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

Optimal. Leaf size=252 \[ \frac{1}{3} d^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \csc ^{-1}(c x)\right )+\frac{b x^2 \sqrt{c^2 x^2-1} \left (280 c^4 d^2+252 c^2 d e+75 e^2\right )}{1680 c^5 \sqrt{c^2 x^2}}+\frac{b x \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{1680 c^6 \sqrt{c^2 x^2}}+\frac{b e x^4 \sqrt{c^2 x^2-1} \left (84 c^2 d+25 e\right )}{840 c^3 \sqrt{c^2 x^2}}+\frac{b e^2 x^6 \sqrt{c^2 x^2-1}}{42 c \sqrt{c^2 x^2}} \]

[Out]

(b*(280*c^4*d^2 + 252*c^2*d*e + 75*e^2)*x^2*Sqrt[-1 + c^2*x^2])/(1680*c^5*Sqrt[c^2*x^2]) + (b*e*(84*c^2*d + 25
*e)*x^4*Sqrt[-1 + c^2*x^2])/(840*c^3*Sqrt[c^2*x^2]) + (b*e^2*x^6*Sqrt[-1 + c^2*x^2])/(42*c*Sqrt[c^2*x^2]) + (d
^2*x^3*(a + b*ArcCsc[c*x]))/3 + (2*d*e*x^5*(a + b*ArcCsc[c*x]))/5 + (e^2*x^7*(a + b*ArcCsc[c*x]))/7 + (b*(280*
c^4*d^2 + 252*c^2*d*e + 75*e^2)*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(1680*c^6*Sqrt[c^2*x^2])

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Rubi [A]  time = 0.237142, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {270, 5239, 12, 1267, 459, 321, 217, 206} \[ \frac{1}{3} d^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \csc ^{-1}(c x)\right )+\frac{b x^2 \sqrt{c^2 x^2-1} \left (280 c^4 d^2+252 c^2 d e+75 e^2\right )}{1680 c^5 \sqrt{c^2 x^2}}+\frac{b x \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{1680 c^6 \sqrt{c^2 x^2}}+\frac{b e x^4 \sqrt{c^2 x^2-1} \left (84 c^2 d+25 e\right )}{840 c^3 \sqrt{c^2 x^2}}+\frac{b e^2 x^6 \sqrt{c^2 x^2-1}}{42 c \sqrt{c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(280*c^4*d^2 + 252*c^2*d*e + 75*e^2)*x^2*Sqrt[-1 + c^2*x^2])/(1680*c^5*Sqrt[c^2*x^2]) + (b*e*(84*c^2*d + 25
*e)*x^4*Sqrt[-1 + c^2*x^2])/(840*c^3*Sqrt[c^2*x^2]) + (b*e^2*x^6*Sqrt[-1 + c^2*x^2])/(42*c*Sqrt[c^2*x^2]) + (d
^2*x^3*(a + b*ArcCsc[c*x]))/3 + (2*d*e*x^5*(a + b*ArcCsc[c*x]))/5 + (e^2*x^7*(a + b*ArcCsc[c*x]))/7 + (b*(280*
c^4*d^2 + 252*c^2*d*e + 75*e^2)*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(1680*c^6*Sqrt[c^2*x^2])

Rule 270

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

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]))

Rule 12

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

Rule 1267

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Si
mp[(c^p*(f*x)^(m + 4*p - 1)*(d + e*x^2)^(q + 1))/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1)), x] + Dist[1/(e*(m + 4*p
+ 2*q + 1)), Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p))
- d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] &&
IGtQ[p, 0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]

Rule 459

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

Rule 321

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int x^2 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac{1}{3} d^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \csc ^{-1}(c x)\right )+\frac{(b c x) \int \frac{x^2 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{105 \sqrt{-1+c^2 x^2}} \, dx}{\sqrt{c^2 x^2}}\\ &=\frac{1}{3} d^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \csc ^{-1}(c x)\right )+\frac{(b c x) \int \frac{x^2 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{\sqrt{-1+c^2 x^2}} \, dx}{105 \sqrt{c^2 x^2}}\\ &=\frac{b e^2 x^6 \sqrt{-1+c^2 x^2}}{42 c \sqrt{c^2 x^2}}+\frac{1}{3} d^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \csc ^{-1}(c x)\right )+\frac{(b x) \int \frac{x^2 \left (210 c^2 d^2+3 e \left (84 c^2 d+25 e\right ) x^2\right )}{\sqrt{-1+c^2 x^2}} \, dx}{630 c \sqrt{c^2 x^2}}\\ &=\frac{b e \left (84 c^2 d+25 e\right ) x^4 \sqrt{-1+c^2 x^2}}{840 c^3 \sqrt{c^2 x^2}}+\frac{b e^2 x^6 \sqrt{-1+c^2 x^2}}{42 c \sqrt{c^2 x^2}}+\frac{1}{3} d^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \csc ^{-1}(c x)\right )+-\frac{\left (b \left (-840 c^4 d^2-9 e \left (84 c^2 d+25 e\right )\right ) x\right ) \int \frac{x^2}{\sqrt{-1+c^2 x^2}} \, dx}{2520 c^3 \sqrt{c^2 x^2}}\\ &=\frac{b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x^2 \sqrt{-1+c^2 x^2}}{1680 c^5 \sqrt{c^2 x^2}}+\frac{b e \left (84 c^2 d+25 e\right ) x^4 \sqrt{-1+c^2 x^2}}{840 c^3 \sqrt{c^2 x^2}}+\frac{b e^2 x^6 \sqrt{-1+c^2 x^2}}{42 c \sqrt{c^2 x^2}}+\frac{1}{3} d^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \csc ^{-1}(c x)\right )+-\frac{\left (b \left (-840 c^4 d^2-9 e \left (84 c^2 d+25 e\right )\right ) x\right ) \int \frac{1}{\sqrt{-1+c^2 x^2}} \, dx}{5040 c^5 \sqrt{c^2 x^2}}\\ &=\frac{b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x^2 \sqrt{-1+c^2 x^2}}{1680 c^5 \sqrt{c^2 x^2}}+\frac{b e \left (84 c^2 d+25 e\right ) x^4 \sqrt{-1+c^2 x^2}}{840 c^3 \sqrt{c^2 x^2}}+\frac{b e^2 x^6 \sqrt{-1+c^2 x^2}}{42 c \sqrt{c^2 x^2}}+\frac{1}{3} d^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \csc ^{-1}(c x)\right )+-\frac{\left (b \left (-840 c^4 d^2-9 e \left (84 c^2 d+25 e\right )\right ) x\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x^2} \, dx,x,\frac{x}{\sqrt{-1+c^2 x^2}}\right )}{5040 c^5 \sqrt{c^2 x^2}}\\ &=\frac{b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x^2 \sqrt{-1+c^2 x^2}}{1680 c^5 \sqrt{c^2 x^2}}+\frac{b e \left (84 c^2 d+25 e\right ) x^4 \sqrt{-1+c^2 x^2}}{840 c^3 \sqrt{c^2 x^2}}+\frac{b e^2 x^6 \sqrt{-1+c^2 x^2}}{42 c \sqrt{c^2 x^2}}+\frac{1}{3} d^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \csc ^{-1}(c x)\right )+\frac{b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) x \tanh ^{-1}\left (\frac{c x}{\sqrt{-1+c^2 x^2}}\right )}{1680 c^6 \sqrt{c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.350116, size = 184, normalized size = 0.73 \[ \frac{c^2 x^2 \left (16 a c^5 x \left (35 d^2+42 d e x^2+15 e^2 x^4\right )+b \sqrt{1-\frac{1}{c^2 x^2}} \left (8 c^4 \left (35 d^2+21 d e x^2+5 e^2 x^4\right )+2 c^2 e \left (126 d+25 e x^2\right )+75 e^2\right )\right )+b \left (280 c^4 d^2+252 c^2 d e+75 e^2\right ) \log \left (x \left (\sqrt{1-\frac{1}{c^2 x^2}}+1\right )\right )+16 b c^7 x^3 \csc ^{-1}(c x) \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{1680 c^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*x^2*(16*a*c^5*x*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4) + b*Sqrt[1 - 1/(c^2*x^2)]*(75*e^2 + 2*c^2*e*(126*d + 2
5*e*x^2) + 8*c^4*(35*d^2 + 21*d*e*x^2 + 5*e^2*x^4))) + 16*b*c^7*x^3*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4)*ArcCsc[
c*x] + b*(280*c^4*d^2 + 252*c^2*d*e + 75*e^2)*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/(1680*c^7)

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Maple [B]  time = 0.184, size = 494, normalized size = 2. \begin{align*}{\frac{a{e}^{2}{x}^{7}}{7}}+{\frac{2\,aed{x}^{5}}{5}}+{\frac{a{x}^{3}{d}^{2}}{3}}+{\frac{b{\rm arccsc} \left (cx\right ){e}^{2}{x}^{7}}{7}}+{\frac{2\,b{\rm arccsc} \left (cx\right )ed{x}^{5}}{5}}+{\frac{b{\rm arccsc} \left (cx\right ){x}^{3}{d}^{2}}{3}}+{\frac{b{x}^{6}{e}^{2}}{42\,c}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{b{x}^{4}{e}^{2}}{168\,{c}^{3}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{b{x}^{4}ed}{10\,c}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{be{x}^{2}d}{20\,{c}^{3}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{b{d}^{2}{x}^{2}}{6\,c}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{b{d}^{2}}{6\,{c}^{3}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{b{d}^{2}}{6\,{c}^{4}x}\sqrt{{c}^{2}{x}^{2}-1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{5\,b{x}^{2}{e}^{2}}{336\,{c}^{5}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{3\,bed}{20\,{c}^{5}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{3\,bed}{20\,{c}^{6}x}\sqrt{{c}^{2}{x}^{2}-1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{5\,b{e}^{2}}{112\,{c}^{7}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{5\,b{e}^{2}}{112\,{c}^{8}x}\sqrt{{c}^{2}{x}^{2}-1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a*e^2*x^7+2/5*a*e*d*x^5+1/3*a*x^3*d^2+1/7*b*arccsc(c*x)*e^2*x^7+2/5*b*arccsc(c*x)*e*d*x^5+1/3*b*arccsc(c*x
)*x^3*d^2+1/42/c*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^6*e^2+1/168/c^3*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^4*e^2+1/10/c*
b/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^4*e*d+1/20/c^3*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^2*e*d+1/6/c*b/((c^2*x^2-1)/c^2/
x^2)^(1/2)*d^2*x^2-1/6/c^3*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*d^2+1/6/c^4*b*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)
^(1/2)/x*d^2*ln(c*x+(c^2*x^2-1)^(1/2))+5/336/c^5*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^2*e^2-3/20/c^5*b/((c^2*x^2-1)
/c^2/x^2)^(1/2)*e*d+3/20/c^6*b*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*e*d*ln(c*x+(c^2*x^2-1)^(1/2))-5
/112/c^7*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*e^2+5/112/c^8*b*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*e^2*ln(
c*x+(c^2*x^2-1)^(1/2))

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Maxima [A]  time = 1.03673, size = 545, normalized size = 2.16 \begin{align*} \frac{1}{7} \, a e^{2} x^{7} + \frac{2}{5} \, a d e x^{5} + \frac{1}{3} \, a d^{2} x^{3} + \frac{1}{12} \,{\left (4 \, x^{3} \operatorname{arccsc}\left (c x\right ) + \frac{\frac{2 \, \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c^{2}{\left (\frac{1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac{\log \left (\sqrt{-\frac{1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac{\log \left (\sqrt{-\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b d^{2} + \frac{1}{40} \,{\left (16 \, x^{5} \operatorname{arccsc}\left (c x\right ) - \frac{\frac{2 \,{\left (3 \,{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{-\frac{1}{c^{2} x^{2}} + 1}\right )}}{c^{4}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4}{\left (\frac{1}{c^{2} x^{2}} - 1\right )} + c^{4}} - \frac{3 \, \log \left (\sqrt{-\frac{1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac{3 \, \log \left (\sqrt{-\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b d e + \frac{1}{672} \,{\left (96 \, x^{7} \operatorname{arccsc}\left (c x\right ) + \frac{\frac{2 \,{\left (15 \,{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{5}{2}} - 40 \,{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + 33 \, \sqrt{-\frac{1}{c^{2} x^{2}} + 1}\right )}}{c^{6}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{3} + 3 \, c^{6}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{6}{\left (\frac{1}{c^{2} x^{2}} - 1\right )} + c^{6}} + \frac{15 \, \log \left (\sqrt{-\frac{1}{c^{2} x^{2}} + 1} + 1\right )}{c^{6}} - \frac{15 \, \log \left (\sqrt{-\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c^{6}}}{c}\right )} b e^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a*e^2*x^7 + 2/5*a*d*e*x^5 + 1/3*a*d^2*x^3 + 1/12*(4*x^3*arccsc(c*x) + (2*sqrt(-1/(c^2*x^2) + 1)/(c^2*(1/(c
^2*x^2) - 1) + c^2) + log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^2 - log(sqrt(-1/(c^2*x^2) + 1) - 1)/c^2)/c)*b*d^2 + 1/
40*(16*x^5*arccsc(c*x) - (2*(3*(-1/(c^2*x^2) + 1)^(3/2) - 5*sqrt(-1/(c^2*x^2) + 1))/(c^4*(1/(c^2*x^2) - 1)^2 +
2*c^4*(1/(c^2*x^2) - 1) + c^4) - 3*log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^4 + 3*log(sqrt(-1/(c^2*x^2) + 1) - 1)/c^
4)/c)*b*d*e + 1/672*(96*x^7*arccsc(c*x) + (2*(15*(-1/(c^2*x^2) + 1)^(5/2) - 40*(-1/(c^2*x^2) + 1)^(3/2) + 33*s
qrt(-1/(c^2*x^2) + 1))/(c^6*(1/(c^2*x^2) - 1)^3 + 3*c^6*(1/(c^2*x^2) - 1)^2 + 3*c^6*(1/(c^2*x^2) - 1) + c^6) +
15*log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^6 - 15*log(sqrt(-1/(c^2*x^2) + 1) - 1)/c^6)/c)*b*e^2

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Fricas [A]  time = 6.06912, size = 641, normalized size = 2.54 \begin{align*} \frac{240 \, a c^{7} e^{2} x^{7} + 672 \, a c^{7} d e x^{5} + 560 \, a c^{7} d^{2} x^{3} + 16 \,{\left (15 \, b c^{7} e^{2} x^{7} + 42 \, b c^{7} d e x^{5} + 35 \, b c^{7} d^{2} x^{3} - 35 \, b c^{7} d^{2} - 42 \, b c^{7} d e - 15 \, b c^{7} e^{2}\right )} \operatorname{arccsc}\left (c x\right ) - 32 \,{\left (35 \, b c^{7} d^{2} + 42 \, b c^{7} d e + 15 \, b c^{7} e^{2}\right )} \arctan \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (280 \, b c^{4} d^{2} + 252 \, b c^{2} d e + 75 \, b e^{2}\right )} \log \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) +{\left (40 \, b c^{5} e^{2} x^{5} + 2 \,{\left (84 \, b c^{5} d e + 25 \, b c^{3} e^{2}\right )} x^{3} +{\left (280 \, b c^{5} d^{2} + 252 \, b c^{3} d e + 75 \, b c e^{2}\right )} x\right )} \sqrt{c^{2} x^{2} - 1}}{1680 \, c^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1680*(240*a*c^7*e^2*x^7 + 672*a*c^7*d*e*x^5 + 560*a*c^7*d^2*x^3 + 16*(15*b*c^7*e^2*x^7 + 42*b*c^7*d*e*x^5 +
35*b*c^7*d^2*x^3 - 35*b*c^7*d^2 - 42*b*c^7*d*e - 15*b*c^7*e^2)*arccsc(c*x) - 32*(35*b*c^7*d^2 + 42*b*c^7*d*e +
15*b*c^7*e^2)*arctan(-c*x + sqrt(c^2*x^2 - 1)) - (280*b*c^4*d^2 + 252*b*c^2*d*e + 75*b*e^2)*log(-c*x + sqrt(c
^2*x^2 - 1)) + (40*b*c^5*e^2*x^5 + 2*(84*b*c^5*d*e + 25*b*c^3*e^2)*x^3 + (280*b*c^5*d^2 + 252*b*c^3*d*e + 75*b
*c*e^2)*x)*sqrt(c^2*x^2 - 1))/c^7

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (a + b \operatorname{acsc}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{2} + d\right )}^{2}{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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