\(\int \frac {x^5 (a+b \csc ^{-1}(c x))}{\sqrt {d+e x^2}} \, dx\) [138]

Optimal result

Integrand size = 23, antiderivative size = 321 \[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=-\frac {b \left (19 c^2 d-9 e\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{120 c^3 e^2 \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e^2 \sqrt {c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {8 b c d^{5/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{15 e^3 \sqrt {c^2 x^2}}+\frac {b \left (45 c^4 d^2-10 c^2 d e+9 e^2\right ) x \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{120 c^4 e^{5/2} \sqrt {c^2 x^2}} \] Output:

-1/120*b*(19*c^2*d-9*e)*x*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/c^3/e^2/(c^2*x 
^2)^(1/2)+1/20*b*x*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(3/2)/c/e^2/(c^2*x^2)^(1/2) 
+d^2*(e*x^2+d)^(1/2)*(a+b*arccsc(c*x))/e^3-2/3*d*(e*x^2+d)^(3/2)*(a+b*arcc 
sc(c*x))/e^3+1/5*(e*x^2+d)^(5/2)*(a+b*arccsc(c*x))/e^3-8/15*b*c*d^(5/2)*x* 
arctan((e*x^2+d)^(1/2)/d^(1/2)/(c^2*x^2-1)^(1/2))/e^3/(c^2*x^2)^(1/2)+1/12 
0*b*(45*c^4*d^2-10*c^2*d*e+9*e^2)*x*arctanh(e^(1/2)*(c^2*x^2-1)^(1/2)/c/(e 
*x^2+d)^(1/2))/c^4/e^(5/2)/(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 = 1.93 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.88 \[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\frac {16 a \left (d+e x^2\right ) \left (8 d^2-4 d e x^2+3 e^2 x^4\right )+\frac {2 b e \sqrt {1-\frac {1}{c^2 x^2}} \left (d+e x^2\right ) \left (9 e x+c^2 \left (-13 d x+6 e x^3\right )\right )}{c^3}+\frac {b \left (-64 c^2 d^3 \sqrt {1+\frac {d}{e x^2}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,\frac {1}{c^2 x^2},-\frac {d}{e x^2}\right )+\frac {e \left (-45 c^4 d^2+10 c^2 d e-9 e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x^4 \sqrt {1+\frac {e x^2}{d}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,c^2 x^2,-\frac {e x^2}{d}\right )}{\sqrt {1-c^2 x^2}}\right )}{c^3 x}+16 b \left (d+e x^2\right ) \left (8 d^2-4 d e x^2+3 e^2 x^4\right ) \csc ^{-1}(c x)}{240 e^3 \sqrt {d+e x^2}} \] Input:

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

Output:

(16*a*(d + e*x^2)*(8*d^2 - 4*d*e*x^2 + 3*e^2*x^4) + (2*b*e*Sqrt[1 - 1/(c^2 
*x^2)]*(d + e*x^2)*(9*e*x + c^2*(-13*d*x + 6*e*x^3)))/c^3 + (b*(-64*c^2*d^ 
3*Sqrt[1 + d/(e*x^2)]*AppellF1[1, 1/2, 1/2, 2, 1/(c^2*x^2), -(d/(e*x^2))] 
+ (e*(-45*c^4*d^2 + 10*c^2*d*e - 9*e^2)*Sqrt[1 - 1/(c^2*x^2)]*x^4*Sqrt[1 + 
 (e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, c^2*x^2, -((e*x^2)/d)])/Sqrt[1 - c^2* 
x^2]))/(c^3*x) + 16*b*(d + e*x^2)*(8*d^2 - 4*d*e*x^2 + 3*e^2*x^4)*ArcCsc[c 
*x])/(240*e^3*Sqrt[d + e*x^2])
 

Rubi [A] (verified)

Time = 1.31 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.92, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {5762, 27, 7282, 2118, 27, 171, 27, 175, 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.

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

\(\Big \downarrow \) 5762

\(\displaystyle \frac {b c x \int \frac {\sqrt {e x^2+d} \left (3 e^2 x^4-4 d e x^2+8 d^2\right )}{15 e^3 x \sqrt {c^2 x^2-1}}dx}{\sqrt {c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b c x \int \frac {\sqrt {e x^2+d} \left (3 e^2 x^4-4 d e x^2+8 d^2\right )}{x \sqrt {c^2 x^2-1}}dx}{15 e^3 \sqrt {c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 7282

\(\displaystyle \frac {b c x \int \frac {\sqrt {e x^2+d} \left (3 e^2 x^4-4 d e x^2+8 d^2\right )}{x^2 \sqrt {c^2 x^2-1}}dx^2}{30 e^3 \sqrt {c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 2118

\(\displaystyle \frac {b c x \left (\frac {\int \frac {e \sqrt {e x^2+d} \left (32 c^2 d^2-\left (19 c^2 d-9 e\right ) e x^2\right )}{2 x^2 \sqrt {c^2 x^2-1}}dx^2}{2 c^2 e}+\frac {3 e \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3 \sqrt {c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b c x \left (\frac {\int \frac {\sqrt {e x^2+d} \left (32 c^2 d^2-\left (19 c^2 d-9 e\right ) e x^2\right )}{x^2 \sqrt {c^2 x^2-1}}dx^2}{4 c^2}+\frac {3 e \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3 \sqrt {c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {b c x \left (\frac {\frac {\int \frac {64 d^3 c^4+e \left (45 d^2 c^4-10 d e c^2+9 e^2\right ) x^2}{2 x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2}{c^2}-\frac {e \sqrt {c^2 x^2-1} \left (19 c^2 d-9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}+\frac {3 e \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3 \sqrt {c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b c x \left (\frac {\frac {\int \frac {64 d^3 c^4+e \left (45 d^2 c^4-10 d e c^2+9 e^2\right ) x^2}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2}{2 c^2}-\frac {e \sqrt {c^2 x^2-1} \left (19 c^2 d-9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}+\frac {3 e \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3 \sqrt {c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 175

\(\displaystyle \frac {b c x \left (\frac {\frac {64 c^4 d^3 \int \frac {1}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2+e \left (45 c^4 d^2-10 c^2 d e+9 e^2\right ) \int \frac {1}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2}{2 c^2}-\frac {e \sqrt {c^2 x^2-1} \left (19 c^2 d-9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}+\frac {3 e \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3 \sqrt {c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {b c x \left (\frac {\frac {64 c^4 d^3 \int \frac {1}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2+2 e \left (45 c^4 d^2-10 c^2 d e+9 e^2\right ) \int \frac {1}{c^2-e x^4}d\frac {\sqrt {c^2 x^2-1}}{\sqrt {e x^2+d}}}{2 c^2}-\frac {e \sqrt {c^2 x^2-1} \left (19 c^2 d-9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}+\frac {3 e \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3 \sqrt {c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {b c x \left (\frac {\frac {128 c^4 d^3 \int \frac {1}{-x^4-d}d\frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}+2 e \left (45 c^4 d^2-10 c^2 d e+9 e^2\right ) \int \frac {1}{c^2-e x^4}d\frac {\sqrt {c^2 x^2-1}}{\sqrt {e x^2+d}}}{2 c^2}-\frac {e \sqrt {c^2 x^2-1} \left (19 c^2 d-9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}+\frac {3 e \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3 \sqrt {c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {b c x \left (\frac {\frac {2 e \left (45 c^4 d^2-10 c^2 d e+9 e^2\right ) \int \frac {1}{c^2-e x^4}d\frac {\sqrt {c^2 x^2-1}}{\sqrt {e x^2+d}}-128 c^4 d^{5/2} \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{2 c^2}-\frac {e \sqrt {c^2 x^2-1} \left (19 c^2 d-9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}+\frac {3 e \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3 \sqrt {c^2 x^2}}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d^2 \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {b c x \left (\frac {\frac {\frac {2 \sqrt {e} \left (45 c^4 d^2-10 c^2 d e+9 e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{c}-128 c^4 d^{5/2} \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{2 c^2}-\frac {e \sqrt {c^2 x^2-1} \left (19 c^2 d-9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}+\frac {3 e \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3 \sqrt {c^2 x^2}}\)

Input:

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

Output:

(d^2*Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/e^3 - (2*d*(d + e*x^2)^(3/2)*(a 
+ b*ArcCsc[c*x]))/(3*e^3) + ((d + e*x^2)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e^3 
) + (b*c*x*((3*e*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(3/2))/(2*c^2) + (-(((19*c 
^2*d - 9*e)*e*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/c^2) + (-128*c^4*d^(5/2) 
*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])] + (2*Sqrt[e]*(45*c^4 
*d^2 - 10*c^2*d*e + 9*e^2)*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d 
+ e*x^2])])/c)/(2*c^2))/(4*c^2)))/(30*e^3*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 171

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && IntegersQ[2*m, 2*n, 2*p]
 

rule 175

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ 
)))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b   Int[(c + d*x)^n*(e + f*x)^p, x] 
, x] + Simp[(b*g - a*h)/b   Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] 
 /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
 

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 2118

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f 
_.)*(x_))^(p_.), x_Symbol] :> With[{q = Expon[Px, x], k = Coeff[Px, x, Expo 
n[Px, x]]}, Simp[k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 
1)/(d*f*b^(q - 1)*(m + n + p + q + 1))), x] + Simp[1/(d*f*b^q*(m + n + p + 
q + 1))   Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + 
n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a + b*x)^(q 
- 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + 
 c*f*(p + 1))) + b*(a*d*f*(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m 
 + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; FreeQ[{a, b, c, 
 d, e, f, m, n, p}, x] && PolyQ[Px, 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]))
 

rule 7282

Int[(u_)/(x_), x_Symbol] :> With[{lst = PowerVariableExpn[u, 0, x]}, Simp[1 
/lst[[2]]   Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, ( 
lst[[3]]*x)^lst[[2]]], x] /;  !FalseQ[lst] && NeQ[lst[[2]], 0]] /; NonsumQ[ 
u] &&  !RationalFunctionQ[u, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 1.09 (sec) , antiderivative size = 1383, normalized size of antiderivative = 4.31 \[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\text {Too large to display} \] Input:

integrate(x^5*(a+b*arccsc(c*x))/(e*x^2+d)^(1/2),x, algorithm="fricas")
 

Output:

[1/480*(64*b*c^5*sqrt(-d)*d^2*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) + (45*b*c^4*d^2 - 10*b*c^2*d*e + 9*b*e^2)*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)*sqrt(e) + 
e^2) + 4*(24*a*c^5*e^2*x^4 - 32*a*c^5*d*e*x^2 + 64*a*c^5*d^2 + 8*(3*b*c^5* 
e^2*x^4 - 4*b*c^5*d*e*x^2 + 8*b*c^5*d^2)*arccsc(c*x) + (6*b*c^3*e^2*x^2 - 
13*b*c^3*d*e + 9*b*c*e^2)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d))/(c^5*e^3), - 
1/480*(128*b*c^5*d^(5/2)*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)) - 
(45*b*c^4*d^2 - 10*b*c^2*d*e + 9*b*e^2)*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)*sqrt(e) + e^2) - 4*(24*a*c^5*e^2*x^4 - 3 
2*a*c^5*d*e*x^2 + 64*a*c^5*d^2 + 8*(3*b*c^5*e^2*x^4 - 4*b*c^5*d*e*x^2 + 8* 
b*c^5*d^2)*arccsc(c*x) + (6*b*c^3*e^2*x^2 - 13*b*c^3*d*e + 9*b*c*e^2)*sqrt 
(c^2*x^2 - 1))*sqrt(e*x^2 + d))/(c^5*e^3), 1/240*(32*b*c^5*sqrt(-d)*d^2*lo 
g(((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 
5*b*c^4*d^2 - 10*b*c^2*d*e + 9*b*e^2)*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...
 

Sympy [F]

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

integrate(x**5*(a+b*acsc(c*x))/(e*x**2+d)**(1/2),x)
 

Output:

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

Maxima [F(-2)]

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

integrate(x^5*(a+b*arccsc(c*x))/(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]

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

integrate(x^5*(a+b*arccsc(c*x))/(e*x^2+d)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\frac {8 \sqrt {e \,x^{2}+d}\, a \,d^{2}-4 \sqrt {e \,x^{2}+d}\, a d e \,x^{2}+3 \sqrt {e \,x^{2}+d}\, a \,e^{2} x^{4}+15 \left (\int \frac {\mathit {acsc} \left (c x \right ) x^{5}}{\sqrt {e \,x^{2}+d}}d x \right ) b \,e^{3}}{15 e^{3}} \] Input:

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

Output:

(8*sqrt(d + e*x**2)*a*d**2 - 4*sqrt(d + e*x**2)*a*d*e*x**2 + 3*sqrt(d + e* 
x**2)*a*e**2*x**4 + 15*int((acsc(c*x)*x**5)/sqrt(d + e*x**2),x)*b*e**3)/(1 
5*e**3)