\(\int \frac {x (a+b \csc ^{-1}(c x))}{(d+e x^2)^{5/2}} \, dx\) [158]

Optimal result

Integrand size = 21, antiderivative size = 138 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {b c x \sqrt {-1+c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {a+b \csc ^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}+\frac {b c x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{3 d^{3/2} e \sqrt {c^2 x^2}} \] Output:

1/3*b*c*x*(c^2*x^2-1)^(1/2)/d/(c^2*d+e)/(c^2*x^2)^(1/2)/(e*x^2+d)^(1/2)-1/ 
3*(a+b*arccsc(c*x))/e/(e*x^2+d)^(3/2)+1/3*b*c*x*arctan((e*x^2+d)^(1/2)/d^( 
1/2)/(c^2*x^2-1)^(1/2))/d^(3/2)/e/(c^2*x^2)^(1/2)
 

Mathematica [C] (warning: unable to verify)

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

Time = 0.34 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.94 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {-\frac {2 a}{e}+\frac {2 b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+e x^2\right )}{d \left (c^2 d+e\right )}+\frac {b \sqrt {1+\frac {d}{e x^2}} \left (d+e x^2\right ) \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,\frac {1}{c^2 x^2},-\frac {d}{e x^2}\right )}{c d e x}-\frac {2 b \csc ^{-1}(c x)}{e}}{6 \left (d+e x^2\right )^{3/2}} \] Input:

Integrate[(x*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(5/2),x]
 

Output:

((-2*a)/e + (2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x^2))/(d*(c^2*d + e)) + 
(b*Sqrt[1 + d/(e*x^2)]*(d + e*x^2)*AppellF1[1, 1/2, 1/2, 2, 1/(c^2*x^2), - 
(d/(e*x^2))])/(c*d*e*x) - (2*b*ArcCsc[c*x])/e)/(6*(d + e*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.91, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5760, 354, 107, 104, 217}

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.

\(\displaystyle \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 5760

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

\(\Big \downarrow \) 354

\(\displaystyle -\frac {b c x \int \frac {1}{x^2 \sqrt {c^2 x^2-1} \left (e x^2+d\right )^{3/2}}dx^2}{6 e \sqrt {c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 107

\(\displaystyle -\frac {b c x \left (\frac {\int \frac {1}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2}{d}-\frac {2 e \sqrt {c^2 x^2-1}}{d \left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{6 e \sqrt {c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {b c x \left (\frac {2 \int \frac {1}{-x^4-d}d\frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}}{d}-\frac {2 e \sqrt {c^2 x^2-1}}{d \left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{6 e \sqrt {c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {a+b \csc ^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}-\frac {b c x \left (-\frac {2 \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{d^{3/2}}-\frac {2 e \sqrt {c^2 x^2-1}}{d \left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{6 e \sqrt {c^2 x^2}}\)

Input:

Int[(x*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(5/2),x]
 

Output:

-1/3*(a + b*ArcCsc[c*x])/(e*(d + e*x^2)^(3/2)) - (b*c*x*((-2*e*Sqrt[-1 + c 
^2*x^2])/(d*(c^2*d + e)*Sqrt[d + e*x^2]) - (2*ArcTan[Sqrt[d + e*x^2]/(Sqrt 
[d]*Sqrt[-1 + c^2*x^2])])/d^(3/2)))/(6*e*Sqrt[c^2*x^2])
 

Defintions of rubi rules used

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 107

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 
 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 
 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x 
] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 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 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]

\[\int \frac {x \left (a +b \,\operatorname {arccsc}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]

Input:

int(x*(a+b*arccsc(c*x))/(e*x^2+d)^(5/2),x)
 

Output:

int(x*(a+b*arccsc(c*x))/(e*x^2+d)^(5/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 276 vs. \(2 (114) = 228\).

Time = 0.20 (sec) , antiderivative size = 573, normalized size of antiderivative = 4.15 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\left [-\frac {{\left (b c^{2} d^{3} + {\left (b c^{2} d e^{2} + b e^{3}\right )} x^{4} + b d^{2} e + 2 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x^{2}\right )} \sqrt {-d} \log \left (\frac {{\left (c^{4} d^{2} - 6 \, c^{2} d e + e^{2}\right )} x^{4} - 8 \, {\left (c^{2} d^{2} - d e\right )} x^{2} + 4 \, \sqrt {c^{2} x^{2} - 1} {\left ({\left (c^{2} d - e\right )} x^{2} - 2 \, d\right )} \sqrt {e x^{2} + d} \sqrt {-d} + 8 \, d^{2}}{x^{4}}\right ) + 4 \, {\left (a c^{2} d^{3} + a d^{2} e + {\left (b c^{2} d^{3} + b d^{2} e\right )} \operatorname {arccsc}\left (c x\right ) - {\left (b d e^{2} x^{2} + b d^{2} e\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d}}{12 \, {\left (c^{2} d^{5} e + d^{4} e^{2} + {\left (c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{4} + 2 \, {\left (c^{2} d^{4} e^{2} + d^{3} e^{3}\right )} x^{2}\right )}}, \frac {{\left (b c^{2} d^{3} + {\left (b c^{2} d e^{2} + b e^{3}\right )} x^{4} + b d^{2} e + 2 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x^{2}\right )} \sqrt {d} \arctan \left (-\frac {\sqrt {c^{2} x^{2} - 1} {\left ({\left (c^{2} d - e\right )} x^{2} - 2 \, d\right )} \sqrt {e x^{2} + d} \sqrt {d}}{2 \, {\left (c^{2} d e x^{4} + {\left (c^{2} d^{2} - d e\right )} x^{2} - d^{2}\right )}}\right ) - 2 \, {\left (a c^{2} d^{3} + a d^{2} e + {\left (b c^{2} d^{3} + b d^{2} e\right )} \operatorname {arccsc}\left (c x\right ) - {\left (b d e^{2} x^{2} + b d^{2} e\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d}}{6 \, {\left (c^{2} d^{5} e + d^{4} e^{2} + {\left (c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{4} + 2 \, {\left (c^{2} d^{4} e^{2} + d^{3} e^{3}\right )} x^{2}\right )}}\right ] \] Input:

integrate(x*(a+b*arccsc(c*x))/(e*x^2+d)^(5/2),x, algorithm="fricas")
 

Output:

[-1/12*((b*c^2*d^3 + (b*c^2*d*e^2 + b*e^3)*x^4 + b*d^2*e + 2*(b*c^2*d^2*e 
+ b*d*e^2)*x^2)*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) + 4*(a*c^2*d^3 + a*d^2*e + (b*c^2*d^3 + b*d^2*e)*ar 
ccsc(c*x) - (b*d*e^2*x^2 + b*d^2*e)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d))/(c 
^2*d^5*e + d^4*e^2 + (c^2*d^3*e^3 + d^2*e^4)*x^4 + 2*(c^2*d^4*e^2 + d^3*e^ 
3)*x^2), 1/6*((b*c^2*d^3 + (b*c^2*d*e^2 + b*e^3)*x^4 + b*d^2*e + 2*(b*c^2* 
d^2*e + b*d*e^2)*x^2)*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 
)) - 2*(a*c^2*d^3 + a*d^2*e + (b*c^2*d^3 + b*d^2*e)*arccsc(c*x) - (b*d*e^2 
*x^2 + b*d^2*e)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d))/(c^2*d^5*e + d^4*e^2 + 
 (c^2*d^3*e^3 + d^2*e^4)*x^4 + 2*(c^2*d^4*e^2 + d^3*e^3)*x^2)]
 

Sympy [F]

\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {x \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(x*(a+b*acsc(c*x))/(e*x**2+d)**(5/2),x)
 

Output:

Integral(x*(a + b*acsc(c*x))/(d + e*x**2)**(5/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x*(a+b*arccsc(c*x))/(e*x^2+d)^(5/2),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]

\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x*(a+b*arccsc(c*x))/(e*x^2+d)^(5/2),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

integrate((b*arccsc(c*x) + a)*x/(e*x^2 + d)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {x\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \] Input:

int((x*(a + b*asin(1/(c*x))))/(d + e*x^2)^(5/2),x)
 

Output:

int((x*(a + b*asin(1/(c*x))))/(d + e*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {-\sqrt {e \,x^{2}+d}\, a +3 \left (\int \frac {\mathit {acsc} \left (c x \right ) x}{\sqrt {e \,x^{2}+d}\, d^{2}+2 \sqrt {e \,x^{2}+d}\, d e \,x^{2}+\sqrt {e \,x^{2}+d}\, e^{2} x^{4}}d x \right ) b \,d^{2} e +6 \left (\int \frac {\mathit {acsc} \left (c x \right ) x}{\sqrt {e \,x^{2}+d}\, d^{2}+2 \sqrt {e \,x^{2}+d}\, d e \,x^{2}+\sqrt {e \,x^{2}+d}\, e^{2} x^{4}}d x \right ) b d \,e^{2} x^{2}+3 \left (\int \frac {\mathit {acsc} \left (c x \right ) x}{\sqrt {e \,x^{2}+d}\, d^{2}+2 \sqrt {e \,x^{2}+d}\, d e \,x^{2}+\sqrt {e \,x^{2}+d}\, e^{2} x^{4}}d x \right ) b \,e^{3} x^{4}}{3 e \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

int(x*(a+b*acsc(c*x))/(e*x^2+d)^(5/2),x)
 

Output:

( - sqrt(d + e*x**2)*a + 3*int((acsc(c*x)*x)/(sqrt(d + e*x**2)*d**2 + 2*sq 
rt(d + e*x**2)*d*e*x**2 + sqrt(d + e*x**2)*e**2*x**4),x)*b*d**2*e + 6*int( 
(acsc(c*x)*x)/(sqrt(d + e*x**2)*d**2 + 2*sqrt(d + e*x**2)*d*e*x**2 + sqrt( 
d + e*x**2)*e**2*x**4),x)*b*d*e**2*x**2 + 3*int((acsc(c*x)*x)/(sqrt(d + e* 
x**2)*d**2 + 2*sqrt(d + e*x**2)*d*e*x**2 + sqrt(d + e*x**2)*e**2*x**4),x)* 
b*e**3*x**4)/(3*e*(d**2 + 2*d*e*x**2 + e**2*x**4))