\(\int \frac {x (a+b \csc ^{-1}(c x))}{\sqrt {d+e x}} \, dx\) [59]

Optimal result

Integrand size = 19, antiderivative size = 344 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=-\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {4 b \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {8 b d \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {8 b d^2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \] Output:

-2*d*(e*x+d)^(1/2)*(a+b*arccsc(c*x))/e^2+2/3*(e*x+d)^(3/2)*(a+b*arccsc(c*x 
))/e^2-4/3*b*(e*x+d)^(1/2)*(-c^2*x^2+1)^(1/2)*EllipticE(1/2*(-c*x+1)^(1/2) 
*2^(1/2),2^(1/2)*(e/(c*d+e))^(1/2))/c^2/e/(1-1/c^2/x^2)^(1/2)/x/(c*(e*x+d) 
/(c*d+e))^(1/2)+8/3*b*d*(c*(e*x+d)/(c*d+e))^(1/2)*(-c^2*x^2+1)^(1/2)*Ellip 
ticF(1/2*(-c*x+1)^(1/2)*2^(1/2),2^(1/2)*(e/(c*d+e))^(1/2))/c^2/e/(1-1/c^2/ 
x^2)^(1/2)/x/(e*x+d)^(1/2)+8/3*b*d^2*(c*(e*x+d)/(c*d+e))^(1/2)*(-c^2*x^2+1 
)^(1/2)*EllipticPi(1/2*(-c*x+1)^(1/2)*2^(1/2),2,2^(1/2)*(e/(c*d+e))^(1/2)) 
/c/e^2/(1-1/c^2/x^2)^(1/2)/x/(e*x+d)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.81 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.11 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\frac {2 \left (a (-2 d+e x) (d+e x)+b (-2 d+e x) (d+e x) \csc ^{-1}(c x)-\frac {2 b d e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 b e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d-e}} \sqrt {\frac {e-c e x}{c d+e}} \left ((c d+e) E\left (\arcsin \left (\sqrt {\frac {c (d+e x)}{c d-e}}\right )|\frac {c d-e}{c d+e}\right )-e \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {c (d+e x)}{c d-e}}\right ),\frac {c d-e}{c d+e}\right )\right )}{c (-1+c x) \sqrt {\frac {e (1+c x)}{-c d+e}}}-\frac {4 b c d^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {1-c^2 x^2}}\right )}{3 e^2 \sqrt {d+e x}} \] Input:

Integrate[(x*(a + b*ArcCsc[c*x]))/Sqrt[d + e*x],x]
 

Output:

(2*(a*(-2*d + e*x)*(d + e*x) + b*(-2*d + e*x)*(d + e*x)*ArcCsc[c*x] - (2*b 
*d*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticF[ArcSi 
n[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/Sqrt[1 - c^2*x^2] + (2*b*e*Sqr 
t[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d - e)]*Sqrt[(e - c*e*x)/(c*d + 
 e)]*((c*d + e)*EllipticE[ArcSin[Sqrt[(c*(d + e*x))/(c*d - e)]], (c*d - e) 
/(c*d + e)] - e*EllipticF[ArcSin[Sqrt[(c*(d + e*x))/(c*d - e)]], (c*d - e) 
/(c*d + e)]))/(c*(-1 + c*x)*Sqrt[(e*(1 + c*x))/(-(c*d) + e)]) - (4*b*c*d^2 
*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticPi[2, ArcSi 
n[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/Sqrt[1 - c^2*x^2]))/(3*e^2*Sqr 
t[d + e*x])
 

Rubi [A] (verified)

Time = 1.69 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.80, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {5770, 27, 7272, 2351, 25, 27, 508, 327, 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.

\(\displaystyle \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx\)

\(\Big \downarrow \) 5770

\(\displaystyle \frac {b \int -\frac {2 (2 d-e x) \sqrt {d+e x}}{3 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}dx}{c}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 b \int \frac {(2 d-e x) \sqrt {d+e x}}{\sqrt {1-\frac {1}{c^2 x^2}} x^2}dx}{3 c e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}\)

\(\Big \downarrow \) 7272

\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \int \frac {(2 d-e x) \sqrt {d+e x}}{x \sqrt {1-c^2 x^2}}dx}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}\)

\(\Big \downarrow \) 2351

\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\int -\frac {e \sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx+2 d \int \frac {\sqrt {d+e x}}{x \sqrt {1-c^2 x^2}}dx\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \int \frac {\sqrt {d+e x}}{x \sqrt {1-c^2 x^2}}dx-\int \frac {e \sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \int \frac {\sqrt {d+e x}}{x \sqrt {1-c^2 x^2}}dx-e \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}\)

\(\Big \downarrow \) 508

\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \int \frac {\sqrt {d+e x}}{x \sqrt {1-c^2 x^2}}dx+\frac {2 e \sqrt {d+e x} \int \frac {\sqrt {1-\frac {e (1-c x)}{c d+e}}}{\sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}\)

\(\Big \downarrow \) 327

\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \int \frac {\sqrt {d+e x}}{x \sqrt {1-c^2 x^2}}dx+\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}\)

\(\Big \downarrow \) 637

\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \int \left (\frac {d}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}+\frac {e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}\right )dx+\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}+2 d \left (-\frac {2 e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 d \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {d+e x}}\right )\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}\)

Input:

Int[(x*(a + b*ArcCsc[c*x]))/Sqrt[d + e*x],x]
 

Output:

(-2*d*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/e^2 + (2*(d + e*x)^(3/2)*(a + b*A 
rcCsc[c*x]))/(3*e^2) - (2*b*Sqrt[1 - c^2*x^2]*((2*e*Sqrt[d + e*x]*Elliptic 
E[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*Sqrt[(c*(d + e*x))/( 
c*d + e)]) + 2*d*((-2*e*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticF[ArcSin[Sqr 
t[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*Sqrt[d + e*x]) - (2*d*Sqrt[(c*(d 
 + e*x))/(c*d + e)]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c* 
d + e)])/Sqrt[d + e*x])))/(3*c*e^2*Sqrt[1 - 1/(c^2*x^2)]*x)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 327

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]
 

rule 508

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q 
 = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c 
*q))]))   Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr 
t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
 

rule 637

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2351

Int[((Px_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.))/(x_), x_S 
ymbol] :> Int[PolynomialQuotient[Px, x, x]*(c + d*x)^n*(a + b*x^2)^p, x] + 
Simp[PolynomialRemainder[Px, x, x]   Int[(c + d*x)^n*((a + b*x^2)^p/x), x], 
 x] /; FreeQ[{a, b, c, d, n, p}, x] && PolynomialQ[Px, x]
 

rule 5770

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*(u_), x_Symbol] :> With[{v = IntHide 
[u, x]}, Simp[(a + b*ArcCsc[c*x])   v, x] + Simp[b/c   Int[SimplifyIntegran 
d[v/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x], x] /; InverseFunctionFreeQ[v, x]] 
/; FreeQ[{a, b, c}, x]
 

rule 7272

Int[(u_.)*((a_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[b^IntPart[p]*(( 
a + b*x^n)^FracPart[p]/(x^(n*FracPart[p])*(1 + a*(1/(x^n*b)))^FracPart[p])) 
   Int[u*x^(n*p)*(1 + a*(1/(x^n*b)))^p, x], x] /; FreeQ[{a, b, p}, x] &&  ! 
IntegerQ[p] && ILtQ[n, 0] &&  !RationalFunctionQ[u, x] && IntegerQ[p + 1/2]
 

Maple [A] (verified)

Time = 9.85 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.19

method	result	size
derivativedivides	\(\frac {-2 a \left (-\frac {\left (e x +d \right )^{\frac {3}{2}}}{3}+d \sqrt {e x +d}\right )-2 b \left (-\frac {\left (e x +d \right )^{\frac {3}{2}} \operatorname {arccsc}\left (c x \right )}{3}+\operatorname {arccsc}\left (c x \right ) d \sqrt {e x +d}+\frac {2 \left (d \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c +\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d -2 d \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c -\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e +\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e \right ) \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}}{3 c^{2} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e^{2}}\)	\(410\)
default	\(\frac {-2 a \left (-\frac {\left (e x +d \right )^{\frac {3}{2}}}{3}+d \sqrt {e x +d}\right )-2 b \left (-\frac {\left (e x +d \right )^{\frac {3}{2}} \operatorname {arccsc}\left (c x \right )}{3}+\operatorname {arccsc}\left (c x \right ) d \sqrt {e x +d}+\frac {2 \left (d \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c +\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d -2 d \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c -\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e +\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e \right ) \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}}{3 c^{2} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e^{2}}\)	\(410\)
parts	\(\frac {2 a \left (\frac {\left (e x +d \right )^{\frac {3}{2}}}{3}-d \sqrt {e x +d}\right )}{e^{2}}+\frac {2 b \left (\frac {\left (e x +d \right )^{\frac {3}{2}} \operatorname {arccsc}\left (c x \right )}{3}-\operatorname {arccsc}\left (c x \right ) d \sqrt {e x +d}-\frac {2 \left (d \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c +\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d -2 d \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c -\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e +\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e \right ) \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}}{3 c^{2} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e^{2}}\)	\(415\)

Input:

int(x*(a+b*arccsc(c*x))/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

2/e^2*(-a*(-1/3*(e*x+d)^(3/2)+d*(e*x+d)^(1/2))-b*(-1/3*(e*x+d)^(3/2)*arccs 
c(c*x)+arccsc(c*x)*d*(e*x+d)^(1/2)+2/3/c^2*(d*EllipticF((e*x+d)^(1/2)*(c/( 
c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c+EllipticE((e*x+d)^(1/2)*(c/(c*d-e 
))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c*d-2*d*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-EllipticF( 
(e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*e+EllipticE((e*x+ 
d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*e)*((-c*(e*x+d)+c*d+e) 
/(c*d+e))^(1/2)*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)/(c/(c*d-e))^(1/2)/x/((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)))
 

Fricas [F]

\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x}{\sqrt {e x + d}} \,d x } \] Input:

integrate(x*(a+b*arccsc(c*x))/(e*x+d)^(1/2),x, algorithm="fricas")
 

Output:

integral((b*x*arccsc(c*x) + a*x)/sqrt(e*x + d), x)
 

Sympy [F]

\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int \frac {x \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\sqrt {d + e x}}\, dx \] Input:

integrate(x*(a+b*acsc(c*x))/(e*x+d)**(1/2),x)
 

Output:

Integral(x*(a + b*acsc(c*x))/sqrt(d + e*x), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x*(a+b*arccsc(c*x))/(e*x+d)^(1/2),x, algorithm="maxima")
 

Output:

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)
 

Giac [F]

\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x}{\sqrt {e x + d}} \,d x } \] Input:

integrate(x*(a+b*arccsc(c*x))/(e*x+d)^(1/2),x, algorithm="giac")
 

Output:

integrate((b*arccsc(c*x) + a)*x/sqrt(e*x + d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int \frac {x\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {d+e\,x}} \,d x \] Input:

int((x*(a + b*asin(1/(c*x))))/(d + e*x)^(1/2),x)
 

Output:

int((x*(a + b*asin(1/(c*x))))/(d + e*x)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\frac {-4 \sqrt {e x +d}\, a d +2 \sqrt {e x +d}\, a e x +3 \left (\int \frac {\mathit {acsc} \left (c x \right ) x}{\sqrt {e x +d}}d x \right ) b \,e^{2}}{3 e^{2}} \] Input:

int(x*(a+b*acsc(c*x))/(e*x+d)^(1/2),x)
 

Output:

( - 4*sqrt(d + e*x)*a*d + 2*sqrt(d + e*x)*a*e*x + 3*int((acsc(c*x)*x)/sqrt 
(d + e*x),x)*b*e**2)/(3*e**2)