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

Optimal. Leaf size=242 \[ \frac {1}{4} d^2 x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {b e x \left (c^2 x^2-1\right )^{5/2} \left (8 c^2 d+9 e\right )}{120 c^7 \sqrt {c^2 x^2}}+\frac {b e^2 x \left (c^2 x^2-1\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}}+\frac {b x \left (c^2 x^2-1\right )^{3/2} \left (6 c^4 d^2+16 c^2 d e+9 e^2\right )}{72 c^7 \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \left (6 c^4 d^2+8 c^2 d e+3 e^2\right )}{24 c^7 \sqrt {c^2 x^2}} \]

[Out]

1/4*d^2*x^4*(a+b*arccsc(c*x))+1/3*d*e*x^6*(a+b*arccsc(c*x))+1/8*e^2*x^8*(a+b*arccsc(c*x))+1/72*b*(6*c^4*d^2+16
*c^2*d*e+9*e^2)*x*(c^2*x^2-1)^(3/2)/c^7/(c^2*x^2)^(1/2)+1/120*b*e*(8*c^2*d+9*e)*x*(c^2*x^2-1)^(5/2)/c^7/(c^2*x
^2)^(1/2)+1/56*b*e^2*x*(c^2*x^2-1)^(7/2)/c^7/(c^2*x^2)^(1/2)+1/24*b*(6*c^4*d^2+8*c^2*d*e+3*e^2)*x*(c^2*x^2-1)^
(1/2)/c^7/(c^2*x^2)^(1/2)

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Rubi [A]  time = 0.22, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {266, 43, 5239, 12, 1251, 771} \[ \frac {1}{4} d^2 x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {b x \left (c^2 x^2-1\right )^{3/2} \left (6 c^4 d^2+16 c^2 d e+9 e^2\right )}{72 c^7 \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \left (6 c^4 d^2+8 c^2 d e+3 e^2\right )}{24 c^7 \sqrt {c^2 x^2}}+\frac {b e x \left (c^2 x^2-1\right )^{5/2} \left (8 c^2 d+9 e\right )}{120 c^7 \sqrt {c^2 x^2}}+\frac {b e^2 x \left (c^2 x^2-1\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(6*c^4*d^2 + 8*c^2*d*e + 3*e^2)*x*Sqrt[-1 + c^2*x^2])/(24*c^7*Sqrt[c^2*x^2]) + (b*(6*c^4*d^2 + 16*c^2*d*e +
9*e^2)*x*(-1 + c^2*x^2)^(3/2))/(72*c^7*Sqrt[c^2*x^2]) + (b*e*(8*c^2*d + 9*e)*x*(-1 + c^2*x^2)^(5/2))/(120*c^7
*Sqrt[c^2*x^2]) + (b*e^2*x*(-1 + c^2*x^2)^(7/2))/(56*c^7*Sqrt[c^2*x^2]) + (d^2*x^4*(a + b*ArcCsc[c*x]))/4 + (d
*e*x^6*(a + b*ArcCsc[c*x]))/3 + (e^2*x^8*(a + b*ArcCsc[c*x]))/8

Rule 12

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

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

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

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 x^3 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac {1}{4} d^2 x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{24 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=\frac {1}{4} d^2 x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{24 \sqrt {c^2 x^2}}\\ &=\frac {1}{4} d^2 x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \operatorname {Subst}\left (\int \frac {x \left (6 d^2+8 d e x+3 e^2 x^2\right )}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{48 \sqrt {c^2 x^2}}\\ &=\frac {1}{4} d^2 x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \operatorname {Subst}\left (\int \left (\frac {6 c^4 d^2+8 c^2 d e+3 e^2}{c^6 \sqrt {-1+c^2 x}}+\frac {\left (6 c^4 d^2+16 c^2 d e+9 e^2\right ) \sqrt {-1+c^2 x}}{c^6}+\frac {e \left (8 c^2 d+9 e\right ) \left (-1+c^2 x\right )^{3/2}}{c^6}+\frac {3 e^2 \left (-1+c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )}{48 \sqrt {c^2 x^2}}\\ &=\frac {b \left (6 c^4 d^2+8 c^2 d e+3 e^2\right ) x \sqrt {-1+c^2 x^2}}{24 c^7 \sqrt {c^2 x^2}}+\frac {b \left (6 c^4 d^2+16 c^2 d e+9 e^2\right ) x \left (-1+c^2 x^2\right )^{3/2}}{72 c^7 \sqrt {c^2 x^2}}+\frac {b e \left (8 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{5/2}}{120 c^7 \sqrt {c^2 x^2}}+\frac {b e^2 x \left (-1+c^2 x^2\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}}+\frac {1}{4} d^2 x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \csc ^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]  time = 0.34, size = 159, normalized size = 0.66 \[ \frac {x \left (105 a x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} \left (3 c^6 \left (70 d^2 x^2+56 d e x^4+15 e^2 x^6\right )+c^4 \left (420 d^2+224 d e x^2+54 e^2 x^4\right )+8 c^2 e \left (56 d+9 e x^2\right )+144 e^2\right )}{c^7}+105 b x^3 \csc ^{-1}(c x) \left (6 d^2+8 d e x^2+3 e^2 x^4\right )\right )}{2520} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(105*a*x^3*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4) + (b*Sqrt[1 - 1/(c^2*x^2)]*(144*e^2 + 8*c^2*e*(56*d + 9*e*x^2) +
c^4*(420*d^2 + 224*d*e*x^2 + 54*e^2*x^4) + 3*c^6*(70*d^2*x^2 + 56*d*e*x^4 + 15*e^2*x^6)))/c^7 + 105*b*x^3*(6*
d^2 + 8*d*e*x^2 + 3*e^2*x^4)*ArcCsc[c*x]))/2520

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fricas [A]  time = 0.75, size = 186, normalized size = 0.77 \[ \frac {315 \, a c^{8} e^{2} x^{8} + 840 \, a c^{8} d e x^{6} + 630 \, a c^{8} d^{2} x^{4} + 105 \, {\left (3 \, b c^{8} e^{2} x^{8} + 8 \, b c^{8} d e x^{6} + 6 \, b c^{8} d^{2} x^{4}\right )} \operatorname {arccsc}\left (c x\right ) + {\left (45 \, b c^{6} e^{2} x^{6} + 420 \, b c^{4} d^{2} + 448 \, b c^{2} d e + 6 \, {\left (28 \, b c^{6} d e + 9 \, b c^{4} e^{2}\right )} x^{4} + 144 \, b e^{2} + 2 \, {\left (105 \, b c^{6} d^{2} + 112 \, b c^{4} d e + 36 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{2520 \, c^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2520*(315*a*c^8*e^2*x^8 + 840*a*c^8*d*e*x^6 + 630*a*c^8*d^2*x^4 + 105*(3*b*c^8*e^2*x^8 + 8*b*c^8*d*e*x^6 + 6
*b*c^8*d^2*x^4)*arccsc(c*x) + (45*b*c^6*e^2*x^6 + 420*b*c^4*d^2 + 448*b*c^2*d*e + 6*(28*b*c^6*d*e + 9*b*c^4*e^
2)*x^4 + 144*b*e^2 + 2*(105*b*c^6*d^2 + 112*b*c^4*d*e + 36*b*c^2*e^2)*x^2)*sqrt(c^2*x^2 - 1))/c^8

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giac [B]  time = 0.45, size = 1700, normalized size = 7.02 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/645120*(315*b*x^8*(sqrt(-1/(c^2*x^2) + 1) + 1)^8*arcsin(1/(c*x))*e^2/c + 315*a*x^8*(sqrt(-1/(c^2*x^2) + 1) +
1)^8*e^2/c + 3360*b*d*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6*arcsin(1/(c*x))*e/c + 90*b*x^7*(sqrt(-1/(c^2*x^2) +
1) + 1)^7*e^2/c^2 + 3360*a*d*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6*e/c + 2520*b*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^
6*arcsin(1/(c*x))*e^2/c^3 + 10080*b*d^2*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*arcsin(1/(c*x))/c + 2520*a*x^6*(sqr
t(-1/(c^2*x^2) + 1) + 1)^6*e^2/c^3 + 1344*b*d*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5*e/c^2 + 10080*a*d^2*x^4*(sqrt
(-1/(c^2*x^2) + 1) + 1)^4/c + 20160*b*d*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*arcsin(1/(c*x))*e/c^3 + 882*b*x^5*(
sqrt(-1/(c^2*x^2) + 1) + 1)^5*e^2/c^4 + 20160*a*d*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*e/c^3 + 6720*b*d^2*x^3*(s
qrt(-1/(c^2*x^2) + 1) + 1)^3/c^2 + 8820*b*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*arcsin(1/(c*x))*e^2/c^5 + 40320*b
*d^2*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*arcsin(1/(c*x))/c^3 + 8820*a*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*e^2/c^
5 + 11200*b*d*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3*e/c^4 + 40320*a*d^2*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2/c^3 +
50400*b*d*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*arcsin(1/(c*x))*e/c^5 + 4410*b*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3
*e^2/c^6 + 50400*a*d*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*e/c^5 + 60480*b*d^2*x*(sqrt(-1/(c^2*x^2) + 1) + 1)/c^4
+ 17640*b*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*arcsin(1/(c*x))*e^2/c^7 + 60480*b*d^2*arcsin(1/(c*x))/c^5 + 1764
0*a*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*e^2/c^7 + 67200*b*d*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*e/c^6 + 60480*a*d^2/
c^5 + 67200*b*d*arcsin(1/(c*x))*e/c^7 + 22050*b*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*e^2/c^8 + 67200*a*d*e/c^7 - 604
80*b*d^2/(c^6*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 22050*b*arcsin(1/(c*x))*e^2/c^9 + 40320*b*d^2*arcsin(1/(c*x))/
(c^7*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) + 22050*a*e^2/c^9 - 67200*b*d*e/(c^8*x*(sqrt(-1/(c^2*x^2) + 1) + 1))
+ 40320*a*d^2/(c^7*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) + 50400*b*d*arcsin(1/(c*x))*e/(c^9*x^2*(sqrt(-1/(c^2*x^
2) + 1) + 1)^2) - 22050*b*e^2/(c^10*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 50400*a*d*e/(c^9*x^2*(sqrt(-1/(c^2*x^2)
+ 1) + 1)^2) - 6720*b*d^2/(c^8*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 17640*b*arcsin(1/(c*x))*e^2/(c^11*x^2*(sq
rt(-1/(c^2*x^2) + 1) + 1)^2) + 10080*b*d^2*arcsin(1/(c*x))/(c^9*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4) + 17640*a*
e^2/(c^11*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) - 11200*b*d*e/(c^10*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 10080*
a*d^2/(c^9*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4) + 20160*b*d*arcsin(1/(c*x))*e/(c^11*x^4*(sqrt(-1/(c^2*x^2) + 1)
+ 1)^4) - 4410*b*e^2/(c^12*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 20160*a*d*e/(c^11*x^4*(sqrt(-1/(c^2*x^2) + 1
) + 1)^4) + 8820*b*arcsin(1/(c*x))*e^2/(c^13*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4) + 8820*a*e^2/(c^13*x^4*(sqrt(
-1/(c^2*x^2) + 1) + 1)^4) - 1344*b*d*e/(c^12*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 3360*b*d*arcsin(1/(c*x))*e/
(c^13*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6) - 882*b*e^2/(c^14*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 3360*a*d*e/(
c^13*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6) + 2520*b*arcsin(1/(c*x))*e^2/(c^15*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6
) + 2520*a*e^2/(c^15*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6) - 90*b*e^2/(c^16*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7)
+ 315*b*arcsin(1/(c*x))*e^2/(c^17*x^8*(sqrt(-1/(c^2*x^2) + 1) + 1)^8) + 315*a*e^2/(c^17*x^8*(sqrt(-1/(c^2*x^2)
+ 1) + 1)^8))*c

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maple [A]  time = 0.06, size = 214, normalized size = 0.88 \[ \frac {\frac {a \left (\frac {1}{8} e^{2} c^{8} x^{8}+\frac {1}{3} c^{8} e d \,x^{6}+\frac {1}{4} x^{4} c^{8} d^{2}\right )}{c^{4}}+\frac {b \left (\frac {\mathrm {arccsc}\left (c x \right ) e^{2} c^{8} x^{8}}{8}+\frac {\mathrm {arccsc}\left (c x \right ) c^{8} e d \,x^{6}}{3}+\frac {\mathrm {arccsc}\left (c x \right ) c^{8} x^{4} d^{2}}{4}+\frac {\left (c^{2} x^{2}-1\right ) \left (45 e^{2} c^{6} x^{6}+168 c^{6} e d \,x^{4}+210 x^{2} c^{6} d^{2}+54 c^{4} e^{2} x^{4}+224 c^{4} d e \,x^{2}+420 d^{2} c^{4}+72 c^{2} e^{2} x^{2}+448 c^{2} e d +144 e^{2}\right )}{2520 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}}{c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^4*(a/c^4*(1/8*e^2*c^8*x^8+1/3*c^8*e*d*x^6+1/4*x^4*c^8*d^2)+b/c^4*(1/8*arccsc(c*x)*e^2*c^8*x^8+1/3*arccsc(c
*x)*c^8*e*d*x^6+1/4*arccsc(c*x)*c^8*x^4*d^2+1/2520*(c^2*x^2-1)*(45*c^6*e^2*x^6+168*c^6*d*e*x^4+210*c^6*d^2*x^2
+54*c^4*e^2*x^4+224*c^4*d*e*x^2+420*c^4*d^2+72*c^2*e^2*x^2+448*c^2*d*e+144*e^2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/c/
x))

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maxima [A]  time = 0.35, size = 253, normalized size = 1.05 \[ \frac {1}{8} \, a e^{2} x^{8} + \frac {1}{3} \, a d e x^{6} + \frac {1}{4} \, a d^{2} x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arccsc}\left (c x\right ) + \frac {c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 3 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b d^{2} + \frac {1}{45} \, {\left (15 \, x^{6} \operatorname {arccsc}\left (c x\right ) + \frac {3 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 10 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b d e + \frac {1}{280} \, {\left (35 \, x^{8} \operatorname {arccsc}\left (c x\right ) + \frac {5 \, c^{6} x^{7} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {7}{2}} + 21 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 35 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 35 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{7}}\right )} b e^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a*e^2*x^8 + 1/3*a*d*e*x^6 + 1/4*a*d^2*x^4 + 1/12*(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^2 + 1/45*(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*d*e + 1/280*(35*x^8*arccsc(c*x) + (5*c^6
*x^7*(-1/(c^2*x^2) + 1)^(7/2) + 21*c^4*x^5*(-1/(c^2*x^2) + 1)^(5/2) + 35*c^2*x^3*(-1/(c^2*x^2) + 1)^(3/2) + 35
*x*sqrt(-1/(c^2*x^2) + 1))/c^7)*b*e^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 9.22, size = 493, normalized size = 2.04 \[ \frac {a d^{2} x^{4}}{4} + \frac {a d e x^{6}}{3} + \frac {a e^{2} x^{8}}{8} + \frac {b d^{2} x^{4} \operatorname {acsc}{\left (c x \right )}}{4} + \frac {b d e x^{6} \operatorname {acsc}{\left (c x \right )}}{3} + \frac {b e^{2} x^{8} \operatorname {acsc}{\left (c x \right )}}{8} + \frac {b d^{2} \left (\begin {cases} \frac {x^{2} \sqrt {c^{2} x^{2} - 1}}{3 c} + \frac {2 \sqrt {c^{2} x^{2} - 1}}{3 c^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {2 i \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} & \text {otherwise} \end {cases}\right )}{4 c} + \frac {b d e \left (\begin {cases} \frac {x^{4} \sqrt {c^{2} x^{2} - 1}}{5 c} + \frac {4 x^{2} \sqrt {c^{2} x^{2} - 1}}{15 c^{3}} + \frac {8 \sqrt {c^{2} x^{2} - 1}}{15 c^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{4} \sqrt {- c^{2} x^{2} + 1}}{5 c} + \frac {4 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{15 c^{3}} + \frac {8 i \sqrt {- c^{2} x^{2} + 1}}{15 c^{5}} & \text {otherwise} \end {cases}\right )}{3 c} + \frac {b e^{2} \left (\begin {cases} \frac {x^{6} \sqrt {c^{2} x^{2} - 1}}{7 c} + \frac {6 x^{4} \sqrt {c^{2} x^{2} - 1}}{35 c^{3}} + \frac {8 x^{2} \sqrt {c^{2} x^{2} - 1}}{35 c^{5}} + \frac {16 \sqrt {c^{2} x^{2} - 1}}{35 c^{7}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{6} \sqrt {- c^{2} x^{2} + 1}}{7 c} + \frac {6 i x^{4} \sqrt {- c^{2} x^{2} + 1}}{35 c^{3}} + \frac {8 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{35 c^{5}} + \frac {16 i \sqrt {- c^{2} x^{2} + 1}}{35 c^{7}} & \text {otherwise} \end {cases}\right )}{8 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d**2*x**4/4 + a*d*e*x**6/3 + a*e**2*x**8/8 + b*d**2*x**4*acsc(c*x)/4 + b*d*e*x**6*acsc(c*x)/3 + b*e**2*x**8*
acsc(c*x)/8 + b*d**2*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))/(4*c) + b*d*e*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))/(3*c) + b*e**2*Piecewise((x**6*sqrt(c**2*x**2 - 1)/(7*c) + 6*x**4*sqrt(c**2*x**2
- 1)/(35*c**3) + 8*x**2*sqrt(c**2*x**2 - 1)/(35*c**5) + 16*sqrt(c**2*x**2 - 1)/(35*c**7), Abs(c**2*x**2) > 1)
, (I*x**6*sqrt(-c**2*x**2 + 1)/(7*c) + 6*I*x**4*sqrt(-c**2*x**2 + 1)/(35*c**3) + 8*I*x**2*sqrt(-c**2*x**2 + 1)
/(35*c**5) + 16*I*sqrt(-c**2*x**2 + 1)/(35*c**7), True))/(8*c)

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