\(\int \frac {(d+e x^2) (a+b \text {csch}^{-1}(c x))}{x^6} \, dx\) [82]

Optimal result

Integrand size = 19, antiderivative size = 158 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^6} \, dx=\frac {2 b c^3 \left (12 c^2 d-25 e\right ) \sqrt {-1-c^2 x^2}}{225 \sqrt {-c^2 x^2}}+\frac {b c d \sqrt {-1-c^2 x^2}}{25 x^4 \sqrt {-c^2 x^2}}-\frac {b c \left (12 c^2 d-25 e\right ) \sqrt {-1-c^2 x^2}}{225 x^2 \sqrt {-c^2 x^2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3} \] Output:

2/225*b*c^3*(12*c^2*d-25*e)*(-c^2*x^2-1)^(1/2)/(-c^2*x^2)^(1/2)+1/25*b*c*d 
*(-c^2*x^2-1)^(1/2)/x^4/(-c^2*x^2)^(1/2)-1/225*b*c*(12*c^2*d-25*e)*(-c^2*x 
^2-1)^(1/2)/x^2/(-c^2*x^2)^(1/2)-1/5*d*(a+b*arccsch(c*x))/x^5-1/3*e*(a+b*a 
rccsch(c*x))/x^3
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.59 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^6} \, dx=\frac {-15 a \left (3 d+5 e x^2\right )+b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (25 e x^2 \left (1-2 c^2 x^2\right )+3 d \left (3-4 c^2 x^2+8 c^4 x^4\right )\right )-15 b \left (3 d+5 e x^2\right ) \text {csch}^{-1}(c x)}{225 x^5} \] Input:

Integrate[((d + e*x^2)*(a + b*ArcCsch[c*x]))/x^6,x]
 

Output:

(-15*a*(3*d + 5*e*x^2) + b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(25*e*x^2*(1 - 2*c^2* 
x^2) + 3*d*(3 - 4*c^2*x^2 + 8*c^4*x^4)) - 15*b*(3*d + 5*e*x^2)*ArcCsch[c*x 
])/(225*x^5)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.85, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6856, 27, 359, 245, 242}

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.

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

\(\Big \downarrow \) 6856

\(\displaystyle -\frac {b c x \int -\frac {5 e x^2+3 d}{15 x^6 \sqrt {-c^2 x^2-1}}dx}{\sqrt {-c^2 x^2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b c x \int \frac {5 e x^2+3 d}{x^6 \sqrt {-c^2 x^2-1}}dx}{15 \sqrt {-c^2 x^2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}\)

\(\Big \downarrow \) 359

\(\displaystyle \frac {b c x \left (\frac {3 d \sqrt {-c^2 x^2-1}}{5 x^5}-\frac {1}{5} \left (12 c^2 d-25 e\right ) \int \frac {1}{x^4 \sqrt {-c^2 x^2-1}}dx\right )}{15 \sqrt {-c^2 x^2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}\)

\(\Big \downarrow \) 245

\(\displaystyle \frac {b c x \left (\frac {3 d \sqrt {-c^2 x^2-1}}{5 x^5}-\frac {1}{5} \left (12 c^2 d-25 e\right ) \left (\frac {\sqrt {-c^2 x^2-1}}{3 x^3}-\frac {2}{3} c^2 \int \frac {1}{x^2 \sqrt {-c^2 x^2-1}}dx\right )\right )}{15 \sqrt {-c^2 x^2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}\)

\(\Big \downarrow \) 242

\(\displaystyle -\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}+\frac {b c x \left (\frac {3 d \sqrt {-c^2 x^2-1}}{5 x^5}-\frac {1}{5} \left (\frac {\sqrt {-c^2 x^2-1}}{3 x^3}-\frac {2 c^2 \sqrt {-c^2 x^2-1}}{3 x}\right ) \left (12 c^2 d-25 e\right )\right )}{15 \sqrt {-c^2 x^2}}\)

Input:

Int[((d + e*x^2)*(a + b*ArcCsch[c*x]))/x^6,x]
 

Output:

(b*c*x*((3*d*Sqrt[-1 - c^2*x^2])/(5*x^5) - ((12*c^2*d - 25*e)*(Sqrt[-1 - c 
^2*x^2]/(3*x^3) - (2*c^2*Sqrt[-1 - c^2*x^2])/(3*x)))/5))/(15*Sqrt[-(c^2*x^ 
2)]) - (d*(a + b*ArcCsch[c*x]))/(5*x^5) - (e*(a + b*ArcCsch[c*x]))/(3*x^3)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 242

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x 
] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
 

rule 245

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + 
 b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) 
   Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si 
mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
 

rule 359

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + 
Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1))   Int[(e*x)^(m + 2)* 
(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] 
&& LtQ[m, -1] &&  !ILtQ[p, -1]
 

rule 6856

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*( 
x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Si 
mp[(a + b*ArcCsch[c*x])   u, x] - Simp[b*c*(x/Sqrt[(-c^2)*x^2])   Int[Simpl 
ifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), x], x], x]] /; FreeQ[{a, b, c, d, e, 
 f, m, p}, x] && ((IGtQ[p, 0] &&  !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 
0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (I 
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))
 

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.80

method	result	size
parts	\(a \left (-\frac {e}{3 x^{3}}-\frac {d}{5 x^{5}}\right )+b \,c^{5} \left (-\frac {\operatorname {arccsch}\left (c x \right ) e}{3 c^{5} x^{3}}-\frac {\operatorname {arccsch}\left (c x \right ) d}{5 c^{5} x^{5}}+\frac {\left (c^{2} x^{2}+1\right ) \left (24 c^{6} d \,x^{4}-50 c^{4} e \,x^{4}-12 c^{4} d \,x^{2}+25 e \,c^{2} x^{2}+9 c^{2} d \right )}{225 c^{8} \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x^{6}}\right )\)	\(127\)
derivativedivides	\(c^{5} \left (\frac {a \left (-\frac {d}{5 c^{3} x^{5}}-\frac {e}{3 c^{3} x^{3}}\right )}{c^{2}}+\frac {b \left (-\frac {\operatorname {arccsch}\left (c x \right ) d}{5 c^{3} x^{5}}-\frac {\operatorname {arccsch}\left (c x \right ) e}{3 c^{3} x^{3}}+\frac {\left (c^{2} x^{2}+1\right ) \left (24 c^{6} d \,x^{4}-50 c^{4} e \,x^{4}-12 c^{4} d \,x^{2}+25 e \,c^{2} x^{2}+9 c^{2} d \right )}{225 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{6} x^{6}}\right )}{c^{2}}\right )\)	\(140\)
default	\(c^{5} \left (\frac {a \left (-\frac {d}{5 c^{3} x^{5}}-\frac {e}{3 c^{3} x^{3}}\right )}{c^{2}}+\frac {b \left (-\frac {\operatorname {arccsch}\left (c x \right ) d}{5 c^{3} x^{5}}-\frac {\operatorname {arccsch}\left (c x \right ) e}{3 c^{3} x^{3}}+\frac {\left (c^{2} x^{2}+1\right ) \left (24 c^{6} d \,x^{4}-50 c^{4} e \,x^{4}-12 c^{4} d \,x^{2}+25 e \,c^{2} x^{2}+9 c^{2} d \right )}{225 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{6} x^{6}}\right )}{c^{2}}\right )\)	\(140\)

Input:

int((e*x^2+d)*(a+b*arccsch(c*x))/x^6,x,method=_RETURNVERBOSE)
 

Output:

a*(-1/3*e/x^3-1/5*d/x^5)+b*c^5*(-1/3/c^5*arccsch(c*x)*e/x^3-1/5*arccsch(c* 
x)*d/c^5/x^5+1/225/c^8*(c^2*x^2+1)*(24*c^6*d*x^4-50*c^4*e*x^4-12*c^4*d*x^2 
+25*c^2*e*x^2+9*c^2*d)/((c^2*x^2+1)/c^2/x^2)^(1/2)/x^6)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.80 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^6} \, dx=-\frac {75 \, a e x^{2} + 45 \, a d + 15 \, {\left (5 \, b e x^{2} + 3 \, b d\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (2 \, {\left (12 \, b c^{5} d - 25 \, b c^{3} e\right )} x^{5} + 9 \, b c d x - {\left (12 \, b c^{3} d - 25 \, b c e\right )} x^{3}\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{225 \, x^{5}} \] Input:

integrate((e*x^2+d)*(a+b*arccsch(c*x))/x^6,x, algorithm="fricas")
 

Output:

-1/225*(75*a*e*x^2 + 45*a*d + 15*(5*b*e*x^2 + 3*b*d)*log((c*x*sqrt((c^2*x^ 
2 + 1)/(c^2*x^2)) + 1)/(c*x)) - (2*(12*b*c^5*d - 25*b*c^3*e)*x^5 + 9*b*c*d 
*x - (12*b*c^3*d - 25*b*c*e)*x^3)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))/x^5
                                                                                    
                                                                                    
 

Sympy [F]

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

integrate((e*x**2+d)*(a+b*acsch(c*x))/x**6,x)
 

Output:

Integral((a + b*acsch(c*x))*(d + e*x**2)/x**6, x)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.84 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^6} \, dx=\frac {1}{75} \, b d {\left (\frac {3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 10 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, c^{6} \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {15 \, \operatorname {arcsch}\left (c x\right )}{x^{5}}\right )} + \frac {1}{9} \, b e {\left (\frac {c^{4} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {3 \, \operatorname {arcsch}\left (c x\right )}{x^{3}}\right )} - \frac {a e}{3 \, x^{3}} - \frac {a d}{5 \, x^{5}} \] Input:

integrate((e*x^2+d)*(a+b*arccsch(c*x))/x^6,x, algorithm="maxima")
 

Output:

1/75*b*d*((3*c^6*(1/(c^2*x^2) + 1)^(5/2) - 10*c^6*(1/(c^2*x^2) + 1)^(3/2) 
+ 15*c^6*sqrt(1/(c^2*x^2) + 1))/c - 15*arccsch(c*x)/x^5) + 1/9*b*e*((c^4*( 
1/(c^2*x^2) + 1)^(3/2) - 3*c^4*sqrt(1/(c^2*x^2) + 1))/c - 3*arccsch(c*x)/x 
^3) - 1/3*a*e/x^3 - 1/5*a*d/x^5
 

Giac [F]

\[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^6} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{x^{6}} \,d x } \] Input:

integrate((e*x^2+d)*(a+b*arccsch(c*x))/x^6,x, algorithm="giac")
 

Output:

integrate((e*x^2 + d)*(b*arccsch(c*x) + a)/x^6, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^6} \, dx=\int \frac {\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x^6} \,d x \] Input:

int(((d + e*x^2)*(a + b*asinh(1/(c*x))))/x^6,x)
 

Output:

int(((d + e*x^2)*(a + b*asinh(1/(c*x))))/x^6, x)
 

Reduce [F]

\[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^6} \, dx=\frac {15 \left (\int \frac {\mathit {acsch} \left (c x \right )}{x^{6}}d x \right ) b d \,x^{5}+15 \left (\int \frac {\mathit {acsch} \left (c x \right )}{x^{4}}d x \right ) b e \,x^{5}-3 a d -5 a e \,x^{2}}{15 x^{5}} \] Input:

int((e*x^2+d)*(a+b*acsch(c*x))/x^6,x)
 

Output:

(15*int(acsch(c*x)/x**6,x)*b*d*x**5 + 15*int(acsch(c*x)/x**4,x)*b*e*x**5 - 
 3*a*d - 5*a*e*x**2)/(15*x**5)