\(\int \frac {x^3 (a+b \csc ^{-1}(c x))}{(d+e x^2)^3} \, dx\) [112]

Optimal result

Integrand size = 21, antiderivative size = 157 \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=-\frac {b c x \sqrt {-1+c^2 x^2}}{8 e \left (c^2 d+e\right ) \sqrt {c^2 x^2} \left (d+e x^2\right )}+\frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {b c \left (c^2 d+2 e\right ) x \arctan \left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{8 d e^{3/2} \left (c^2 d+e\right )^{3/2} \sqrt {c^2 x^2}} \]

[Out]

1/4*x^4*(a+b*arccsc(c*x))/d/(e*x^2+d)^2+1/8*b*c*(c^2*d+2*e)*x*arctan(e^(1/2)*(c^2*x^2-1)^(1/2)/(c^2*d+e)^(1/2)
)/d/e^(3/2)/(c^2*d+e)^(3/2)/(c^2*x^2)^(1/2)-1/8*b*c*x*(c^2*x^2-1)^(1/2)/e/(c^2*d+e)/(e*x^2+d)/(c^2*x^2)^(1/2)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {270, 5347, 12, 457, 79, 65, 211} \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {b c x \left (c^2 d+2 e\right ) \arctan \left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{\sqrt {c^2 d+e}}\right )}{8 d e^{3/2} \sqrt {c^2 x^2} \left (c^2 d+e\right )^{3/2}}-\frac {b c x \sqrt {c^2 x^2-1}}{8 e \sqrt {c^2 x^2} \left (c^2 d+e\right ) \left (d+e x^2\right )} \]

[In]

Int[(x^3*(a + b*ArcCsc[c*x]))/(d + e*x^2)^3,x]

[Out]

-1/8*(b*c*x*Sqrt[-1 + c^2*x^2])/(e*(c^2*d + e)*Sqrt[c^2*x^2]*(d + e*x^2)) + (x^4*(a + b*ArcCsc[c*x]))/(4*d*(d
+ e*x^2)^2) + (b*c*(c^2*d + 2*e)*x*ArcTan[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/Sqrt[c^2*d + e]])/(8*d*e^(3/2)*(c^2*d +
e)^(3/2)*Sqrt[c^2*x^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 211

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 457

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

Rule 5347

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]}, Dist[a + b*ArcCsc[c*x], u, x] + Dist[b*c*(x/Sqrt[c^2*x^2]), Int[SimplifyI
ntegrand[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])) || (I
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rubi steps \begin{align*} \text {integral}& = \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {(b c x) \int \frac {x^3}{4 d \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^2} \, dx}{\sqrt {c^2 x^2}} \\ & = \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {(b c x) \int \frac {x^3}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )^2} \, dx}{4 d \sqrt {c^2 x^2}} \\ & = \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {(b c x) \text {Subst}\left (\int \frac {x}{\sqrt {-1+c^2 x} (d+e x)^2} \, dx,x,x^2\right )}{8 d \sqrt {c^2 x^2}} \\ & = -\frac {b c x \sqrt {-1+c^2 x^2}}{8 e \left (c^2 d+e\right ) \sqrt {c^2 x^2} \left (d+e x^2\right )}+\frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {\left (b c \left (c^2 d+2 e\right ) x\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+c^2 x} (d+e x)} \, dx,x,x^2\right )}{16 d e \left (c^2 d+e\right ) \sqrt {c^2 x^2}} \\ & = -\frac {b c x \sqrt {-1+c^2 x^2}}{8 e \left (c^2 d+e\right ) \sqrt {c^2 x^2} \left (d+e x^2\right )}+\frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {\left (b \left (c^2 d+2 e\right ) x\right ) \text {Subst}\left (\int \frac {1}{d+\frac {e}{c^2}+\frac {e x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{8 c d e \left (c^2 d+e\right ) \sqrt {c^2 x^2}} \\ & = -\frac {b c x \sqrt {-1+c^2 x^2}}{8 e \left (c^2 d+e\right ) \sqrt {c^2 x^2} \left (d+e x^2\right )}+\frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {b c \left (c^2 d+2 e\right ) x \arctan \left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{8 d e^{3/2} \left (c^2 d+e\right )^{3/2} \sqrt {c^2 x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.76 (sec) , antiderivative size = 390, normalized size of antiderivative = 2.48 \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\frac {\frac {4 a d}{\left (d+e x^2\right )^2}-\frac {8 a}{d+e x^2}-\frac {2 b c e \sqrt {1-\frac {1}{c^2 x^2}} x}{\left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {4 b \left (d+2 e x^2\right ) \csc ^{-1}(c x)}{\left (d+e x^2\right )^2}+\frac {4 b \arcsin \left (\frac {1}{c x}\right )}{d}+\frac {b \sqrt {e} \left (c^2 d+2 e\right ) \log \left (\frac {16 d \sqrt {-c^2 d-e} e^{3/2} \left (i \sqrt {e}+c \left (c \sqrt {d}-i \sqrt {-c^2 d-e} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{b \left (c^2 d+2 e\right ) \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{d \left (-c^2 d-e\right )^{3/2}}+\frac {b \sqrt {e} \left (c^2 d+2 e\right ) \log \left (-\frac {16 d \sqrt {-c^2 d-e} e^{3/2} \left (-\sqrt {e}+c \left (-i c \sqrt {d}+\sqrt {-c^2 d-e} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{b \left (c^2 d+2 e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{d \left (-c^2 d-e\right )^{3/2}}}{16 e^2} \]

[In]

Integrate[(x^3*(a + b*ArcCsc[c*x]))/(d + e*x^2)^3,x]

[Out]

((4*a*d)/(d + e*x^2)^2 - (8*a)/(d + e*x^2) - (2*b*c*e*Sqrt[1 - 1/(c^2*x^2)]*x)/((c^2*d + e)*(d + e*x^2)) - (4*
b*(d + 2*e*x^2)*ArcCsc[c*x])/(d + e*x^2)^2 + (4*b*ArcSin[1/(c*x)])/d + (b*Sqrt[e]*(c^2*d + 2*e)*Log[(16*d*Sqrt
[-(c^2*d) - e]*e^(3/2)*(I*Sqrt[e] + c*(c*Sqrt[d] - I*Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(b*(c^2*d +
2*e)*(Sqrt[d] + I*Sqrt[e]*x))])/(d*(-(c^2*d) - e)^(3/2)) + (b*Sqrt[e]*(c^2*d + 2*e)*Log[(-16*d*Sqrt[-(c^2*d)
- e]*e^(3/2)*(-Sqrt[e] + c*((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(b*(c^2*d + 2*e)*(I
*Sqrt[d] + Sqrt[e]*x))])/(d*(-(c^2*d) - e)^(3/2)))/(16*e^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(944\) vs. \(2(135)=270\).

Time = 6.54 (sec) , antiderivative size = 945, normalized size of antiderivative = 6.02

method	result	size
parts	\(a \left (\frac {d}{4 e^{2} \left (e \,x^{2}+d \right )^{2}}-\frac {1}{2 e^{2} \left (e \,x^{2}+d \right )}\right )+\frac {b \left (-\frac {c^{6} \operatorname {arccsc}\left (c x \right )}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {c^{8} \operatorname {arccsc}\left (c x \right ) d}{4 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}-\frac {c^{3} \sqrt {c^{2} x^{2}-1}\, \left (4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{4} d e \,x^{2}+4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{4} d^{2}-\ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e -2 \sqrt {-c^{2} d e}\, c x -2 e}{c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}-\ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e -2 \sqrt {-c^{2} d e}\, c x -2 e}{c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}-\ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}-\ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}+4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, e^{2} c^{2} x^{2}+4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{2} d e -2 \sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, c^{2} d e -2 \ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e -2 \sqrt {-c^{2} d e}\, c x -2 e}{c e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}-2 \ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e -2 \sqrt {-c^{2} d e}\, c x -2 e}{c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e -2 \ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}-2 \ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e \right )}{16 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d \sqrt {-\frac {c^{2} d +e}{e}}\, \left (c^{2} d +e \right ) \left (-c e x +\sqrt {-c^{2} d e}\right ) \left (c e x +\sqrt {-c^{2} d e}\right )}\right )}{c^{4}}\)	\(945\)
derivativedivides	\(\frac {a \,c^{6} \left (-\frac {1}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {d \,c^{2}}{4 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}\right )+b \,c^{6} \left (-\frac {\operatorname {arccsc}\left (c x \right )}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {\operatorname {arccsc}\left (c x \right ) d \,c^{2}}{4 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (-4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{4} d^{2}-4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{4} d e \,x^{2}+\ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}+\ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}+\ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}+\ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}+2 \sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, c^{2} d e -4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{2} d e -4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, e^{2} c^{2} x^{2}+2 \ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e +2 \ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}+2 \ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e +2 \ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}\right )}{16 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x d \sqrt {-\frac {c^{2} d +e}{e}}\, \left (c^{2} d +e \right ) \left (-c e x +\sqrt {-c^{2} d e}\right ) \left (c e x +\sqrt {-c^{2} d e}\right )}\right )}{c^{4}}\)	\(954\)
default	\(\frac {a \,c^{6} \left (-\frac {1}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {d \,c^{2}}{4 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}\right )+b \,c^{6} \left (-\frac {\operatorname {arccsc}\left (c x \right )}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {\operatorname {arccsc}\left (c x \right ) d \,c^{2}}{4 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (-4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{4} d^{2}-4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{4} d e \,x^{2}+\ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}+\ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}+\ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}+\ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}+2 \sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, c^{2} d e -4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{2} d e -4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, e^{2} c^{2} x^{2}+2 \ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e +2 \ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}+2 \ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e +2 \ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{c e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}\right )}{16 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x d \sqrt {-\frac {c^{2} d +e}{e}}\, \left (c^{2} d +e \right ) \left (-c e x +\sqrt {-c^{2} d e}\right ) \left (c e x +\sqrt {-c^{2} d e}\right )}\right )}{c^{4}}\)	\(954\)

[In]

int(x^3*(a+b*arccsc(c*x))/(e*x^2+d)^3,x,method=_RETURNVERBOSE)

[Out]

a*(1/4*d/e^2/(e*x^2+d)^2-1/2/e^2/(e*x^2+d))+b/c^4*(-1/2*c^6*arccsc(c*x)/e^2/(c^2*e*x^2+c^2*d)+1/4*c^8*arccsc(c
*x)*d/e^2/(c^2*e*x^2+c^2*d)^2-1/16*c^3*(c^2*x^2-1)^(1/2)/e*(4*arctan(1/(c^2*x^2-1)^(1/2))*(-(c^2*d+e)/e)^(1/2)
*c^4*d*e*x^2+4*arctan(1/(c^2*x^2-1)^(1/2))*(-(c^2*d+e)/e)^(1/2)*c^4*d^2-ln(2*((c^2*x^2-1)^(1/2)*(-(c^2*d+e)/e)
^(1/2)*e-(-c^2*d*e)^(1/2)*c*x-e)/(c*e*x+(-c^2*d*e)^(1/2)))*c^4*d*e*x^2-ln(2*((c^2*x^2-1)^(1/2)*(-(c^2*d+e)/e)^
(1/2)*e-(-c^2*d*e)^(1/2)*c*x-e)/(c*e*x+(-c^2*d*e)^(1/2)))*c^4*d^2-ln(-2*((c^2*x^2-1)^(1/2)*(-(c^2*d+e)/e)^(1/2
)*e+(-c^2*d*e)^(1/2)*c*x-e)/(-c*e*x+(-c^2*d*e)^(1/2)))*c^4*d*e*x^2-ln(-2*((c^2*x^2-1)^(1/2)*(-(c^2*d+e)/e)^(1/
2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(-c*e*x+(-c^2*d*e)^(1/2)))*c^4*d^2+4*arctan(1/(c^2*x^2-1)^(1/2))*(-(c^2*d+e)/e)^(
1/2)*e^2*c^2*x^2+4*arctan(1/(c^2*x^2-1)^(1/2))*(-(c^2*d+e)/e)^(1/2)*c^2*d*e-2*(c^2*x^2-1)^(1/2)*(-(c^2*d+e)/e)
^(1/2)*c^2*d*e-2*ln(2*((c^2*x^2-1)^(1/2)*(-(c^2*d+e)/e)^(1/2)*e-(-c^2*d*e)^(1/2)*c*x-e)/(c*e*x+(-c^2*d*e)^(1/2
)))*e^2*c^2*x^2-2*ln(2*((c^2*x^2-1)^(1/2)*(-(c^2*d+e)/e)^(1/2)*e-(-c^2*d*e)^(1/2)*c*x-e)/(c*e*x+(-c^2*d*e)^(1/
2)))*c^2*d*e-2*ln(-2*((c^2*x^2-1)^(1/2)*(-(c^2*d+e)/e)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(-c*e*x+(-c^2*d*e)^(1/2
)))*e^2*c^2*x^2-2*ln(-2*((c^2*x^2-1)^(1/2)*(-(c^2*d+e)/e)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(-c*e*x+(-c^2*d*e)^(
1/2)))*c^2*d*e)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x/d/(-(c^2*d+e)/e)^(1/2)/(c^2*d+e)/(-c*e*x+(-c^2*d*e)^(1/2))/(c*e*
x+(-c^2*d*e)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (135) = 270\).

Time = 0.50 (sec) , antiderivative size = 1015, normalized size of antiderivative = 6.46 \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\left [-\frac {4 \, a c^{4} d^{4} + 8 \, a c^{2} d^{3} e + 4 \, a d^{2} e^{2} + 8 \, {\left (a c^{4} d^{3} e + 2 \, a c^{2} d^{2} e^{2} + a d e^{3}\right )} x^{2} + {\left (b c^{2} d^{3} + {\left (b c^{2} d e^{2} + 2 \, b e^{3}\right )} x^{4} + 2 \, b d^{2} e + 2 \, {\left (b c^{2} d^{2} e + 2 \, b d e^{2}\right )} x^{2}\right )} \sqrt {-c^{2} d e - e^{2}} \log \left (\frac {c^{2} e x^{2} - c^{2} d - 2 \, \sqrt {-c^{2} d e - e^{2}} \sqrt {c^{2} x^{2} - 1} - 2 \, e}{e x^{2} + d}\right ) + 4 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2} + 2 \, {\left (b c^{4} d^{3} e + 2 \, b c^{2} d^{2} e^{2} + b d e^{3}\right )} x^{2}\right )} \operatorname {arccsc}\left (c x\right ) + 8 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2} + {\left (b c^{4} d^{2} e^{2} + 2 \, b c^{2} d e^{3} + b e^{4}\right )} x^{4} + 2 \, {\left (b c^{4} d^{3} e + 2 \, b c^{2} d^{2} e^{2} + b d e^{3}\right )} x^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, {\left (b c^{2} d^{3} e + b d^{2} e^{2} + {\left (b c^{2} d^{2} e^{2} + b d e^{3}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{16 \, {\left (c^{4} d^{5} e^{2} + 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4} + {\left (c^{4} d^{3} e^{4} + 2 \, c^{2} d^{2} e^{5} + d e^{6}\right )} x^{4} + 2 \, {\left (c^{4} d^{4} e^{3} + 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{2}\right )}}, -\frac {2 \, a c^{4} d^{4} + 4 \, a c^{2} d^{3} e + 2 \, a d^{2} e^{2} + 4 \, {\left (a c^{4} d^{3} e + 2 \, a c^{2} d^{2} e^{2} + a d e^{3}\right )} x^{2} - {\left (b c^{2} d^{3} + {\left (b c^{2} d e^{2} + 2 \, b e^{3}\right )} x^{4} + 2 \, b d^{2} e + 2 \, {\left (b c^{2} d^{2} e + 2 \, b d e^{2}\right )} x^{2}\right )} \sqrt {c^{2} d e + e^{2}} \arctan \left (\frac {\sqrt {c^{2} d e + e^{2}} \sqrt {c^{2} x^{2} - 1}}{c^{2} d + e}\right ) + 2 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2} + 2 \, {\left (b c^{4} d^{3} e + 2 \, b c^{2} d^{2} e^{2} + b d e^{3}\right )} x^{2}\right )} \operatorname {arccsc}\left (c x\right ) + 4 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2} + {\left (b c^{4} d^{2} e^{2} + 2 \, b c^{2} d e^{3} + b e^{4}\right )} x^{4} + 2 \, {\left (b c^{4} d^{3} e + 2 \, b c^{2} d^{2} e^{2} + b d e^{3}\right )} x^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (b c^{2} d^{3} e + b d^{2} e^{2} + {\left (b c^{2} d^{2} e^{2} + b d e^{3}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{8 \, {\left (c^{4} d^{5} e^{2} + 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4} + {\left (c^{4} d^{3} e^{4} + 2 \, c^{2} d^{2} e^{5} + d e^{6}\right )} x^{4} + 2 \, {\left (c^{4} d^{4} e^{3} + 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{2}\right )}}\right ] \]

[In]

integrate(x^3*(a+b*arccsc(c*x))/(e*x^2+d)^3,x, algorithm="fricas")

[Out]

[-1/16*(4*a*c^4*d^4 + 8*a*c^2*d^3*e + 4*a*d^2*e^2 + 8*(a*c^4*d^3*e + 2*a*c^2*d^2*e^2 + a*d*e^3)*x^2 + (b*c^2*d
^3 + (b*c^2*d*e^2 + 2*b*e^3)*x^4 + 2*b*d^2*e + 2*(b*c^2*d^2*e + 2*b*d*e^2)*x^2)*sqrt(-c^2*d*e - e^2)*log((c^2*
e*x^2 - c^2*d - 2*sqrt(-c^2*d*e - e^2)*sqrt(c^2*x^2 - 1) - 2*e)/(e*x^2 + d)) + 4*(b*c^4*d^4 + 2*b*c^2*d^3*e +
b*d^2*e^2 + 2*(b*c^4*d^3*e + 2*b*c^2*d^2*e^2 + b*d*e^3)*x^2)*arccsc(c*x) + 8*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^
2*e^2 + (b*c^4*d^2*e^2 + 2*b*c^2*d*e^3 + b*e^4)*x^4 + 2*(b*c^4*d^3*e + 2*b*c^2*d^2*e^2 + b*d*e^3)*x^2)*arctan(
-c*x + sqrt(c^2*x^2 - 1)) + 2*(b*c^2*d^3*e + b*d^2*e^2 + (b*c^2*d^2*e^2 + b*d*e^3)*x^2)*sqrt(c^2*x^2 - 1))/(c^
4*d^5*e^2 + 2*c^2*d^4*e^3 + d^3*e^4 + (c^4*d^3*e^4 + 2*c^2*d^2*e^5 + d*e^6)*x^4 + 2*(c^4*d^4*e^3 + 2*c^2*d^3*e
^4 + d^2*e^5)*x^2), -1/8*(2*a*c^4*d^4 + 4*a*c^2*d^3*e + 2*a*d^2*e^2 + 4*(a*c^4*d^3*e + 2*a*c^2*d^2*e^2 + a*d*e
^3)*x^2 - (b*c^2*d^3 + (b*c^2*d*e^2 + 2*b*e^3)*x^4 + 2*b*d^2*e + 2*(b*c^2*d^2*e + 2*b*d*e^2)*x^2)*sqrt(c^2*d*e
+ e^2)*arctan(sqrt(c^2*d*e + e^2)*sqrt(c^2*x^2 - 1)/(c^2*d + e)) + 2*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2 +
2*(b*c^4*d^3*e + 2*b*c^2*d^2*e^2 + b*d*e^3)*x^2)*arccsc(c*x) + 4*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2 + (b*
c^4*d^2*e^2 + 2*b*c^2*d*e^3 + b*e^4)*x^4 + 2*(b*c^4*d^3*e + 2*b*c^2*d^2*e^2 + b*d*e^3)*x^2)*arctan(-c*x + sqrt
(c^2*x^2 - 1)) + (b*c^2*d^3*e + b*d^2*e^2 + (b*c^2*d^2*e^2 + b*d*e^3)*x^2)*sqrt(c^2*x^2 - 1))/(c^4*d^5*e^2 + 2
*c^2*d^4*e^3 + d^3*e^4 + (c^4*d^3*e^4 + 2*c^2*d^2*e^5 + d*e^6)*x^4 + 2*(c^4*d^4*e^3 + 2*c^2*d^3*e^4 + d^2*e^5)
*x^2)]

Sympy [F(-1)]

Timed out. \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(x**3*(a+b*acsc(c*x))/(e*x**2+d)**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]

[In]

integrate(x^3*(a+b*arccsc(c*x))/(e*x^2+d)^3,x, algorithm="maxima")

[Out]

-1/4*(2*e*x^2 + d)*a/(e^4*x^4 + 2*d*e^3*x^2 + d^2*e^2) - 1/4*(2*e*x^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))
+ d*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + 4*(e^4*x^4 + 2*d*e^3*x^2 + d^2*e^2)*integrate(1/4*(2*c^2*e*x^3 +
c^2*d*x)*e^(1/2*log(c*x + 1) + 1/2*log(c*x - 1))/(c^2*e^4*x^6 + (2*c^2*d*e^3 - e^4)*x^4 - d^2*e^2 + (c^2*d^2*
e^2 - 2*d*e^3)*x^2 + (c^2*e^4*x^6 + (2*c^2*d*e^3 - e^4)*x^4 - d^2*e^2 + (c^2*d^2*e^2 - 2*d*e^3)*x^2)*e^(log(c*
x + 1) + log(c*x - 1))), x))*b/(e^4*x^4 + 2*d*e^3*x^2 + d^2*e^2)

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(x^3*(a+b*arccsc(c*x))/(e*x^2+d)^3,x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]

[In]

int((x^3*(a + b*asin(1/(c*x))))/(d + e*x^2)^3,x)

[Out]

int((x^3*(a + b*asin(1/(c*x))))/(d + e*x^2)^3, x)