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

Optimal. Leaf size=139 \[ -\frac{i b^2 \text{PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )}{3 c^3}+\frac{i b^2 \text{PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )}{3 c^3}+\frac{b x^2 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{3 c}+\frac{2 b \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{3 c^3}+\frac{1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^2+\frac{b^2 x}{3 c^2} \]

[Out]

(b^2*x)/(3*c^2) + (b*Sqrt[1 - 1/(c^2*x^2)]*x^2*(a + b*ArcCsc[c*x]))/(3*c) + (x^3*(a + b*ArcCsc[c*x])^2)/3 + (2
*b*(a + b*ArcCsc[c*x])*ArcTanh[E^(I*ArcCsc[c*x])])/(3*c^3) - ((I/3)*b^2*PolyLog[2, -E^(I*ArcCsc[c*x])])/c^3 +
((I/3)*b^2*PolyLog[2, E^(I*ArcCsc[c*x])])/c^3

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Rubi [A]  time = 0.119006, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5223, 4410, 4185, 4183, 2279, 2391} \[ -\frac{i b^2 \text{PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )}{3 c^3}+\frac{i b^2 \text{PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )}{3 c^3}+\frac{b x^2 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{3 c}+\frac{2 b \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{3 c^3}+\frac{1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^2+\frac{b^2 x}{3 c^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcCsc[c*x])^2,x]

[Out]

(b^2*x)/(3*c^2) + (b*Sqrt[1 - 1/(c^2*x^2)]*x^2*(a + b*ArcCsc[c*x]))/(3*c) + (x^3*(a + b*ArcCsc[c*x])^2)/3 + (2
*b*(a + b*ArcCsc[c*x])*ArcTanh[E^(I*ArcCsc[c*x])])/(3*c^3) - ((I/3)*b^2*PolyLog[2, -E^(I*ArcCsc[c*x])])/c^3 +
((I/3)*b^2*PolyLog[2, E^(I*ArcCsc[c*x])])/c^3

Rule 5223

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b*
x)^n*Csc[x]^(m + 1)*Cot[x], x], x, ArcCsc[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G
tQ[n, 0] || LtQ[m, -1])

Rule 4410

Int[Cot[(a_.) + (b_.)*(x_)]^(p_.)*Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp
[((c + d*x)^m*Csc[a + b*x]^n)/(b*n), x] + Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Csc[a + b*x]^n, x], x] /; Fr
eeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 4185

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> -Simp[(b^2*(c + d*x)*Cot[e + f*x]*
(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2
), x], x] - Simp[(b^2*d*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && G
tQ[n, 1] && NeQ[n, 2]

Rule 4183

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E^(I*(e + f*
x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[(d*m)/f, Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int x^2 \left (a+b \csc ^{-1}(c x)\right )^2 \, dx &=-\frac{\operatorname{Subst}\left (\int (a+b x)^2 \cot (x) \csc ^3(x) \, dx,x,\csc ^{-1}(c x)\right )}{c^3}\\ &=\frac{1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^2-\frac{(2 b) \operatorname{Subst}\left (\int (a+b x) \csc ^3(x) \, dx,x,\csc ^{-1}(c x)\right )}{3 c^3}\\ &=\frac{b^2 x}{3 c^2}+\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )}{3 c}+\frac{1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^2-\frac{b \operatorname{Subst}\left (\int (a+b x) \csc (x) \, dx,x,\csc ^{-1}(c x)\right )}{3 c^3}\\ &=\frac{b^2 x}{3 c^2}+\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )}{3 c}+\frac{1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^2+\frac{2 b \left (a+b \csc ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{3 c^3}+\frac{b^2 \operatorname{Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{3 c^3}-\frac{b^2 \operatorname{Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{3 c^3}\\ &=\frac{b^2 x}{3 c^2}+\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )}{3 c}+\frac{1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^2+\frac{2 b \left (a+b \csc ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{3 c^3}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{3 c^3}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{3 c^3}\\ &=\frac{b^2 x}{3 c^2}+\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )}{3 c}+\frac{1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^2+\frac{2 b \left (a+b \csc ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{3 c^3}-\frac{i b^2 \text{Li}_2\left (-e^{i \csc ^{-1}(c x)}\right )}{3 c^3}+\frac{i b^2 \text{Li}_2\left (e^{i \csc ^{-1}(c x)}\right )}{3 c^3}\\ \end{align*}

Mathematica [A]  time = 1.33869, size = 210, normalized size = 1.51 \[ \frac{1}{3} \left (\frac{b^2 \left (i \text{PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )+c^3 x^3 \csc ^{-1}(c x)^2+\csc ^{-1}(c x) \left (c^2 x^2 \sqrt{1-\frac{1}{c^2 x^2}}-\log \left (1-e^{i \csc ^{-1}(c x)}\right )+\log \left (1+e^{i \csc ^{-1}(c x)}\right )\right )+c x\right )}{c^3}-\frac{i b^2 \text{PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )}{c^3}+a^2 x^3+\frac{a b \left (c^3 x^3+\sqrt{c^2 x^2-1} \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )-c x\right )}{c^4 x \sqrt{1-\frac{1}{c^2 x^2}}}+2 a b x^3 \csc ^{-1}(c x)\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^2*x^3 + 2*a*b*x^3*ArcCsc[c*x] + (a*b*(-(c*x) + c^3*x^3 + Sqrt[-1 + c^2*x^2]*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2
]]))/(c^4*Sqrt[1 - 1/(c^2*x^2)]*x) - (I*b^2*PolyLog[2, -E^(I*ArcCsc[c*x])])/c^3 + (b^2*(c*x + c^3*x^3*ArcCsc[c
*x]^2 + ArcCsc[c*x]*(c^2*Sqrt[1 - 1/(c^2*x^2)]*x^2 - Log[1 - E^(I*ArcCsc[c*x])] + Log[1 + E^(I*ArcCsc[c*x])])
+ I*PolyLog[2, E^(I*ArcCsc[c*x])]))/c^3)/3

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Maple [B]  time = 0.352, size = 327, normalized size = 2.4 \begin{align*}{\frac{{x}^{3}{a}^{2}}{3}}+{\frac{{x}^{3}{b}^{2} \left ({\rm arccsc} \left (cx\right ) \right ) ^{2}}{3}}+{\frac{{b}^{2}{\rm arccsc} \left (cx\right ){x}^{2}}{3\,c}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}+{\frac{{b}^{2}x}{3\,{c}^{2}}}+{\frac{{b}^{2}{\rm arccsc} \left (cx\right )}{3\,{c}^{3}}\ln \left ( 1+{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }-{\frac{{\frac{i}{3}}{b}^{2}}{{c}^{3}}{\it polylog} \left ( 2,{\frac{-i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }-{\frac{{b}^{2}{\rm arccsc} \left (cx\right )}{3\,{c}^{3}}\ln \left ( 1-{\frac{i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }+{\frac{{\frac{i}{3}}{b}^{2}}{{c}^{3}}{\it polylog} \left ( 2,{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }+{\frac{2\,{x}^{3}ab{\rm arccsc} \left (cx\right )}{3}}+{\frac{ab{x}^{2}}{3\,c}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{ab}{3\,{c}^{3}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{ab}{3\,{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}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*a^2+1/3*x^3*b^2*arccsc(c*x)^2+1/3/c*b^2*((c^2*x^2-1)/c^2/x^2)^(1/2)*arccsc(c*x)*x^2+1/3*b^2*x/c^2+1/3/
c^3*b^2*arccsc(c*x)*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2))-1/3*I*b^2*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2))/c^3-1/3/c^
3*b^2*arccsc(c*x)*ln(1-I/c/x-(1-1/c^2/x^2)^(1/2))+1/3*I*b^2*polylog(2,I/c/x+(1-1/c^2/x^2)^(1/2))/c^3+2/3*x^3*a
*b*arccsc(c*x)+1/3/c*a*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^2-1/3/c^3*a*b/((c^2*x^2-1)/c^2/x^2)^(1/2)+1/3/c^4*a*b*(
c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*ln(c*x+(c^2*x^2-1)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*x^3 + 1/6*(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)*a*b + 1/12*(4*x^3*arctan2(1, sqrt(c*x + 1)*sq
rt(c*x - 1))^2 - x^3*log(c^2*x^2)^2 - 2*c^2*(2*(c^2*x^3 + 3*x)/c^4 - 3*log(c*x + 1)/c^5 + 3*log(c*x - 1)/c^5)*
log(c)^2 + 36*c^2*integrate(1/3*x^4*log(c^2*x^2)/(c^2*x^2 - 1), x)*log(c) - 72*c^2*integrate(1/3*x^4*log(x)/(c
^2*x^2 - 1), x)*log(c) + 36*c^2*integrate(1/3*x^4*log(c^2*x^2)*log(x)/(c^2*x^2 - 1), x) - 36*c^2*integrate(1/3
*x^4*log(x)^2/(c^2*x^2 - 1), x) + 12*c^2*integrate(1/3*x^4*log(c^2*x^2)/(c^2*x^2 - 1), x) + 6*(2*x/c^2 - log(c
*x + 1)/c^3 + log(c*x - 1)/c^3)*log(c)^2 - 36*integrate(1/3*x^2*log(c^2*x^2)/(c^2*x^2 - 1), x)*log(c) + 72*int
egrate(1/3*x^2*log(x)/(c^2*x^2 - 1), x)*log(c) + 24*integrate(1/3*sqrt(c*x + 1)*sqrt(c*x - 1)*x^2*arctan(1/(sq
rt(c*x + 1)*sqrt(c*x - 1)))/(c^2*x^2 - 1), x) - 36*integrate(1/3*x^2*log(c^2*x^2)*log(x)/(c^2*x^2 - 1), x) + 3
6*integrate(1/3*x^2*log(x)^2/(c^2*x^2 - 1), x) - 12*integrate(1/3*x^2*log(c^2*x^2)/(c^2*x^2 - 1), x))*b^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(b^2*x^2*arccsc(c*x)^2 + 2*a*b*x^2*arccsc(c*x) + a^2*x^2, x)

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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 )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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