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

Optimal. Leaf size=195 \[ \frac {\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac {b c d^3 x \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )}{6 e \sqrt {c^2 x^2}}+\frac {b e x \left (c^2 x^2-1\right )^{3/2} \left (3 c^2 d+2 e\right )}{18 c^5 \sqrt {c^2 x^2}}+\frac {b e^2 x \left (c^2 x^2-1\right )^{5/2}}{30 c^5 \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \left (3 c^4 d^2+3 c^2 d e+e^2\right )}{6 c^5 \sqrt {c^2 x^2}} \]

[Out]

1/6*(e*x^2+d)^3*(a+b*arccsc(c*x))/e+1/18*b*e*(3*c^2*d+2*e)*x*(c^2*x^2-1)^(3/2)/c^5/(c^2*x^2)^(1/2)+1/30*b*e^2*
x*(c^2*x^2-1)^(5/2)/c^5/(c^2*x^2)^(1/2)+1/6*b*c*d^3*x*arctan((c^2*x^2-1)^(1/2))/e/(c^2*x^2)^(1/2)+1/6*b*(3*c^4
*d^2+3*c^2*d*e+e^2)*x*(c^2*x^2-1)^(1/2)/c^5/(c^2*x^2)^(1/2)

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Rubi [A]  time = 0.14, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5237, 446, 88, 63, 205} \[ \frac {\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac {b x \sqrt {c^2 x^2-1} \left (3 c^4 d^2+3 c^2 d e+e^2\right )}{6 c^5 \sqrt {c^2 x^2}}+\frac {b c d^3 x \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )}{6 e \sqrt {c^2 x^2}}+\frac {b e x \left (c^2 x^2-1\right )^{3/2} \left (3 c^2 d+2 e\right )}{18 c^5 \sqrt {c^2 x^2}}+\frac {b e^2 x \left (c^2 x^2-1\right )^{5/2}}{30 c^5 \sqrt {c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(3*c^4*d^2 + 3*c^2*d*e + e^2)*x*Sqrt[-1 + c^2*x^2])/(6*c^5*Sqrt[c^2*x^2]) + (b*e*(3*c^2*d + 2*e)*x*(-1 + c^
2*x^2)^(3/2))/(18*c^5*Sqrt[c^2*x^2]) + (b*e^2*x*(-1 + c^2*x^2)^(5/2))/(30*c^5*Sqrt[c^2*x^2]) + ((d + e*x^2)^3*
(a + b*ArcCsc[c*x]))/(6*e) + (b*c*d^3*x*ArcTan[Sqrt[-1 + c^2*x^2]])/(6*e*Sqrt[c^2*x^2])

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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

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 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 5237

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] + Dist[(b*c*x)/(2*e*(p + 1)*Sqrt[c^2*x^2]), Int[(d + e*x^2)^(p + 1)/
(x*Sqrt[c^2*x^2 - 1]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]

Rubi steps

\begin {align*} \int x \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac {\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac {(b c x) \int \frac {\left (d+e x^2\right )^3}{x \sqrt {-1+c^2 x^2}} \, dx}{6 e \sqrt {c^2 x^2}}\\ &=\frac {\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac {(b c x) \operatorname {Subst}\left (\int \frac {(d+e x)^3}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{12 e \sqrt {c^2 x^2}}\\ &=\frac {\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac {(b c x) \operatorname {Subst}\left (\int \left (\frac {e \left (3 c^4 d^2+3 c^2 d e+e^2\right )}{c^4 \sqrt {-1+c^2 x}}+\frac {d^3}{x \sqrt {-1+c^2 x}}+\frac {e^2 \left (3 c^2 d+2 e\right ) \sqrt {-1+c^2 x}}{c^4}+\frac {e^3 \left (-1+c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )}{12 e \sqrt {c^2 x^2}}\\ &=\frac {b \left (3 c^4 d^2+3 c^2 d e+e^2\right ) x \sqrt {-1+c^2 x^2}}{6 c^5 \sqrt {c^2 x^2}}+\frac {b e \left (3 c^2 d+2 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{18 c^5 \sqrt {c^2 x^2}}+\frac {b e^2 x \left (-1+c^2 x^2\right )^{5/2}}{30 c^5 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac {\left (b c d^3 x\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{12 e \sqrt {c^2 x^2}}\\ &=\frac {b \left (3 c^4 d^2+3 c^2 d e+e^2\right ) x \sqrt {-1+c^2 x^2}}{6 c^5 \sqrt {c^2 x^2}}+\frac {b e \left (3 c^2 d+2 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{18 c^5 \sqrt {c^2 x^2}}+\frac {b e^2 x \left (-1+c^2 x^2\right )^{5/2}}{30 c^5 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac {\left (b d^3 x\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{6 c e \sqrt {c^2 x^2}}\\ &=\frac {b \left (3 c^4 d^2+3 c^2 d e+e^2\right ) x \sqrt {-1+c^2 x^2}}{6 c^5 \sqrt {c^2 x^2}}+\frac {b e \left (3 c^2 d+2 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{18 c^5 \sqrt {c^2 x^2}}+\frac {b e^2 x \left (-1+c^2 x^2\right )^{5/2}}{30 c^5 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac {b c d^3 x \tan ^{-1}\left (\sqrt {-1+c^2 x^2}\right )}{6 e \sqrt {c^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.29, size = 124, normalized size = 0.64 \[ \frac {1}{90} x \left (15 a x \left (3 d^2+3 d e x^2+e^2 x^4\right )+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} \left (3 c^4 \left (15 d^2+5 d e x^2+e^2 x^4\right )+2 c^2 e \left (15 d+2 e x^2\right )+8 e^2\right )}{c^5}+15 b x \csc ^{-1}(c x) \left (3 d^2+3 d e x^2+e^2 x^4\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(15*a*x*(3*d^2 + 3*d*e*x^2 + e^2*x^4) + (b*Sqrt[1 - 1/(c^2*x^2)]*(8*e^2 + 2*c^2*e*(15*d + 2*e*x^2) + 3*c^4*
(15*d^2 + 5*d*e*x^2 + e^2*x^4)))/c^5 + 15*b*x*(3*d^2 + 3*d*e*x^2 + e^2*x^4)*ArcCsc[c*x]))/90

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fricas [A]  time = 0.48, size = 152, normalized size = 0.78 \[ \frac {15 \, a c^{6} e^{2} x^{6} + 45 \, a c^{6} d e x^{4} + 45 \, a c^{6} d^{2} x^{2} + 15 \, {\left (b c^{6} e^{2} x^{6} + 3 \, b c^{6} d e x^{4} + 3 \, b c^{6} d^{2} x^{2}\right )} \operatorname {arccsc}\left (c x\right ) + {\left (3 \, b c^{4} e^{2} x^{4} + 45 \, b c^{4} d^{2} + 30 \, b c^{2} d e + 8 \, b e^{2} + {\left (15 \, b c^{4} d e + 4 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{90 \, c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/90*(15*a*c^6*e^2*x^6 + 45*a*c^6*d*e*x^4 + 45*a*c^6*d^2*x^2 + 15*(b*c^6*e^2*x^6 + 3*b*c^6*d*e*x^4 + 3*b*c^6*d
^2*x^2)*arccsc(c*x) + (3*b*c^4*e^2*x^4 + 45*b*c^4*d^2 + 30*b*c^2*d*e + 8*b*e^2 + (15*b*c^4*d*e + 4*b*c^2*e^2)*
x^2)*sqrt(c^2*x^2 - 1))/c^6

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giac [B]  time = 0.32, size = 1154, normalized size = 5.92 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5760*(15*b*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6*arcsin(1/(c*x))*e^2/c + 15*a*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^
6*e^2/c + 180*b*d*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*arcsin(1/(c*x))*e/c + 6*b*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1
)^5*e^2/c^2 + 180*a*d*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*e/c + 90*b*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*arcsin(
1/(c*x))*e^2/c^3 + 720*b*d^2*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*arcsin(1/(c*x))/c + 90*a*x^4*(sqrt(-1/(c^2*x^2
) + 1) + 1)^4*e^2/c^3 + 120*b*d*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3*e/c^2 + 720*a*d^2*x^2*(sqrt(-1/(c^2*x^2) +
1) + 1)^2/c + 720*b*d*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*arcsin(1/(c*x))*e/c^3 + 50*b*x^3*(sqrt(-1/(c^2*x^2) +
1) + 1)^3*e^2/c^4 + 720*a*d*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*e/c^3 + 1440*b*d^2*x*(sqrt(-1/(c^2*x^2) + 1) +
1)/c^2 + 225*b*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*arcsin(1/(c*x))*e^2/c^5 + 1440*b*d^2*arcsin(1/(c*x))/c^3 +
225*a*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*e^2/c^5 + 1080*b*d*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*e/c^4 + 1440*a*d^2/
c^3 + 1080*b*d*arcsin(1/(c*x))*e/c^5 + 300*b*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*e^2/c^6 + 1080*a*d*e/c^5 - 1440*b*
d^2/(c^4*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 300*b*arcsin(1/(c*x))*e^2/c^7 + 720*b*d^2*arcsin(1/(c*x))/(c^5*x^2*
(sqrt(-1/(c^2*x^2) + 1) + 1)^2) + 300*a*e^2/c^7 - 1080*b*d*e/(c^6*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 720*a*d^2/
(c^5*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) + 720*b*d*arcsin(1/(c*x))*e/(c^7*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2)
- 300*b*e^2/(c^8*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 720*a*d*e/(c^7*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) + 225*b*
arcsin(1/(c*x))*e^2/(c^9*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) + 225*a*e^2/(c^9*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)
^2) - 120*b*d*e/(c^8*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 180*b*d*arcsin(1/(c*x))*e/(c^9*x^4*(sqrt(-1/(c^2*x^
2) + 1) + 1)^4) - 50*b*e^2/(c^10*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 180*a*d*e/(c^9*x^4*(sqrt(-1/(c^2*x^2) +
1) + 1)^4) + 90*b*arcsin(1/(c*x))*e^2/(c^11*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4) + 90*a*e^2/(c^11*x^4*(sqrt(-1
/(c^2*x^2) + 1) + 1)^4) - 6*b*e^2/(c^12*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 15*b*arcsin(1/(c*x))*e^2/(c^13*x
^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6) + 15*a*e^2/(c^13*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6))*c

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maple [A]  time = 0.06, size = 182, normalized size = 0.93 \[ \frac {\frac {a \left (\frac {1}{6} e^{2} c^{6} x^{6}+\frac {1}{2} c^{6} e d \,x^{4}+\frac {1}{2} x^{2} c^{6} d^{2}\right )}{c^{4}}+\frac {b \left (\frac {\mathrm {arccsc}\left (c x \right ) e^{2} c^{6} x^{6}}{6}+\frac {\mathrm {arccsc}\left (c x \right ) c^{6} x^{4} d e}{2}+\frac {\mathrm {arccsc}\left (c x \right ) c^{6} x^{2} d^{2}}{2}+\frac {\left (c^{2} x^{2}-1\right ) \left (3 c^{4} e^{2} x^{4}+15 c^{4} d e \,x^{2}+45 d^{2} c^{4}+4 c^{2} e^{2} x^{2}+30 c^{2} e d +8 e^{2}\right )}{90 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^2*(a/c^4*(1/6*e^2*c^6*x^6+1/2*c^6*e*d*x^4+1/2*x^2*c^6*d^2)+b/c^4*(1/6*arccsc(c*x)*e^2*c^6*x^6+1/2*arccsc(c
*x)*c^6*x^4*d*e+1/2*arccsc(c*x)*c^6*x^2*d^2+1/90*(c^2*x^2-1)*(3*c^4*e^2*x^4+15*c^4*d*e*x^2+45*c^4*d^2+4*c^2*e^
2*x^2+30*c^2*d*e+8*e^2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/c/x))

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maxima [A]  time = 0.34, size = 189, normalized size = 0.97 \[ \frac {1}{6} \, a e^{2} x^{6} + \frac {1}{2} \, a d e x^{4} + \frac {1}{2} \, a d^{2} x^{2} + \frac {1}{2} \, {\left (x^{2} \operatorname {arccsc}\left (c x\right ) + \frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b d^{2} + \frac {1}{6} \, {\left (3 \, x^{4} \operatorname {arccsc}\left (c x\right ) + \frac {c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 3 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b d e + \frac {1}{90} \, {\left (15 \, x^{6} \operatorname {arccsc}\left (c x\right ) + \frac {3 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 10 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b e^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*a*e^2*x^6 + 1/2*a*d*e*x^4 + 1/2*a*d^2*x^2 + 1/2*(x^2*arccsc(c*x) + x*sqrt(-1/(c^2*x^2) + 1)/c)*b*d^2 + 1/6
*(3*x^4*arccsc(c*x) + (c^2*x^3*(-1/(c^2*x^2) + 1)^(3/2) + 3*x*sqrt(-1/(c^2*x^2) + 1))/c^3)*b*d*e + 1/90*(15*x^
6*arccsc(c*x) + (3*c^4*x^5*(-1/(c^2*x^2) + 1)^(5/2) + 10*c^2*x^3*(-1/(c^2*x^2) + 1)^(3/2) + 15*x*sqrt(-1/(c^2*
x^2) + 1))/c^5)*b*e^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 6.38, size = 352, normalized size = 1.81 \[ \frac {a d^{2} x^{2}}{2} + \frac {a d e x^{4}}{2} + \frac {a e^{2} x^{6}}{6} + \frac {b d^{2} x^{2} \operatorname {acsc}{\left (c x \right )}}{2} + \frac {b d e x^{4} \operatorname {acsc}{\left (c x \right )}}{2} + \frac {b e^{2} x^{6} \operatorname {acsc}{\left (c x \right )}}{6} + \frac {b d^{2} \left (\begin {cases} \frac {\sqrt {c^{2} x^{2} - 1}}{c} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i \sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right )}{2 c} + \frac {b d e \left (\begin {cases} \frac {x^{2} \sqrt {c^{2} x^{2} - 1}}{3 c} + \frac {2 \sqrt {c^{2} x^{2} - 1}}{3 c^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {2 i \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} & \text {otherwise} \end {cases}\right )}{2 c} + \frac {b e^{2} \left (\begin {cases} \frac {x^{4} \sqrt {c^{2} x^{2} - 1}}{5 c} + \frac {4 x^{2} \sqrt {c^{2} x^{2} - 1}}{15 c^{3}} + \frac {8 \sqrt {c^{2} x^{2} - 1}}{15 c^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{4} \sqrt {- c^{2} x^{2} + 1}}{5 c} + \frac {4 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{15 c^{3}} + \frac {8 i \sqrt {- c^{2} x^{2} + 1}}{15 c^{5}} & \text {otherwise} \end {cases}\right )}{6 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d**2*x**2/2 + a*d*e*x**4/2 + a*e**2*x**6/6 + b*d**2*x**2*acsc(c*x)/2 + b*d*e*x**4*acsc(c*x)/2 + b*e**2*x**6*
acsc(c*x)/6 + b*d**2*Piecewise((sqrt(c**2*x**2 - 1)/c, Abs(c**2*x**2) > 1), (I*sqrt(-c**2*x**2 + 1)/c, True))/
(2*c) + b*d*e*Piecewise((x**2*sqrt(c**2*x**2 - 1)/(3*c) + 2*sqrt(c**2*x**2 - 1)/(3*c**3), Abs(c**2*x**2) > 1),
(I*x**2*sqrt(-c**2*x**2 + 1)/(3*c) + 2*I*sqrt(-c**2*x**2 + 1)/(3*c**3), True))/(2*c) + b*e**2*Piecewise((x**4
*sqrt(c**2*x**2 - 1)/(5*c) + 4*x**2*sqrt(c**2*x**2 - 1)/(15*c**3) + 8*sqrt(c**2*x**2 - 1)/(15*c**5), Abs(c**2*
x**2) > 1), (I*x**4*sqrt(-c**2*x**2 + 1)/(5*c) + 4*I*x**2*sqrt(-c**2*x**2 + 1)/(15*c**3) + 8*I*sqrt(-c**2*x**2
+ 1)/(15*c**5), True))/(6*c)

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