3.26 \(\int x (a+b \csc ^{-1}(c x))^3 \, dx\)

Optimal. Leaf size=126 \[ \frac{3 i b^3 \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 c^2}-\frac{3 b^2 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac{3 b x \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac{3 i b \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^2}+\frac{1}{2} x^2 \left (a+b \csc ^{-1}(c x)\right )^3 \]

[Out]

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

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Rubi [A]  time = 0.147319, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {5223, 4410, 4184, 3717, 2190, 2279, 2391} \[ \frac{3 i b^3 \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 c^2}-\frac{3 b^2 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac{3 b x \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac{3 i b \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^2}+\frac{1}{2} x^2 \left (a+b \csc ^{-1}(c x)\right )^3 \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcCsc[c*x])^3,x]

[Out]

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

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 4184

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

Rule 3717

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[(I*(c + d*x)^(m + 1))/(d*
(m + 1)), x] - Dist[2*I, Int[((c + d*x)^m*E^(2*I*k*Pi)*E^(2*I*(e + f*x)))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, 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 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx &=-\frac{\operatorname{Subst}\left (\int (a+b x)^3 \cot (x) \csc ^2(x) \, dx,x,\csc ^{-1}(c x)\right )}{c^2}\\ &=\frac{1}{2} x^2 \left (a+b \csc ^{-1}(c x)\right )^3-\frac{(3 b) \operatorname{Subst}\left (\int (a+b x)^2 \csc ^2(x) \, dx,x,\csc ^{-1}(c x)\right )}{2 c^2}\\ &=\frac{3 b \sqrt{1-\frac{1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac{1}{2} x^2 \left (a+b \csc ^{-1}(c x)\right )^3-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int (a+b x) \cot (x) \, dx,x,\csc ^{-1}(c x)\right )}{c^2}\\ &=\frac{3 i b \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^2}+\frac{3 b \sqrt{1-\frac{1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac{1}{2} x^2 \left (a+b \csc ^{-1}(c x)\right )^3+\frac{\left (6 i b^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(c x)\right )}{c^2}\\ &=\frac{3 i b \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^2}+\frac{3 b \sqrt{1-\frac{1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac{1}{2} x^2 \left (a+b \csc ^{-1}(c x)\right )^3-\frac{3 b^2 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{c^2}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c^2}\\ &=\frac{3 i b \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^2}+\frac{3 b \sqrt{1-\frac{1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac{1}{2} x^2 \left (a+b \csc ^{-1}(c x)\right )^3-\frac{3 b^2 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{c^2}-\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-1}(c x)}\right )}{2 c^2}\\ &=\frac{3 i b \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^2}+\frac{3 b \sqrt{1-\frac{1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac{1}{2} x^2 \left (a+b \csc ^{-1}(c x)\right )^3-\frac{3 b^2 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{c^2}+\frac{3 i b^3 \text{Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )}{2 c^2}\\ \end{align*}

Mathematica [A]  time = 0.46718, size = 182, normalized size = 1.44 \[ \frac{3 i b^3 \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )+a \left (a c x \left (a c x+3 b \sqrt{1-\frac{1}{c^2 x^2}}\right )-6 b^2 \log \left (\frac{1}{c x}\right )\right )+3 b^2 \csc ^{-1}(c x)^2 \left (a c^2 x^2+b \left (c x \sqrt{1-\frac{1}{c^2 x^2}}+i\right )\right )+3 b \csc ^{-1}(c x) \left (a c x \left (a c x+2 b \sqrt{1-\frac{1}{c^2 x^2}}\right )-2 b^2 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )\right )+b^3 c^2 x^2 \csc ^{-1}(c x)^3}{2 c^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x*(a + b*ArcCsc[c*x])^3,x]

[Out]

(3*b^2*(a*c^2*x^2 + b*(I + c*Sqrt[1 - 1/(c^2*x^2)]*x))*ArcCsc[c*x]^2 + b^3*c^2*x^2*ArcCsc[c*x]^3 + 3*b*ArcCsc[
c*x]*(a*c*x*(2*b*Sqrt[1 - 1/(c^2*x^2)] + a*c*x) - 2*b^2*Log[1 - E^((2*I)*ArcCsc[c*x])]) + a*(a*c*x*(3*b*Sqrt[1
- 1/(c^2*x^2)] + a*c*x) - 6*b^2*Log[1/(c*x)]) + (3*I)*b^3*PolyLog[2, E^((2*I)*ArcCsc[c*x])])/(2*c^2)

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Maple [B]  time = 0.33, size = 349, normalized size = 2.8 \begin{align*}{\frac{{x}^{2}{a}^{3}}{2}}+{\frac{{x}^{2}{b}^{3} \left ({\rm arccsc} \left (cx\right ) \right ) ^{3}}{2}}+{\frac{3\,{b}^{3} \left ({\rm arccsc} \left (cx\right ) \right ) ^{2}x}{2\,c}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}+{\frac{{\frac{3\,i}{2}}{b}^{3} \left ({\rm arccsc} \left (cx\right ) \right ) ^{2}}{{c}^{2}}}-3\,{\frac{{b}^{3}{\rm arccsc} \left (cx\right )}{{c}^{2}}\ln \left ( 1+{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }-3\,{\frac{{b}^{3}{\rm arccsc} \left (cx\right )}{{c}^{2}}\ln \left ( 1-{\frac{i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }+{\frac{3\,i{b}^{3}}{{c}^{2}}{\it polylog} \left ( 2,{\frac{-i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }+{\frac{3\,i{b}^{3}}{{c}^{2}}{\it polylog} \left ( 2,{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }+{\frac{3\,{x}^{2}a{b}^{2} \left ({\rm arccsc} \left (cx\right ) \right ) ^{2}}{2}}+3\,{\frac{a{b}^{2}{\rm arccsc} \left (cx\right )x}{c}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}-3\,{\frac{a{b}^{2}}{{c}^{2}}\ln \left ({\frac{1}{cx}} \right ) }+{\frac{3\,{a}^{2}b{x}^{2}{\rm arccsc} \left (cx\right )}{2}}+{\frac{3\,{a}^{2}bx}{2\,c}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{3\,{a}^{2}b}{2\,{c}^{3}x}{\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*(a+b*arccsc(c*x))^3,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{3}{2} \, a b^{2} x^{2} \operatorname{arccsc}\left (c x\right )^{2} + \frac{1}{2} \, a^{3} x^{2} + \frac{3}{2} \,{\left (x^{2} \operatorname{arccsc}\left (c x\right ) + \frac{x \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c}\right )} a^{2} b + 3 \,{\left (\frac{x \sqrt{-\frac{1}{c^{2} x^{2}} + 1} \operatorname{arccsc}\left (c x\right )}{c} + \frac{\log \left (x\right )}{c^{2}}\right )} a b^{2} + \frac{1}{8} \,{\left (4 \, x^{2} \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right )^{3} - 3 \, x^{2} \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right ) \log \left (c^{2} x^{2}\right )^{2} - 8 \, \int \frac{3 \,{\left (8 \, c^{2} x^{3} \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right ) \log \left (c\right )^{2} - 8 \, x \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right ) \log \left (c\right )^{2} + 8 \,{\left (c^{2} x^{3} \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right ) - x \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right )\right )} \log \left (x\right )^{2} -{\left (4 \, x \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right )^{2} - x \log \left (c^{2} x^{2}\right )^{2}\right )} \sqrt{c x + 1} \sqrt{c x - 1} - 4 \,{\left ({\left (2 \, c^{2} \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right ) \log \left (c\right ) + c^{2} \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right )\right )} x^{3} -{\left (2 \, \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right ) \log \left (c\right ) + \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right )\right )} x + 2 \,{\left (c^{2} x^{3} \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right ) - x \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right )\right )} \log \left (x\right )\right )} \log \left (c^{2} x^{2}\right ) + 16 \,{\left (c^{2} x^{3} \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right ) \log \left (c\right ) - x \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right ) \log \left (c\right )\right )} \log \left (x\right )\right )}}{8 \,{\left (c^{2} x^{2} - 1\right )}}\,{d x}\right )} b^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*a*b^2*x^2*arccsc(c*x)^2 + 1/2*a^3*x^2 + 3/2*(x^2*arccsc(c*x) + x*sqrt(-1/(c^2*x^2) + 1)/c)*a^2*b + 3*(x*sq
rt(-1/(c^2*x^2) + 1)*arccsc(c*x)/c + log(x)/c^2)*a*b^2 + 1/8*(4*x^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^3
- 3*x^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c^2*x^2)^2 - 8*integrate(3/8*(8*c^2*x^3*arctan2(1, sqrt(c*
x + 1)*sqrt(c*x - 1))*log(c)^2 - 8*x*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c)^2 + 8*(c^2*x^3*arctan2(1,
sqrt(c*x + 1)*sqrt(c*x - 1)) - x*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)^2 - (4*x*arctan2(1, sqrt(c*x
+ 1)*sqrt(c*x - 1))^2 - x*log(c^2*x^2)^2)*sqrt(c*x + 1)*sqrt(c*x - 1) - 4*((2*c^2*arctan2(1, sqrt(c*x + 1)*sqr
t(c*x - 1))*log(c) + c^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*x^3 - (2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x -
1))*log(c) + arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*x + 2*(c^2*x^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))
- x*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x))*log(c^2*x^2) + 16*(c^2*x^3*arctan2(1, sqrt(c*x + 1)*sqrt(
c*x - 1))*log(c) - x*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c))*log(x))/(c^2*x^2 - 1), x))*b^3

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{3} x \operatorname{arccsc}\left (c x\right )^{3} + 3 \, a b^{2} x \operatorname{arccsc}\left (c x\right )^{2} + 3 \, a^{2} b x \operatorname{arccsc}\left (c x\right ) + a^{3} x, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(b^3*x*arccsc(c*x)^3 + 3*a*b^2*x*arccsc(c*x)^2 + 3*a^2*b*x*arccsc(c*x) + a^3*x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (a + b \operatorname{acsc}{\left (c x \right )}\right )^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*(a + b*acsc(c*x))**3, 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 )}^{3} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arccsc(c*x) + a)^3*x, x)