\(\int \frac {x (a+b \csc ^{-1}(c x))}{\sqrt {d+e x^2}} \, dx\) [140]

Optimal result

Integrand size = 21, antiderivative size = 132 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}-\frac {b c \sqrt {d} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{e \sqrt {c^2 x^2}}+\frac {b x \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{\sqrt {e} \sqrt {c^2 x^2}} \] Output:

(e*x^2+d)^(1/2)*(a+b*arccsc(c*x))/e-b*c*d^(1/2)*x*arctan((e*x^2+d)^(1/2)/d 
^(1/2)/(c^2*x^2-1)^(1/2))/e/(c^2*x^2)^(1/2)+b*x*arctanh(e^(1/2)*(c^2*x^2-1 
)^(1/2)/c/(e*x^2+d)^(1/2))/e^(1/2)/(c^2*x^2)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.35 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.81 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\frac {\sqrt {d+e x^2} \left (a+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} x \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},1,\frac {3}{2},\frac {e-c^2 e x^2}{c^2 d+e},1-c^2 x^2\right )}{\sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d+e}}}+b \csc ^{-1}(c x)\right )}{e} \] Input:

Integrate[(x*(a + b*ArcCsc[c*x]))/Sqrt[d + e*x^2],x]
 

Output:

(Sqrt[d + e*x^2]*(a + (b*c*Sqrt[1 - 1/(c^2*x^2)]*x*AppellF1[1/2, -1/2, 1, 
3/2, (e - c^2*e*x^2)/(c^2*d + e), 1 - c^2*x^2])/Sqrt[(c^2*(d + e*x^2))/(c^ 
2*d + e)] + b*ArcCsc[c*x]))/e
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {5760, 354, 140, 27, 66, 104, 217, 221}

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.

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

\(\Big \downarrow \) 5760

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 140

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 66

\(\displaystyle \frac {b c x \left (d \int \frac {1}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2+2 e \int \frac {1}{c^2-e x^4}d\frac {\sqrt {c^2 x^2-1}}{\sqrt {e x^2+d}}\right )}{2 e \sqrt {c^2 x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {b c x \left (2 d \int \frac {1}{-x^4-d}d\frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}+2 e \int \frac {1}{c^2-e x^4}d\frac {\sqrt {c^2 x^2-1}}{\sqrt {e x^2+d}}\right )}{2 e \sqrt {c^2 x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {b c x \left (2 e \int \frac {1}{c^2-e x^4}d\frac {\sqrt {c^2 x^2-1}}{\sqrt {e x^2+d}}-2 \sqrt {d} \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )\right )}{2 e \sqrt {c^2 x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {b c x \left (\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{c}-2 \sqrt {d} \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )\right )}{2 e \sqrt {c^2 x^2}}\)

Input:

Int[(x*(a + b*ArcCsc[c*x]))/Sqrt[d + e*x^2],x]
 

Output:

(Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/e + (b*c*x*(-2*Sqrt[d]*ArcTan[Sqrt[d 
 + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])] + (2*Sqrt[e]*ArcTanh[(Sqrt[e]*Sqrt 
[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/c))/(2*e*Sqrt[c^2*x^2])
 

Defintions of rubi rules used

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 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 140

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*d^(m + n)*f^p   Int[(a + b*x)^(m - 1)/(c + d*x)^m, x] 
, x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandToSum[(a + b*x 
)*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, 
b, c, d, e, f, m, n}, x] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 
0] || SumSimplerQ[m, -1] ||  !(GtQ[n, 0] || SumSimplerQ[n, -1]))
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 5760

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*(x_)*((d_.) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCsc[c*x])/(2*e*(p + 1))), x 
] + Simp[b*c*(x/(2*e*(p + 1)*Sqrt[c^2*x^2]))   Int[(d + e*x^2)^(p + 1)/(x*S 
qrt[c^2*x^2 - 1]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
 

Maple [F]

\[\int \frac {x \left (a +b \,\operatorname {arccsc}\left (c x \right )\right )}{\sqrt {e \,x^{2}+d}}d x\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 870, normalized size of antiderivative = 6.59 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/4*(b*c*sqrt(-d)*log(((c^4*d^2 - 6*c^2*d*e + e^2)*x^4 - 8*(c^2*d^2 - d*e 
)*x^2 + 4*sqrt(c^2*x^2 - 1)*((c^2*d - e)*x^2 - 2*d)*sqrt(e*x^2 + d)*sqrt(- 
d) + 8*d^2)/x^4) + b*sqrt(e)*log(8*c^4*e^2*x^4 + c^4*d^2 - 6*c^2*d*e + 8*( 
c^4*d*e - c^2*e^2)*x^2 + 4*(2*c^3*e*x^2 + c^3*d - c*e)*sqrt(c^2*x^2 - 1)*s 
qrt(e*x^2 + d)*sqrt(e) + e^2) + 4*sqrt(e*x^2 + d)*(b*c*arccsc(c*x) + a*c)) 
/(c*e), -1/4*(2*b*c*sqrt(d)*arctan(-1/2*sqrt(c^2*x^2 - 1)*((c^2*d - e)*x^2 
 - 2*d)*sqrt(e*x^2 + d)*sqrt(d)/(c^2*d*e*x^4 + (c^2*d^2 - d*e)*x^2 - d^2)) 
 - b*sqrt(e)*log(8*c^4*e^2*x^4 + c^4*d^2 - 6*c^2*d*e + 8*(c^4*d*e - c^2*e^ 
2)*x^2 + 4*(2*c^3*e*x^2 + c^3*d - c*e)*sqrt(c^2*x^2 - 1)*sqrt(e*x^2 + d)*s 
qrt(e) + e^2) - 4*sqrt(e*x^2 + d)*(b*c*arccsc(c*x) + a*c))/(c*e), 1/4*(b*c 
*sqrt(-d)*log(((c^4*d^2 - 6*c^2*d*e + e^2)*x^4 - 8*(c^2*d^2 - d*e)*x^2 + 4 
*sqrt(c^2*x^2 - 1)*((c^2*d - e)*x^2 - 2*d)*sqrt(e*x^2 + d)*sqrt(-d) + 8*d^ 
2)/x^4) - 2*b*sqrt(-e)*arctan(1/2*(2*c^2*e*x^2 + c^2*d - e)*sqrt(c^2*x^2 - 
 1)*sqrt(e*x^2 + d)*sqrt(-e)/(c^3*e^2*x^4 - c*d*e + (c^3*d*e - c*e^2)*x^2) 
) + 4*sqrt(e*x^2 + d)*(b*c*arccsc(c*x) + a*c))/(c*e), -1/2*(b*c*sqrt(d)*ar 
ctan(-1/2*sqrt(c^2*x^2 - 1)*((c^2*d - e)*x^2 - 2*d)*sqrt(e*x^2 + d)*sqrt(d 
)/(c^2*d*e*x^4 + (c^2*d^2 - d*e)*x^2 - d^2)) + b*sqrt(-e)*arctan(1/2*(2*c^ 
2*e*x^2 + c^2*d - e)*sqrt(c^2*x^2 - 1)*sqrt(e*x^2 + d)*sqrt(-e)/(c^3*e^2*x 
^4 - c*d*e + (c^3*d*e - c*e^2)*x^2)) - 2*sqrt(e*x^2 + d)*(b*c*arccsc(c*x) 
+ a*c))/(c*e)]
 

Sympy [F]

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

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

Output:

Integral(x*(a + b*acsc(c*x))/sqrt(d + e*x**2), x)
 

Maxima [F]

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

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

Output:

(e*integrate((c^2*e*x^3 + c^2*d*x)*e^(-1/2*log(e*x^2 + d) + 1/2*log(c*x + 
1) + 1/2*log(c*x - 1))/(c^2*e*x^2 + (c^2*e*x^2 - e)*e^(log(c*x + 1) + log( 
c*x - 1)) - e), x) + sqrt(e*x^2 + d)*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1 
)))*b/e + sqrt(e*x^2 + d)*a/e
 

Giac [F]

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

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

Output:

integrate((b*arccsc(c*x) + a)*x/sqrt(e*x^2 + d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int \frac {x\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {e\,x^2+d}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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