Integrand size = 23, antiderivative size = 275 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=-\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^2 \sqrt {c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b \left (c^2 d+2 e\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \]
(-a-b*arccsc(c*x))/d/x/(e*x^2+d)^(1/2)-2*e*x*(a+b*arccsc(c*x))/d^2/(e*x^2+ d)^(1/2)-b*c*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d^2/(c^2*x^2)^(1/2)+b*c^2*x *EllipticE(c*x,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(e*x^2+d)^(1/2)/d^2/(c ^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(1+e*x^2/d)^(1/2)-b*(c^2*d+2*e)*x*Elliptic F(c*x,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(1+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)^(1/2)
Result contains complex when optimal does not.
Time = 4.96 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.77 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\frac {-b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+e x^2\right )-a \left (d+2 e x^2\right )-b \left (d+2 e x^2\right ) \csc ^{-1}(c x)}{d^2 x \sqrt {d+e x^2}}+\frac {i b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (c^2 d E\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )-\left (c^2 d+2 e\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right ),-\frac {e}{c^2 d}\right )\right )}{\sqrt {-c^2} d^2 \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]
(-(b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x^2)) - a*(d + 2*e*x^2) - b*(d + 2*e *x^2)*ArcCsc[c*x])/(d^2*x*Sqrt[d + e*x^2]) + (I*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 ))] - (c^2*d + 2*e)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))/(Sq rt[-c^2]*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])
Time = 0.57 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.94, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {5762, 25, 27, 445, 27, 399, 323, 323, 321, 331, 330, 327}
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.
| \(\displaystyle \int \frac {a+b \csc ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5762 |
| \(\displaystyle \frac {b c x \int -\frac {2 e x^2+d}{d^2 x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx}{\sqrt {c^2 x^2}}-\frac {2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \csc ^{-1}(c x)}{d x \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b c x \int \frac {2 e x^2+d}{d^2 x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx}{\sqrt {c^2 x^2}}-\frac {2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \csc ^{-1}(c x)}{d x \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c x \int \frac {2 e x^2+d}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx}{d^2 \sqrt {c^2 x^2}}-\frac {2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \csc ^{-1}(c x)}{d x \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle -\frac {b c x \left (\frac {\int \frac {d e \left (2-c^2 x^2\right )}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx}{d}+\frac {\sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{x}\right )}{d^2 \sqrt {c^2 x^2}}-\frac {2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \csc ^{-1}(c x)}{d x \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c x \left (e \int \frac {2-c^2 x^2}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx+\frac {\sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{x}\right )}{d^2 \sqrt {c^2 x^2}}-\frac {2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \csc ^{-1}(c x)}{d x \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle -\frac {b c x \left (e \left (\frac {\left (c^2 d+2 e\right ) \int \frac {1}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx}{e}-\frac {c^2 \int \frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}dx}{e}\right )+\frac {\sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{x}\right )}{d^2 \sqrt {c^2 x^2}}-\frac {2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \csc ^{-1}(c x)}{d x \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle -\frac {b c x \left (e \left (\frac {\left (c^2 d+2 e\right ) \sqrt {\frac {e x^2}{d}+1} \int \frac {1}{\sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}dx}{e \sqrt {d+e x^2}}-\frac {c^2 \int \frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}dx}{e}\right )+\frac {\sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{x}\right )}{d^2 \sqrt {c^2 x^2}}-\frac {2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \csc ^{-1}(c x)}{d x \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle -\frac {b c x \left (e \left (\frac {\sqrt {1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt {\frac {e x^2}{d}+1} \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1}}dx}{e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {c^2 \int \frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}dx}{e}\right )+\frac {\sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{x}\right )}{d^2 \sqrt {c^2 x^2}}-\frac {2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \csc ^{-1}(c x)}{d x \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {b c x \left (e \left (\frac {\sqrt {1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {c^2 \int \frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}dx}{e}\right )+\frac {\sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{x}\right )}{d^2 \sqrt {c^2 x^2}}-\frac {2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \csc ^{-1}(c x)}{d x \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 331 |
| \(\displaystyle -\frac {b c x \left (e \left (\frac {\sqrt {1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {c^2 \sqrt {1-c^2 x^2} \int \frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}dx}{e \sqrt {c^2 x^2-1}}\right )+\frac {\sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{x}\right )}{d^2 \sqrt {c^2 x^2}}-\frac {2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \csc ^{-1}(c x)}{d x \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle -\frac {b c x \left (e \left (\frac {\sqrt {1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {c^2 \sqrt {1-c^2 x^2} \sqrt {d+e x^2} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-c^2 x^2}}dx}{e \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}\right )+\frac {\sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{x}\right )}{d^2 \sqrt {c^2 x^2}}-\frac {2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \csc ^{-1}(c x)}{d x \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {2 e x \left (a+b \csc ^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \csc ^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {b c x \left (e \left (\frac {\sqrt {1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {c \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{e \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}\right )+\frac {\sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{x}\right )}{d^2 \sqrt {c^2 x^2}}\) |
-((a + b*ArcCsc[c*x])/(d*x*Sqrt[d + e*x^2])) - (2*e*x*(a + b*ArcCsc[c*x])) /(d^2*Sqrt[d + e*x^2]) - (b*c*x*((Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/x + e*(-((c*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2* d))])/(e*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d])) + ((c^2*d + 2*e)*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(c*e* Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2]))))/(d^2*Sqrt[c^2*x^2])
3.2.55.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && !GtQ[c, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2] Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && !GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && !GtQ[c, 0]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
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_.) + ArcCsc[(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]}, Sim p[(a + b*ArcCsc[c*x]) u, x] + Simp[b*c*(x/Sqrt[c^2*x^2]) Int[SimplifyIn tegrand[u/(x*Sqrt[c^2*x^2 - 1]), 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])) || (ILtQ[(m + 2*p + 1)/2, 0] && !ILtQ[(m - 1)/2, 0]))
\[\int \frac {a +b \,\operatorname {arccsc}\left (c x \right )}{x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]
Time = 0.11 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.68 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=-\frac {{\left (2 \, a c d e x^{2} + a c d^{2} + {\left (2 \, b c d e x^{2} + b c d^{2}\right )} \operatorname {arccsc}\left (c x\right ) + {\left (b c d e x^{2} + b c d^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d} + {\left ({\left (b c^{4} d e x^{3} + b c^{4} d^{2} x\right )} E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left ({\left (b c^{4} d e + 2 \, b e^{2}\right )} x^{3} + {\left (b c^{4} d^{2} + 2 \, b d e\right )} x\right )} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {-d}}{c d^{3} e x^{3} + c d^{4} x} \]
-((2*a*c*d*e*x^2 + a*c*d^2 + (2*b*c*d*e*x^2 + b*c*d^2)*arccsc(c*x) + (b*c* d*e*x^2 + b*c*d^2)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d) + ((b*c^4*d*e*x^3 + b*c^4*d^2*x)*elliptic_e(arcsin(c*x), -e/(c^2*d)) - ((b*c^4*d*e + 2*b*e^2)* x^3 + (b*c^4*d^2 + 2*b*d*e)*x)*elliptic_f(arcsin(c*x), -e/(c^2*d)))*sqrt(- d))/(c*d^3*e*x^3 + c*d^4*x)
\[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {acsc}{\left (c x \right )}}{x^{2} \left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{x^2\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \]