Integrand size = 20, antiderivative size = 278 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}-\frac {b c \sqrt {e} x \sqrt {-1-c^2 x^2} E\left (\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )|1-\frac {c^2 d}{e}\right )}{3 d^{3/2} \left (c^2 d-e\right ) \sqrt {-c^2 x^2} \sqrt {\frac {d \left (1+c^2 x^2\right )}{d+e x^2}} \sqrt {d+e x^2}}-\frac {b \left (3 c^2 d-2 e\right ) x \sqrt {d+e x^2} \operatorname {EllipticF}\left (\arctan (c x),1-\frac {e}{c^2 d}\right )}{3 d^3 \left (c^2 d-e\right ) \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}} \]
1/3*x*(a+b*arccsch(c*x))/d/(e*x^2+d)^(3/2)+2/3*x*(a+b*arccsch(c*x))/d^2/(e *x^2+d)^(1/2)-1/3*b*c*x*(1/(1+e*x^2/d))^(1/2)*(1+e*x^2/d)^(1/2)*EllipticE( x*e^(1/2)/d^(1/2)/(1+e*x^2/d)^(1/2),(1-c^2*d/e)^(1/2))*e^(1/2)*(-c^2*x^2-1 )^(1/2)/d^(3/2)/(c^2*d-e)/(-c^2*x^2)^(1/2)/(d*(c^2*x^2+1)/(e*x^2+d))^(1/2) /(e*x^2+d)^(1/2)-1/3*b*(3*c^2*d-2*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^ 3/(c^2*d-e)/(-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 = 5.87 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.89 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {x \left (-b c e \sqrt {1+\frac {1}{c^2 x^2}} x \left (d+e x^2\right )+a \left (c^2 d-e\right ) \left (3 d+2 e x^2\right )+b \left (c^2 d-e\right ) \left (3 d+2 e x^2\right ) \text {csch}^{-1}(c x)\right )}{3 d^2 \left (c^2 d-e\right ) \left (d+e x^2\right )^{3/2}}-\frac {i b c \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (c^2 d E\left (i \text {arcsinh}\left (\sqrt {c^2} x\right )|\frac {e}{c^2 d}\right )+2 \left (c^2 d-e\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {c^2} x\right ),\frac {e}{c^2 d}\right )\right )}{3 \sqrt {c^2} d^2 \left (c^2 d-e\right ) \sqrt {1+c^2 x^2} \sqrt {d+e x^2}} \]
(x*(-(b*c*e*Sqrt[1 + 1/(c^2*x^2)]*x*(d + e*x^2)) + a*(c^2*d - e)*(3*d + 2* e*x^2) + b*(c^2*d - e)*(3*d + 2*e*x^2)*ArcCsch[c*x]))/(3*d^2*(c^2*d - e)*( d + e*x^2)^(3/2)) - ((I/3)*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d] *(c^2*d*EllipticE[I*ArcSinh[Sqrt[c^2]*x], e/(c^2*d)] + 2*(c^2*d - e)*Ellip ticF[I*ArcSinh[Sqrt[c^2]*x], e/(c^2*d)]))/(Sqrt[c^2]*d^2*(c^2*d - e)*Sqrt[ 1 + c^2*x^2]*Sqrt[d + e*x^2])
Time = 0.42 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6846, 27, 400, 313, 320}
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.
| \(\displaystyle \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6846 |
| \(\displaystyle -\frac {b c x \int \frac {2 e x^2+3 d}{3 d^2 \sqrt {-c^2 x^2-1} \left (e x^2+d\right )^{3/2}}dx}{\sqrt {-c^2 x^2}}+\frac {2 x \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c x \int \frac {2 e x^2+3 d}{\sqrt {-c^2 x^2-1} \left (e x^2+d\right )^{3/2}}dx}{3 d^2 \sqrt {-c^2 x^2}}+\frac {2 x \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle -\frac {b c x \left (\frac {d e \int \frac {\sqrt {-c^2 x^2-1}}{\left (e x^2+d\right )^{3/2}}dx}{c^2 d-e}+\frac {\left (3 c^2 d-2 e\right ) \int \frac {1}{\sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx}{c^2 d-e}\right )}{3 d^2 \sqrt {-c^2 x^2}}+\frac {2 x \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle -\frac {b c x \left (\frac {\left (3 c^2 d-2 e\right ) \int \frac {1}{\sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx}{c^2 d-e}+\frac {\sqrt {d} \sqrt {e} \sqrt {-c^2 x^2-1} E\left (\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )|1-\frac {c^2 d}{e}\right )}{\left (c^2 d-e\right ) \sqrt {d+e x^2} \sqrt {\frac {d \left (c^2 x^2+1\right )}{d+e x^2}}}\right )}{3 d^2 \sqrt {-c^2 x^2}}+\frac {2 x \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {2 x \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b c x \left (\frac {\left (3 c^2 d-2 e\right ) \sqrt {d+e x^2} \operatorname {EllipticF}\left (\arctan (c x),1-\frac {e}{c^2 d}\right )}{c d \sqrt {-c^2 x^2-1} \left (c^2 d-e\right ) \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac {\sqrt {d} \sqrt {e} \sqrt {-c^2 x^2-1} E\left (\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )|1-\frac {c^2 d}{e}\right )}{\left (c^2 d-e\right ) \sqrt {d+e x^2} \sqrt {\frac {d \left (c^2 x^2+1\right )}{d+e x^2}}}\right )}{3 d^2 \sqrt {-c^2 x^2}}\) |
(x*(a + b*ArcCsch[c*x]))/(3*d*(d + e*x^2)^(3/2)) + (2*x*(a + b*ArcCsch[c*x ]))/(3*d^2*Sqrt[d + e*x^2]) - (b*c*x*((Sqrt[d]*Sqrt[e]*Sqrt[-1 - c^2*x^2]* EllipticE[ArcTan[(Sqrt[e]*x)/Sqrt[d]], 1 - (c^2*d)/e])/((c^2*d - e)*Sqrt[( d*(1 + c^2*x^2))/(d + e*x^2)]*Sqrt[d + e*x^2]) + ((3*c^2*d - 2*e)*Sqrt[d + e*x^2]*EllipticF[ArcTan[c*x], 1 - e/(c^2*d)])/(c*d*(c^2*d - e)*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))])))/(3*d^2*Sqrt[-(c^2*x^2)])
3.2.64.3.1 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_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^ (3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(Sqrt[a + b*x^2]* Sqrt[c + d*x^2]), x], x] - Simp[(d*e - c*f)/(b*c - a*d) Int[Sqrt[a + b*x^ 2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] & & PosQ[d/c]
Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(p_.), x_Sym bol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcCsch[c*x]) u , x] - Simp[b*c*(x/Sqrt[(-c^2)*x^2]) Int[SimplifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && (IGtQ[p, 0] || ILtQ [p + 1/2, 0])
\[\int \frac {a +b \,\operatorname {arccsch}\left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
Time = 0.11 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.59 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {{\left (b c^{4} d e^{2} x^{4} + 2 \, b c^{4} d^{2} e x^{2} + b c^{4} d^{3}\right )} \sqrt {-c^{2}} \sqrt {d} E(\arcsin \left (\sqrt {-c^{2}} x\right )\,|\,\frac {e}{c^{2} d}) - {\left ({\left ({\left (b c^{4} + 3 \, b c^{2}\right )} d e^{2} - 2 \, b e^{3}\right )} x^{4} + {\left (b c^{4} + 3 \, b c^{2}\right )} d^{3} - 2 \, b d^{2} e + 2 \, {\left ({\left (b c^{4} + 3 \, b c^{2}\right )} d^{2} e - 2 \, b d e^{2}\right )} x^{2}\right )} \sqrt {-c^{2}} \sqrt {d} F(\arcsin \left (\sqrt {-c^{2}} x\right )\,|\,\frac {e}{c^{2} d}) + {\left (2 \, {\left (b c^{4} d^{2} e - b c^{2} d e^{2}\right )} x^{3} + 3 \, {\left (b c^{4} d^{3} - b c^{2} d^{2} e\right )} x\right )} \sqrt {e x^{2} + d} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (2 \, {\left (a c^{4} d^{2} e - a c^{2} d e^{2}\right )} x^{3} + 3 \, {\left (a c^{4} d^{3} - a c^{2} d^{2} e\right )} x - {\left (b c^{3} d e^{2} x^{4} + b c^{3} d^{2} e x^{2}\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \sqrt {e x^{2} + d}}{3 \, {\left (c^{4} d^{6} - c^{2} d^{5} e + {\left (c^{4} d^{4} e^{2} - c^{2} d^{3} e^{3}\right )} x^{4} + 2 \, {\left (c^{4} d^{5} e - c^{2} d^{4} e^{2}\right )} x^{2}\right )}} \]
1/3*((b*c^4*d*e^2*x^4 + 2*b*c^4*d^2*e*x^2 + b*c^4*d^3)*sqrt(-c^2)*sqrt(d)* elliptic_e(arcsin(sqrt(-c^2)*x), e/(c^2*d)) - (((b*c^4 + 3*b*c^2)*d*e^2 - 2*b*e^3)*x^4 + (b*c^4 + 3*b*c^2)*d^3 - 2*b*d^2*e + 2*((b*c^4 + 3*b*c^2)*d^ 2*e - 2*b*d*e^2)*x^2)*sqrt(-c^2)*sqrt(d)*elliptic_f(arcsin(sqrt(-c^2)*x), e/(c^2*d)) + (2*(b*c^4*d^2*e - b*c^2*d*e^2)*x^3 + 3*(b*c^4*d^3 - b*c^2*d^2 *e)*x)*sqrt(e*x^2 + d)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + (2*(a*c^4*d^2*e - a*c^2*d*e^2)*x^3 + 3*(a*c^4*d^3 - a*c^2*d^2*e)*x - (b* c^3*d*e^2*x^4 + b*c^3*d^2*e*x^2)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))*sqrt(e*x^2 + d))/(c^4*d^6 - c^2*d^5*e + (c^4*d^4*e^2 - c^2*d^3*e^3)*x^4 + 2*(c^4*d^5 *e - c^2*d^4*e^2)*x^2)
Timed out. \[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
1/3*a*(2*x/(sqrt(e*x^2 + d)*d^2) + x/((e*x^2 + d)^(3/2)*d)) + b*integrate( log(sqrt(1/(c^2*x^2) + 1) + 1/(c*x))/(e*x^2 + d)^(5/2), x)
\[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]