\(\int \frac {(d+e x^2)^2 (a+b \text {csch}^{-1}(c x))}{x^6} \, dx\) [93]

Optimal result

Integrand size = 21, antiderivative size = 189 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^6} \, dx=\frac {b c \left (24 c^4 d^2-100 c^2 d e+225 e^2\right ) \sqrt {-1-c^2 x^2}}{225 \sqrt {-c^2 x^2}}+\frac {b c d^2 \sqrt {-1-c^2 x^2}}{25 x^4 \sqrt {-c^2 x^2}}-\frac {2 b c d \left (6 c^2 d-25 e\right ) \sqrt {-1-c^2 x^2}}{225 x^2 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x} \] Output:

1/225*b*c*(24*c^4*d^2-100*c^2*d*e+225*e^2)*(-c^2*x^2-1)^(1/2)/(-c^2*x^2)^( 
1/2)+1/25*b*c*d^2*(-c^2*x^2-1)^(1/2)/x^4/(-c^2*x^2)^(1/2)-2/225*b*c*d*(6*c 
^2*d-25*e)*(-c^2*x^2-1)^(1/2)/x^2/(-c^2*x^2)^(1/2)-1/5*d^2*(a+b*arccsch(c* 
x))/x^5-2/3*d*e*(a+b*arccsch(c*x))/x^3-e^2*(a+b*arccsch(c*x))/x
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.67 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^6} \, dx=\frac {-15 a \left (3 d^2+10 d e x^2+15 e^2 x^4\right )+b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (225 e^2 x^4-50 d e x^2 \left (-1+2 c^2 x^2\right )+3 d^2 \left (3-4 c^2 x^2+8 c^4 x^4\right )\right )-15 b \left (3 d^2+10 d e x^2+15 e^2 x^4\right ) \text {csch}^{-1}(c x)}{225 x^5} \] Input:

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

Output:

(-15*a*(3*d^2 + 10*d*e*x^2 + 15*e^2*x^4) + b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(22 
5*e^2*x^4 - 50*d*e*x^2*(-1 + 2*c^2*x^2) + 3*d^2*(3 - 4*c^2*x^2 + 8*c^4*x^4 
)) - 15*b*(3*d^2 + 10*d*e*x^2 + 15*e^2*x^4)*ArcCsch[c*x])/(225*x^5)
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6856, 27, 1588, 25, 359, 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.

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

\(\Big \downarrow \) 6856

\(\displaystyle -\frac {b c x \int -\frac {15 e^2 x^4+10 d e x^2+3 d^2}{15 x^6 \sqrt {-c^2 x^2-1}}dx}{\sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b c x \int \frac {15 e^2 x^4+10 d e x^2+3 d^2}{x^6 \sqrt {-c^2 x^2-1}}dx}{15 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x}\)

\(\Big \downarrow \) 1588

\(\displaystyle \frac {b c x \left (\frac {1}{5} \int -\frac {2 d \left (6 c^2 d-25 e\right )-75 e^2 x^2}{x^4 \sqrt {-c^2 x^2-1}}dx+\frac {3 d^2 \sqrt {-c^2 x^2-1}}{5 x^5}\right )}{15 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b c x \left (\frac {3 d^2 \sqrt {-c^2 x^2-1}}{5 x^5}-\frac {1}{5} \int \frac {2 d \left (6 c^2 d-25 e\right )-75 e^2 x^2}{x^4 \sqrt {-c^2 x^2-1}}dx\right )}{15 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x}\)

\(\Big \downarrow \) 359

\(\displaystyle \frac {b c x \left (\frac {1}{5} \left (\frac {1}{3} \left (24 c^4 d^2-100 c^2 d e+225 e^2\right ) \int \frac {1}{x^2 \sqrt {-c^2 x^2-1}}dx-\frac {2 d \sqrt {-c^2 x^2-1} \left (6 c^2 d-25 e\right )}{3 x^3}\right )+\frac {3 d^2 \sqrt {-c^2 x^2-1}}{5 x^5}\right )}{15 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x}\)

\(\Big \downarrow \) 242

\(\displaystyle -\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x}+\frac {b c x \left (\frac {3 d^2 \sqrt {-c^2 x^2-1}}{5 x^5}+\frac {1}{5} \left (\frac {\sqrt {-c^2 x^2-1} \left (24 c^4 d^2-100 c^2 d e+225 e^2\right )}{3 x}-\frac {2 d \sqrt {-c^2 x^2-1} \left (6 c^2 d-25 e\right )}{3 x^3}\right )\right )}{15 \sqrt {-c^2 x^2}}\)

Input:

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

Output:

(b*c*x*((3*d^2*Sqrt[-1 - c^2*x^2])/(5*x^5) + ((-2*d*(6*c^2*d - 25*e)*Sqrt[ 
-1 - c^2*x^2])/(3*x^3) + ((24*c^4*d^2 - 100*c^2*d*e + 225*e^2)*Sqrt[-1 - c 
^2*x^2])/(3*x))/5))/(15*Sqrt[-(c^2*x^2)]) - (d^2*(a + b*ArcCsch[c*x]))/(5* 
x^5) - (2*d*e*(a + b*ArcCsch[c*x]))/(3*x^3) - (e^2*(a + b*ArcCsch[c*x]))/x
 

Defintions of rubi rules used

rule 25

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

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 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 1588

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c 
_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + 
c*x^4)^p, f*x, x], R = PolynomialRemainder[(a + b*x^2 + c*x^4)^p, f*x, x]}, 
 Simp[R*(f*x)^(m + 1)*((d + e*x^2)^(q + 1)/(d*f*(m + 1))), x] + Simp[1/(d*f 
^2*(m + 1))   Int[(f*x)^(m + 2)*(d + e*x^2)^q*ExpandToSum[d*f*(m + 1)*(Qx/x 
) - e*R*(m + 2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && Ne 
Q[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -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 = 0.37 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.93

method	result	size
parts	\(a \left (-\frac {2 d e}{3 x^{3}}-\frac {e^{2}}{x}-\frac {d^{2}}{5 x^{5}}\right )+b \,c^{5} \left (-\frac {2 \,\operatorname {arccsch}\left (c x \right ) d e}{3 c^{5} x^{3}}-\frac {\operatorname {arccsch}\left (c x \right ) e^{2}}{c^{5} x}-\frac {\operatorname {arccsch}\left (c x \right ) d^{2}}{5 c^{5} x^{5}}+\frac {\left (c^{2} x^{2}+1\right ) \left (24 c^{8} d^{2} x^{4}-100 c^{6} d e \,x^{4}-12 c^{6} d^{2} x^{2}+225 c^{4} e^{2} x^{4}+50 c^{4} d e \,x^{2}+9 c^{4} d^{2}\right )}{225 c^{10} \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x^{6}}\right )\)	\(175\)
derivativedivides	\(c^{5} \left (\frac {a \left (-\frac {d^{2}}{5 c \,x^{5}}-\frac {2 d e}{3 c \,x^{3}}-\frac {e^{2}}{c x}\right )}{c^{4}}+\frac {b \left (-\frac {\operatorname {arccsch}\left (c x \right ) d^{2}}{5 c \,x^{5}}-\frac {2 \,\operatorname {arccsch}\left (c x \right ) d e}{3 c \,x^{3}}-\frac {\operatorname {arccsch}\left (c x \right ) e^{2}}{c x}+\frac {\left (c^{2} x^{2}+1\right ) \left (24 c^{8} d^{2} x^{4}-100 c^{6} d e \,x^{4}-12 c^{6} d^{2} x^{2}+225 c^{4} e^{2} x^{4}+50 c^{4} d e \,x^{2}+9 c^{4} d^{2}\right )}{225 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{6} x^{6}}\right )}{c^{4}}\right )\)	\(191\)
default	\(c^{5} \left (\frac {a \left (-\frac {d^{2}}{5 c \,x^{5}}-\frac {2 d e}{3 c \,x^{3}}-\frac {e^{2}}{c x}\right )}{c^{4}}+\frac {b \left (-\frac {\operatorname {arccsch}\left (c x \right ) d^{2}}{5 c \,x^{5}}-\frac {2 \,\operatorname {arccsch}\left (c x \right ) d e}{3 c \,x^{3}}-\frac {\operatorname {arccsch}\left (c x \right ) e^{2}}{c x}+\frac {\left (c^{2} x^{2}+1\right ) \left (24 c^{8} d^{2} x^{4}-100 c^{6} d e \,x^{4}-12 c^{6} d^{2} x^{2}+225 c^{4} e^{2} x^{4}+50 c^{4} d e \,x^{2}+9 c^{4} d^{2}\right )}{225 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{6} x^{6}}\right )}{c^{4}}\right )\)	\(191\)

Input:

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

Output:

a*(-2/3*d*e/x^3-e^2/x-1/5*d^2/x^5)+b*c^5*(-2/3/c^5*arccsch(c*x)*d*e/x^3-1/ 
c^5*arccsch(c*x)*e^2/x-1/5*arccsch(c*x)*d^2/c^5/x^5+1/225/c^10*(c^2*x^2+1) 
*(24*c^8*d^2*x^4-100*c^6*d*e*x^4-12*c^6*d^2*x^2+225*c^4*e^2*x^4+50*c^4*d*e 
*x^2+9*c^4*d^2)/((c^2*x^2+1)/c^2/x^2)^(1/2)/x^6)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.87 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^6} \, dx=-\frac {225 \, a e^{2} x^{4} + 150 \, a d e x^{2} + 45 \, a d^{2} + 15 \, {\left (15 \, b e^{2} x^{4} + 10 \, b d e x^{2} + 3 \, b d^{2}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left ({\left (24 \, b c^{5} d^{2} - 100 \, b c^{3} d e + 225 \, b c e^{2}\right )} x^{5} + 9 \, b c d^{2} x - 2 \, {\left (6 \, b c^{3} d^{2} - 25 \, b c d e\right )} x^{3}\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{225 \, x^{5}} \] Input:

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

Output:

-1/225*(225*a*e^2*x^4 + 150*a*d*e*x^2 + 45*a*d^2 + 15*(15*b*e^2*x^4 + 10*b 
*d*e*x^2 + 3*b*d^2)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) - ( 
(24*b*c^5*d^2 - 100*b*c^3*d*e + 225*b*c*e^2)*x^5 + 9*b*c*d^2*x - 2*(6*b*c^ 
3*d^2 - 25*b*c*d*e)*x^3)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))/x^5
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.93 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^6} \, dx={\left (c \sqrt {\frac {1}{c^{2} x^{2}} + 1} - \frac {\operatorname {arcsch}\left (c x\right )}{x}\right )} b e^{2} + \frac {1}{75} \, b d^{2} {\left (\frac {3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 10 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, c^{6} \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {15 \, \operatorname {arcsch}\left (c x\right )}{x^{5}}\right )} + \frac {2}{9} \, b d e {\left (\frac {c^{4} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {3 \, \operatorname {arcsch}\left (c x\right )}{x^{3}}\right )} - \frac {a e^{2}}{x} - \frac {2 \, a d e}{3 \, x^{3}} - \frac {a d^{2}}{5 \, x^{5}} \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^6} \, dx=\frac {15 \left (\int \frac {\mathit {acsch} \left (c x \right )}{x^{6}}d x \right ) b \,d^{2} x^{5}+30 \left (\int \frac {\mathit {acsch} \left (c x \right )}{x^{4}}d x \right ) b d e \,x^{5}+15 \left (\int \frac {\mathit {acsch} \left (c x \right )}{x^{2}}d x \right ) b \,e^{2} x^{5}-3 a \,d^{2}-10 a d e \,x^{2}-15 a \,e^{2} x^{4}}{15 x^{5}} \] Input:

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

Output:

(15*int(acsch(c*x)/x**6,x)*b*d**2*x**5 + 30*int(acsch(c*x)/x**4,x)*b*d*e*x 
**5 + 15*int(acsch(c*x)/x**2,x)*b*e**2*x**5 - 3*a*d**2 - 10*a*d*e*x**2 - 1 
5*a*e**2*x**4)/(15*x**5)