Integrand size = 23, antiderivative size = 389 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx=-\frac {2 b c^3 \left (c^2 d-2 e\right ) x^2 \sqrt {d+e x^2}}{9 d \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}-\frac {2 b c \left (c^2 d-2 e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 x^2 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}+\frac {2 b c^2 \left (c^2 d-2 e\right ) x \sqrt {d+e x^2} E\left (\arctan (c x)\left |1-\frac {e}{c^2 d}\right .\right )}{9 d \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}}-\frac {b \left (c^2 d-3 e\right ) e x \sqrt {d+e x^2} \operatorname {EllipticF}\left (\arctan (c x),1-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}} \]
-1/3*(e*x^2+d)^(3/2)*(a+b*arccsch(c*x))/d/x^3-2/9*b*c^3*(c^2*d-2*e)*x^2*(e *x^2+d)^(1/2)/d/(-c^2*x^2)^(1/2)/(-c^2*x^2-1)^(1/2)-2/9*b*c*(c^2*d-2*e)*(- c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d/(-c^2*x^2)^(1/2)+1/9*b*c*(-c^2*x^2-1)^( 1/2)*(e*x^2+d)^(1/2)/x^2/(-c^2*x^2)^(1/2)+2/9*b*c^2*(c^2*d-2*e)*x*(1/(c^2* x^2+1))^(1/2)*(c^2*x^2+1)^(1/2)*EllipticE(c*x/(c^2*x^2+1)^(1/2),(1-e/c^2/d )^(1/2))*(e*x^2+d)^(1/2)/d/(-c^2*x^2)^(1/2)/(-c^2*x^2-1)^(1/2)/((e*x^2+d)/ d/(c^2*x^2+1))^(1/2)-1/9*b*(c^2*d-3*e)*e*x*(1/(c^2*x^2+1))^(1/2)*(c^2*x^2+ 1)^(1/2)*EllipticF(c*x/(c^2*x^2+1)^(1/2),(1-e/c^2/d)^(1/2))*(e*x^2+d)^(1/2 )/d^2/(-c^2*x^2)^(1/2)/(-c^2*x^2-1)^(1/2)/((e*x^2+d)/d/(c^2*x^2+1))^(1/2)
Result contains complex when optimal does not.
Time = 11.03 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx=-\frac {\sqrt {d+e x^2} \left (b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (-d+2 c^2 d x^2-4 e x^2\right )+3 a \left (d+e x^2\right )+3 b \left (d+e x^2\right ) \text {csch}^{-1}(c x)\right )}{9 d x^3}-\frac {i b c \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (2 c^2 d \left (c^2 d-2 e\right ) E\left (i \text {arcsinh}\left (\sqrt {c^2} x\right )|\frac {e}{c^2 d}\right )+\left (-2 c^4 d^2+5 c^2 d e-3 e^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {c^2} x\right ),\frac {e}{c^2 d}\right )\right )}{9 \sqrt {c^2} d \sqrt {1+c^2 x^2} \sqrt {d+e x^2}} \]
-1/9*(Sqrt[d + e*x^2]*(b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(-d + 2*c^2*d*x^2 - 4*e *x^2) + 3*a*(d + e*x^2) + 3*b*(d + e*x^2)*ArcCsch[c*x]))/(d*x^3) - ((I/9)* b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(2*c^2*d*(c^2*d - 2*e)*Ell ipticE[I*ArcSinh[Sqrt[c^2]*x], e/(c^2*d)] + (-2*c^4*d^2 + 5*c^2*d*e - 3*e^ 2)*EllipticF[I*ArcSinh[Sqrt[c^2]*x], e/(c^2*d)]))/(Sqrt[c^2]*d*Sqrt[1 + c^ 2*x^2]*Sqrt[d + e*x^2])
Time = 0.61 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.87, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6856, 27, 376, 445, 27, 406, 320, 388, 313}
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.
| \(\displaystyle \int \frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx\) |
| \(\Big \downarrow \) 6856 |
| \(\displaystyle -\frac {b c x \int -\frac {\left (e x^2+d\right )^{3/2}}{3 d x^4 \sqrt {-c^2 x^2-1}}dx}{\sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c x \int \frac {\left (e x^2+d\right )^{3/2}}{x^4 \sqrt {-c^2 x^2-1}}dx}{3 d \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}\) |
| \(\Big \downarrow \) 376 |
| \(\displaystyle \frac {b c x \left (\frac {d \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{3 x^3}-\frac {1}{3} \int \frac {\left (c^2 d-3 e\right ) e x^2+2 d \left (c^2 d-2 e\right )}{x^2 \sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx\right )}{3 d \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {b c x \left (\frac {1}{3} \left (-\frac {\int \frac {d e \left (2 \left (c^2 d-2 e\right ) x^2 c^2+d c^2-3 e\right )}{\sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx}{d}-\frac {2 \sqrt {-c^2 x^2-1} \left (c^2 d-2 e\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{3 x^3}\right )}{3 d \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c x \left (\frac {1}{3} \left (-e \int \frac {2 \left (c^2 d-2 e\right ) x^2 c^2+d c^2-3 e}{\sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx-\frac {2 \sqrt {-c^2 x^2-1} \left (c^2 d-2 e\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{3 x^3}\right )}{3 d \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {b c x \left (\frac {1}{3} \left (-e \left (2 c^2 \left (c^2 d-2 e\right ) \int \frac {x^2}{\sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx+\left (c^2 d-3 e\right ) \int \frac {1}{\sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx\right )-\frac {2 \sqrt {-c^2 x^2-1} \left (c^2 d-2 e\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{3 x^3}\right )}{3 d \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b c x \left (\frac {1}{3} \left (-e \left (2 c^2 \left (c^2 d-2 e\right ) \int \frac {x^2}{\sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx+\frac {\left (c^2 d-3 e\right ) \sqrt {d+e x^2} \operatorname {EllipticF}\left (\arctan (c x),1-\frac {e}{c^2 d}\right )}{c d \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}\right )-\frac {2 \sqrt {-c^2 x^2-1} \left (c^2 d-2 e\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{3 x^3}\right )}{3 d \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {b c x \left (\frac {1}{3} \left (-e \left (2 c^2 \left (c^2 d-2 e\right ) \left (\frac {\int \frac {\sqrt {e x^2+d}}{\left (-c^2 x^2-1\right )^{3/2}}dx}{e}+\frac {x \sqrt {d+e x^2}}{e \sqrt {-c^2 x^2-1}}\right )+\frac {\left (c^2 d-3 e\right ) \sqrt {d+e x^2} \operatorname {EllipticF}\left (\arctan (c x),1-\frac {e}{c^2 d}\right )}{c d \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}\right )-\frac {2 \sqrt {-c^2 x^2-1} \left (c^2 d-2 e\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{3 x^3}\right )}{3 d \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b c x \left (\frac {1}{3} \left (-e \left (\frac {\left (c^2 d-3 e\right ) \sqrt {d+e x^2} \operatorname {EllipticF}\left (\arctan (c x),1-\frac {e}{c^2 d}\right )}{c d \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}+2 c^2 \left (c^2 d-2 e\right ) \left (\frac {x \sqrt {d+e x^2}}{e \sqrt {-c^2 x^2-1}}-\frac {\sqrt {d+e x^2} E\left (\arctan (c x)\left |1-\frac {e}{c^2 d}\right .\right )}{c e \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}\right )\right )-\frac {2 \sqrt {-c^2 x^2-1} \left (c^2 d-2 e\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{3 x^3}\right )}{3 d \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}\) |
-1/3*((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/(d*x^3) + (b*c*x*((d*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(3*x^3) + ((-2*(c^2*d - 2*e)*Sqrt[-1 - c^2*x^ 2]*Sqrt[d + e*x^2])/x - e*(2*c^2*(c^2*d - 2*e)*((x*Sqrt[d + e*x^2])/(e*Sqr t[-1 - c^2*x^2]) - (Sqrt[d + e*x^2]*EllipticE[ArcTan[c*x], 1 - e/(c^2*d)]) /(c*e*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))])) + ((c^2*d - 3*e)*Sqrt[d + e*x^2]*EllipticF[ArcTan[c*x], 1 - e/(c^2*d)])/(c*d*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))])))/3))/(3*d*Sqrt[-(c^2*x^2) ])
3.2.26.3.1 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[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> Simp[c*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1 )/(a*e*(m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^ 2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b*c - a*d)*(m + 1) + 2*c*(b*c*(p + 1) + a* d*(q - 1)) + d*((b*c - a*d)*(m + 1) + 2*b*c*(p + q))*x^2, x], x], x] /; Fre eQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && LtQ[m, -1] & & IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[e Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim p[f Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -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]))
\[\int \frac {\left (a +b \,\operatorname {arccsch}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}}{x^{4}}d x\]
Time = 0.11 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx=\frac {2 \, {\left (b c^{6} d^{2} - 2 \, b c^{4} d e\right )} \sqrt {-c^{2}} \sqrt {d} x^{3} E(\arcsin \left (\sqrt {-c^{2}} x\right )\,|\,\frac {e}{c^{2} d}) - {\left (2 \, b c^{6} d^{2} - {\left (4 \, b c^{4} - b c^{2}\right )} d e - 3 \, b e^{2}\right )} \sqrt {-c^{2}} \sqrt {d} x^{3} F(\arcsin \left (\sqrt {-c^{2}} x\right )\,|\,\frac {e}{c^{2} d}) - 3 \, {\left (b c^{2} d e x^{2} + b c^{2} d^{2}\right )} \sqrt {e x^{2} + d} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (3 \, a c^{2} d e x^{2} + 3 \, a c^{2} d^{2} - {\left (b c^{3} d^{2} x - 2 \, {\left (b c^{5} d^{2} - 2 \, b c^{3} d e\right )} x^{3}\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \sqrt {e x^{2} + d}}{9 \, c^{2} d^{2} x^{3}} \]
1/9*(2*(b*c^6*d^2 - 2*b*c^4*d*e)*sqrt(-c^2)*sqrt(d)*x^3*elliptic_e(arcsin( sqrt(-c^2)*x), e/(c^2*d)) - (2*b*c^6*d^2 - (4*b*c^4 - b*c^2)*d*e - 3*b*e^2 )*sqrt(-c^2)*sqrt(d)*x^3*elliptic_f(arcsin(sqrt(-c^2)*x), e/(c^2*d)) - 3*( b*c^2*d*e*x^2 + b*c^2*d^2)*sqrt(e*x^2 + d)*log((c*x*sqrt((c^2*x^2 + 1)/(c^ 2*x^2)) + 1)/(c*x)) - (3*a*c^2*d*e*x^2 + 3*a*c^2*d^2 - (b*c^3*d^2*x - 2*(b *c^5*d^2 - 2*b*c^3*d*e)*x^3)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))*sqrt(e*x^2 + d ))/(c^2*d^2*x^3)
\[ \int \frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx=\int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}}{x^{4}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx=\int { \frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx=\int \frac {\sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x^4} \,d x \]