\(\int \frac {a+b \csc ^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx\) [145]

Optimal result

Integrand size = 23, antiderivative size = 244 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx=-\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{d x}-\frac {b c^2 x \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{\sqrt {c^2 x^2} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}+\frac {b \left (c^2 d+e\right ) x \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{d \sqrt {c^2 x^2} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.57 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx=-\frac {\sqrt {d+e x^2} \left (a+b c \sqrt {1-\frac {1}{c^2 x^2}} x+b \csc ^{-1}(c x)\right )}{d x}+\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} E\left (\arcsin \left (\sqrt {-\frac {e}{d}} x\right )|-\frac {c^2 d}{e}\right )}{d \sqrt {-\frac {e}{d}} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \] Input:

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

Output:

-((Sqrt[d + e*x^2]*(a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x + b*ArcCsc[c*x]))/(d*x 
)) + (b*c*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*EllipticE[ArcSin[S 
qrt[-(e/d)]*x], -((c^2*d)/e)])/(d*Sqrt[-(e/d)]*Sqrt[1 - c^2*x^2]*Sqrt[d + 
e*x^2])
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.95, 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, 377, 27, 326, 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.

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

\(\Big \downarrow \) 5762

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 377

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 326

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 331

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

\(\Big \downarrow \) 330

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

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

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 321

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])
 

rule 323

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]
 

rule 326

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
b/d   Int[Sqrt[c + d*x^2]/Sqrt[a + b*x^2], x], x] - Simp[(b*c - a*d)/d   In 
t[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && 
PosQ[d/c] && NegQ[b/a]
 

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 330

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]
 

rule 331

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]
 

rule 377

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

rule 5762

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]))
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.43 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx=-\frac {{\left (b c d \operatorname {arccsc}\left (c x\right ) + \sqrt {c^{2} x^{2} - 1} b c d + a c d\right )} \sqrt {e x^{2} + d} + {\left (b c^{4} d x E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left (b c^{4} d + b e\right )} x F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {-d}}{c d^{2} x} \] Input:

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

Output:

-((b*c*d*arccsc(c*x) + sqrt(c^2*x^2 - 1)*b*c*d + a*c*d)*sqrt(e*x^2 + d) + 
(b*c^4*d*x*elliptic_e(arcsin(c*x), -e/(c^2*d)) - (b*c^4*d + b*e)*x*ellipti 
c_f(arcsin(c*x), -e/(c^2*d)))*sqrt(-d))/(c*d^2*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+b*arccsc(c*x))/x^2/(e*x^2+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>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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