Integrand size = 19, antiderivative size = 109 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx=-\frac {b c \left (2 c^2 d-9 e\right ) \sqrt {-1-c^2 x^2}}{9 \sqrt {-c^2 x^2}}+\frac {b c d \sqrt {-1-c^2 x^2}}{9 x^2 \sqrt {-c^2 x^2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{x} \]
-1/3*d*(a+b*arccsch(c*x))/x^3-e*(a+b*arccsch(c*x))/x-1/9*b*c*(2*c^2*d-9*e) *(-c^2*x^2-1)^(1/2)/(-c^2*x^2)^(1/2)+1/9*b*c*d*(-c^2*x^2-1)^(1/2)/x^2/(-c^ 2*x^2)^(1/2)
Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.62 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx=\frac {-3 a \left (d+3 e x^2\right )+b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (d-2 c^2 d x^2+9 e x^2\right )-3 b \left (d+3 e x^2\right ) \text {csch}^{-1}(c x)}{9 x^3} \]
(-3*a*(d + 3*e*x^2) + b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(d - 2*c^2*d*x^2 + 9*e*x ^2) - 3*b*(d + 3*e*x^2)*ArcCsch[c*x])/(9*x^3)
Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6856, 27, 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.
\(\displaystyle \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx\) |
\(\Big \downarrow \) 6856 |
\(\displaystyle -\frac {b c x \int -\frac {3 e x^2+d}{3 x^4 \sqrt {-c^2 x^2-1}}dx}{\sqrt {-c^2 x^2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b c x \int \frac {3 e x^2+d}{x^4 \sqrt {-c^2 x^2-1}}dx}{3 \sqrt {-c^2 x^2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{x}\) |
\(\Big \downarrow \) 359 |
\(\displaystyle \frac {b c x \left (\frac {d \sqrt {-c^2 x^2-1}}{3 x^3}-\frac {1}{3} \left (2 c^2 d-9 e\right ) \int \frac {1}{x^2 \sqrt {-c^2 x^2-1}}dx\right )}{3 \sqrt {-c^2 x^2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{x}\) |
\(\Big \downarrow \) 242 |
\(\displaystyle -\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{x}+\frac {b c x \left (\frac {d \sqrt {-c^2 x^2-1}}{3 x^3}-\frac {\sqrt {-c^2 x^2-1} \left (2 c^2 d-9 e\right )}{3 x}\right )}{3 \sqrt {-c^2 x^2}}\) |
(b*c*x*((d*Sqrt[-1 - c^2*x^2])/(3*x^3) - ((2*c^2*d - 9*e)*Sqrt[-1 - c^2*x^ 2])/(3*x)))/(3*Sqrt[-(c^2*x^2)]) - (d*(a + b*ArcCsch[c*x]))/(3*x^3) - (e*( a + b*ArcCsch[c*x]))/x
3.1.80.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]))
Time = 0.35 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00
method | result | size |
parts | \(a \left (-\frac {e}{x}-\frac {d}{3 x^{3}}\right )+b \,c^{3} \left (-\frac {\operatorname {arccsch}\left (c x \right ) e}{c^{3} x}-\frac {\operatorname {arccsch}\left (c x \right ) d}{3 x^{3} c^{3}}-\frac {\left (c^{2} x^{2}+1\right ) \left (2 c^{4} d \,x^{2}-9 e \,c^{2} x^{2}-c^{2} d \right )}{9 c^{6} \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x^{4}}\right )\) | \(109\) |
derivativedivides | \(c^{3} \left (\frac {a \left (-\frac {e}{c x}-\frac {d}{3 c \,x^{3}}\right )}{c^{2}}+\frac {b \left (-\frac {\operatorname {arccsch}\left (c x \right ) e}{c x}-\frac {\operatorname {arccsch}\left (c x \right ) d}{3 c \,x^{3}}-\frac {\left (c^{2} x^{2}+1\right ) \left (2 c^{4} d \,x^{2}-9 e \,c^{2} x^{2}-c^{2} d \right )}{9 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )}{c^{2}}\right )\) | \(122\) |
default | \(c^{3} \left (\frac {a \left (-\frac {e}{c x}-\frac {d}{3 c \,x^{3}}\right )}{c^{2}}+\frac {b \left (-\frac {\operatorname {arccsch}\left (c x \right ) e}{c x}-\frac {\operatorname {arccsch}\left (c x \right ) d}{3 c \,x^{3}}-\frac {\left (c^{2} x^{2}+1\right ) \left (2 c^{4} d \,x^{2}-9 e \,c^{2} x^{2}-c^{2} d \right )}{9 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )}{c^{2}}\right )\) | \(122\) |
a*(-e/x-1/3*d/x^3)+b*c^3*(-1/c^3*arccsch(c*x)*e/x-1/3*arccsch(c*x)*d/x^3/c ^3-1/9/c^6*(c^2*x^2+1)*(2*c^4*d*x^2-9*c^2*e*x^2-c^2*d)/((c^2*x^2+1)/c^2/x^ 2)^(1/2)/x^4)
Time = 0.26 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.96 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx=-\frac {9 \, a e x^{2} + 3 \, a d + 3 \, {\left (3 \, b e x^{2} + b d\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (b c d x - {\left (2 \, b c^{3} d - 9 \, b c e\right )} x^{3}\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{9 \, x^{3}} \]
-1/9*(9*a*e*x^2 + 3*a*d + 3*(3*b*e*x^2 + b*d)*log((c*x*sqrt((c^2*x^2 + 1)/ (c^2*x^2)) + 1)/(c*x)) - (b*c*d*x - (2*b*c^3*d - 9*b*c*e)*x^3)*sqrt((c^2*x ^2 + 1)/(c^2*x^2)))/x^3
\[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx=\int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{4}}\, dx \]
Time = 0.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.83 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx={\left (c \sqrt {\frac {1}{c^{2} x^{2}} + 1} - \frac {\operatorname {arcsch}\left (c x\right )}{x}\right )} b e + \frac {1}{9} \, b d {\left (\frac {c^{4} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {3 \, \operatorname {arcsch}\left (c x\right )}{x^{3}}\right )} - \frac {a e}{x} - \frac {a d}{3 \, x^{3}} \]
(c*sqrt(1/(c^2*x^2) + 1) - arccsch(c*x)/x)*b*e + 1/9*b*d*((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/ x - 1/3*a*d/x^3
\[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx=\int \frac {\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x^4} \,d x \]