Integrand size = 18, antiderivative size = 212 \[ \int \frac {a+b \csc ^{-1}(c x)}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e}-\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b d \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
2*(a+b*arccsc(c*x))*(e*x+d)^(1/2)/e-4*b*EllipticF(1/2*(-c*x+1)^(1/2)*2^(1/ 2),2^(1/2)*(e/(c*d+e))^(1/2))*(c*(e*x+d)/(c*d+e))^(1/2)*(-c^2*x^2+1)^(1/2) /c^2/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)-4*b*d*EllipticPi(1/2*(-c*x+1)^(1/ 2)*2^(1/2),2,2^(1/2)*(e/(c*d+e))^(1/2))*(c*(e*x+d)/(c*d+e))^(1/2)*(-c^2*x^ 2+1)^(1/2)/c/e/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)
Time = 5.23 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.15 \[ \int \frac {a+b \csc ^{-1}(c x)}{\sqrt {d+e x}} \, dx=\frac {2 \left (\frac {a (d+e x)}{e}+\frac {b \left ((d+e x) \csc ^{-1}(c x)+\frac {2 c d \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {1-c^2 x^2}}\right )}{e}+\frac {2 b c x^2 \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right ) \left (\cos \left (\frac {1}{2} \csc ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \csc ^{-1}(c x)\right )\right )^3 \left (\cos \left (\frac {1}{2} \csc ^{-1}(c x)\right )+\sin \left (\frac {1}{2} \csc ^{-1}(c x)\right )\right )}{\sqrt {1-c x} \left (-1+c^2 x^2\right )}\right )}{\sqrt {d+e x}} \]
(2*((a*(d + e*x))/e + (b*((d + e*x)*ArcCsc[c*x] + (2*c*d*Sqrt[1 - 1/(c^2*x ^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sq rt[2]], (2*e)/(c*d + e)])/Sqrt[1 - c^2*x^2]))/e + (2*b*c*x^2*Sqrt[1 + c*x] *Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2 *e)/(c*d + e)]*(Cos[ArcCsc[c*x]/2] - Sin[ArcCsc[c*x]/2])^3*(Cos[ArcCsc[c*x ]/2] + Sin[ArcCsc[c*x]/2]))/(Sqrt[1 - c*x]*(-1 + c^2*x^2))))/Sqrt[d + e*x]
Time = 0.58 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5750, 1898, 637, 2009}
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.
| \(\displaystyle \int \frac {a+b \csc ^{-1}(c x)}{\sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 5750 |
| \(\displaystyle \frac {2 b \int \frac {\sqrt {d+e x}}{\sqrt {1-\frac {1}{c^2 x^2}} x^2}dx}{c e}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e}\) |
| \(\Big \downarrow \) 1898 |
| \(\displaystyle \frac {2 b \sqrt {x^2-\frac {1}{c^2}} \int \frac {\sqrt {d+e x}}{x \sqrt {x^2-\frac {1}{c^2}}}dx}{c e x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e}\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \frac {2 b \sqrt {x^2-\frac {1}{c^2}} \int \left (\frac {d}{x \sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}+\frac {e}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}\right )dx}{c e x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {d+e x}}-\frac {2 d \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {x^2-\frac {1}{c^2}} \sqrt {d+e x}}\right )}{c e x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
(2*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/e + (2*b*Sqrt[-c^(-2) + x^2]*((-2*e* Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*Sqrt[d + e*x]*Sqrt[-c^(-2) + x^2]) - ( 2*d*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[S qrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[d + e*x]*Sqrt[-c^(-2) + x^2 ])))/(c*e*Sqrt[1 - 1/(c^2*x^2)]*x)
3.1.60.3.1 Defintions of rubi rules used
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Int[ExpandIntegrand[(a + b*x^2)^p/Sqrt[c + d*x], x^m*(c + d*x)^(n + 1 /2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[p + 1/2] && IntegerQ[n + 1/2] && IntegerQ[m]
Int[(x_)^(m_.)*((a_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^ (q_.), x_Symbol] :> Simp[x^(2*n*FracPart[p])*((a + c/x^(2*n))^FracPart[p]/( c + a*x^(2*n))^FracPart[p]) Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + a*x^(2*n ))^p, x], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[mn2, -2*n] && !I ntegerQ[p] && !IntegerQ[q] && PosQ[n]
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol ] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcCsc[c*x])/(e*(m + 1))), x] + Simp[b/ (c*e*(m + 1)) Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x] / ; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
Time = 3.32 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.19
| method | result | size |
| derivativedivides | \(\frac {2 a \sqrt {e x +d}+2 b \left (\sqrt {e x +d}\, \operatorname {arccsc}\left (c x \right )+\frac {2 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \left (\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right )-\operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right )\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {c}{c d -e}}}\right )}{e}\) | \(252\) |
| default | \(\frac {2 a \sqrt {e x +d}+2 b \left (\sqrt {e x +d}\, \operatorname {arccsc}\left (c x \right )+\frac {2 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \left (\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right )-\operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right )\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {c}{c d -e}}}\right )}{e}\) | \(252\) |
| parts | \(\frac {2 a \sqrt {e x +d}}{e}+\frac {2 b \left (\sqrt {e x +d}\, \operatorname {arccsc}\left (c x \right )+\frac {2 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \left (\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right )-\operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right )\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {c}{c d -e}}}\right )}{e}\) | \(257\) |
2/e*(a*(e*x+d)^(1/2)+b*((e*x+d)^(1/2)*arccsc(c*x)+2/c*((-c*(e*x+d)+c*d-e)/ (c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*(EllipticF((e*x+d)^(1/2) *(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))-EllipticPi((e*x+d)^(1/2)*(c/(c *d-e))^(1/2),1/c*(c*d-e)/d,(c/(c*d+e))^(1/2)/(c/(c*d-e))^(1/2)))/((c^2*(e* x+d)^2-2*c^2*d*(e*x+d)+c^2*d^2-e^2)/c^2/e^2/x^2)^(1/2)/x/(c/(c*d-e))^(1/2) ))
\[ \int \frac {a+b \csc ^{-1}(c x)}{\sqrt {d+e x}} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{\sqrt {e x + d}} \,d x } \]
\[ \int \frac {a+b \csc ^{-1}(c x)}{\sqrt {d+e x}} \, dx=\int \frac {a + b \operatorname {acsc}{\left (c x \right )}}{\sqrt {d + e x}}\, dx \]
Exception generated. \[ \int \frac {a+b \csc ^{-1}(c x)}{\sqrt {d+e x}} \, dx=\text {Exception raised: ValueError} \]
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+c*d>0)', see `assume?` for mor e details)
\[ \int \frac {a+b \csc ^{-1}(c x)}{\sqrt {d+e x}} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{\sqrt {e x + d}} \,d x } \]
Timed out. \[ \int \frac {a+b \csc ^{-1}(c x)}{\sqrt {d+e x}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{\sqrt {d+e\,x}} \,d x \]