} + 2 \, c d x\right )} \sqrt {e x^{2} + d} \sqrt {d} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 8 \, d^{2}}{x^{4}}\right ) + 4 \, {\left (a c^{2} d^{3} - a d^{2} e - {\left (b c d e^{2} x^{3} + b c d^{2} e x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \sqrt {e x^{2} + d}}{12 \, {\left (c^{2} d^{5} e - d^{4} e^{2} + {\left (c^{2} d^{3} e^{3} - d^{2} e^{4}\right )} x^{4} + 2 \, {\left (c^{2} d^{4} e^{2} - d^{3} e^{3}\right )} x^{2}\right )}}, -\frac {{\left (b c^{2} d^{3} + {\left (b c^{2} d e^{2} - b e^{3}\right )} x^{4} - b d^{2} e + 2 \, {\left (b c^{2} d^{2} e - b d e^{2}\right )} x^{2}\right )} \sqrt {-d} \arctan \left (\frac {{\left ({\left (c^{3} d + c e\right )} x^{3} + 2 \, c d x\right )} \sqrt {e x^{2} + d} \sqrt {-d} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{2 \, {\left (c^{2} d e x^{4} + {\left (c^{2} d^{2} + d e\right )} x^{2} + d^{2}\right )}}\right ) + 2 \, {\left (b c^{2} d^{3} - b d^{2} e\right )} \sqrt {e x^{2} + d} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 2 \, {\left (a c^{2} d^{3} - a d^{2} e - {\left (b c d e^{2} x^{3} + b c d^{2} e x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \sqrt {e x^{2} + d}}{6 \, {\left (c^{2} d^{5} e - d^{4} e^{2} + {\left (c^{2} d^{3} e^{3} - d^{2} e^{4}\right )} x^{4} + 2 \, {\left (c^{2} d^{4} e^{2} - d^{3} e^{3}\right )} x^{2}\right )}}\right ] \] Input:

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

Output:

[-1/12*(4*(b*c^2*d^3 - b*d^2*e)*sqrt(e*x^2 + d)*log((c*x*sqrt((c^2*x^2 + 1 
)/(c^2*x^2)) + 1)/(c*x)) - (b*c^2*d^3 + (b*c^2*d*e^2 - b*e^3)*x^4 - b*d^2* 
e + 2*(b*c^2*d^2*e - b*d*e^2)*x^2)*sqrt(d)*log(((c^4*d^2 + 6*c^2*d*e + e^2 
)*x^4 + 8*(c^2*d^2 + d*e)*x^2 + 4*((c^3*d + c*e)*x^3 + 2*c*d*x)*sqrt(e*x^2 
 + d)*sqrt(d)*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 8*d^2)/x^4) + 4*(a*c^2*d^3 - 
 a*d^2*e - (b*c*d*e^2*x^3 + b*c*d^2*e*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))*sq 
rt(e*x^2 + d))/(c^2*d^5*e - d^4*e^2 + (c^2*d^3*e^3 - d^2*e^4)*x^4 + 2*(c^2 
*d^4*e^2 - d^3*e^3)*x^2), -1/6*((b*c^2*d^3 + (b*c^2*d*e^2 - b*e^3)*x^4 - b 
*d^2*e + 2*(b*c^2*d^2*e - b*d*e^2)*x^2)*sqrt(-d)*arctan(1/2*((c^3*d + c*e) 
*x^3 + 2*c*d*x)*sqrt(e*x^2 + d)*sqrt(-d)*sqrt((c^2*x^2 + 1)/(c^2*x^2))/(c^ 
2*d*e*x^4 + (c^2*d^2 + d*e)*x^2 + d^2)) + 2*(b*c^2*d^3 - b*d^2*e)*sqrt(e*x 
^2 + d)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + 2*(a*c^2*d^3 
- a*d^2*e - (b*c*d*e^2*x^3 + b*c*d^2*e*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))*s 
qrt(e*x^2 + d))/(c^2*d^5*e - d^4*e^2 + (c^2*d^3*e^3 - d^2*e^4)*x^4 + 2*(c^ 
2*d^4*e^2 - d^3*e^3)*x^2)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(x*(a+b*acsch(c*x))/(e*x**2+d)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

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

Output:

b*integrate(x*log(sqrt(1/(c^2*x^2) + 1) + 1/(c*x))/(e*x^2 + d)^(5/2), x) - 
 1/3*a/((e*x^2 + d)^(3/2)*e)
 

Giac [F]

\[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

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

Output:

integrate((b*arccsch(c*x) + a)*x/(e*x^2 + d)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {x\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \] Input:

int((x*(a + b*asinh(1/(c*x))))/(d + e*x^2)^(5/2),x)
 

Output:

int((x*(a + b*asinh(1/(c*x))))/(d + e*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {-\sqrt {e \,x^{2}+d}\, a +3 \left (\int \frac {\mathit {acsch} \left (c x \right ) x}{\sqrt {e \,x^{2}+d}\, d^{2}+2 \sqrt {e \,x^{2}+d}\, d e \,x^{2}+\sqrt {e \,x^{2}+d}\, e^{2} x^{4}}d x \right ) b \,d^{2} e +6 \left (\int \frac {\mathit {acsch} \left (c x \right ) x}{\sqrt {e \,x^{2}+d}\, d^{2}+2 \sqrt {e \,x^{2}+d}\, d e \,x^{2}+\sqrt {e \,x^{2}+d}\, e^{2} x^{4}}d x \right ) b d \,e^{2} x^{2}+3 \left (\int \frac {\mathit {acsch} \left (c x \right ) x}{\sqrt {e \,x^{2}+d}\, d^{2}+2 \sqrt {e \,x^{2}+d}\, d e \,x^{2}+\sqrt {e \,x^{2}+d}\, e^{2} x^{4}}d x \right ) b \,e^{3} x^{4}}{3 e \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

int(x*(a+b*acsch(c*x))/(e*x^2+d)^(5/2),x)
 

Output:

( - sqrt(d + e*x**2)*a + 3*int((acsch(c*x)*x)/(sqrt(d + e*x**2)*d**2 + 2*s 
qrt(d + e*x**2)*d*e*x**2 + sqrt(d + e*x**2)*e**2*x**4),x)*b*d**2*e + 6*int 
((acsch(c*x)*x)/(sqrt(d + e*x**2)*d**2 + 2*sqrt(d + e*x**2)*d*e*x**2 + sqr 
t(d + e*x**2)*e**2*x**4),x)*b*d*e**2*x**2 + 3*int((acsch(c*x)*x)/(sqrt(d + 
 e*x**2)*d**2 + 2*sqrt(d + e*x**2)*d*e*x**2 + sqrt(d + e*x**2)*e**2*x**4), 
x)*b*e**3*x**4)/(3*e*(d**2 + 2*d*e*x**2 + e**2*x**4))