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

Optimal. Leaf size=157 \[ \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {b c x \left (c^2 d+2 e\right ) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{\sqrt {c^2 d+e}}\right )}{8 d e^{3/2} \sqrt {c^2 x^2} \left (c^2 d+e\right )^{3/2}}-\frac {b c x \sqrt {c^2 x^2-1}}{8 e \sqrt {c^2 x^2} \left (c^2 d+e\right ) \left (d+e x^2\right )} \]

[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)

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Rubi [A]  time = 0.18, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {264, 5239, 12, 446, 78, 63, 205} \[ \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {b c x \left (c^2 d+2 e\right ) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{\sqrt {c^2 d+e}}\right )}{8 d e^{3/2} \sqrt {c^2 x^2} \left (c^2 d+e\right )^{3/2}}-\frac {b c x \sqrt {c^2 x^2-1}}{8 e \sqrt {c^2 x^2} \left (c^2 d+e\right ) \left (d+e x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*c*x*Sqrt[-1 + c^2*x^2])/(8*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 63

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 78

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] || LtQ
[p, n]))))

Rule 205

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

Rule 264

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 446

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 5239

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*} \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) \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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 \tan ^{-1}\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*}

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Mathematica [C]  time = 1.12, size = 390, normalized size = 2.48 \[ \frac {-\frac {8 a}{d+e x^2}+\frac {4 a d}{\left (d+e x^2\right )^2}+\frac {b \sqrt {e} \left (c^2 d+2 e\right ) \log \left (\frac {16 d e^{3/2} \sqrt {c^2 (-d)-e} \left (c x \left (c \sqrt {d}-i \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {c^2 (-d)-e}\right )+i \sqrt {e}\right )}{b \left (c^2 d+2 e\right ) \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{d \left (c^2 (-d)-e\right )^{3/2}}+\frac {b \sqrt {e} \left (c^2 d+2 e\right ) \log \left (-\frac {16 d e^{3/2} \sqrt {c^2 (-d)-e} \left (-\sqrt {e}+c x \left (\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {c^2 (-d)-e}-i c \sqrt {d}\right )\right )}{b \left (c^2 d+2 e\right ) \left (\sqrt {e} x+i \sqrt {d}\right )}\right )}{d \left (c^2 (-d)-e\right )^{3/2}}-\frac {2 b c e x \sqrt {1-\frac {1}{c^2 x^2}}}{\left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {4 b \csc ^{-1}(c x) \left (d+2 e x^2\right )}{\left (d+e x^2\right )^2}+\frac {4 b \sin ^{-1}\left (\frac {1}{c x}\right )}{d}}{16 e^2} \]

Antiderivative was successfully verified.

[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)

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fricas [B]  time = 1.39, size = 1015, normalized size = 6.46 \[ \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 ] \]

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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:Warning, integrat
ion of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(x)]sym2poly
/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [B]  time = 0.09, size = 1870, normalized size = 11.91 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (2 \, e x^{2} + d\right )} a}{4 \, {\left (e^{4} x^{4} + 2 \, d e^{3} x^{2} + d^{2} e^{2}\right )}} - \frac {{\left (2 \, e x^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) + d \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left (e^{4} x^{4} + 2 \, d e^{3} x^{2} + d^{2} e^{2}\right )} \int \frac {{\left (2 \, c^{2} e x^{3} + c^{2} d x\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (c x - 1\right )\right )}}{c^{2} e^{4} x^{6} + {\left (2 \, c^{2} d e^{3} - e^{4}\right )} x^{4} - d^{2} e^{2} + {\left (c^{2} e^{4} x^{6} + {\left (2 \, c^{2} d e^{3} - e^{4}\right )} x^{4} - d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} - 2 \, d e^{3}\right )} x^{2}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} + {\left (c^{2} d^{2} e^{2} - 2 \, d e^{3}\right )} x^{2}}\,{d x}\right )} b}{4 \, {\left (e^{4} x^{4} + 2 \, d e^{3} x^{2} + d^{2} e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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