\(\int x^2 (a+b \csc ^{-1}(c x))^2 \, dx\) [16]

Optimal result

Integrand size = 14, antiderivative size = 139 \[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\frac {b^2 x}{3 c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )}{3 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^2+\frac {2 b \left (a+b \csc ^{-1}(c x)\right ) \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right )}{3 c^3}-\frac {i b^2 \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )}{3 c^3}+\frac {i b^2 \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )}{3 c^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5331, 4495, 4270, 4268, 2317, 2438} \[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\frac {2 b \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{3 c^3}+\frac {b x^2 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{3 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^2-\frac {i b^2 \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )}{3 c^3}+\frac {i b^2 \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )}{3 c^3}+\frac {b^2 x}{3 c^2} \]

[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 2317

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 2438

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]

Rule 4268

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 4270

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] &&
GtQ[n, 1] && NeQ[n, 2]

Rule 4495

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] /;
FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 5331

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])

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {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) \text {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 \text {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 ) \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right )}{3 c^3}+\frac {b^2 \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{3 c^3}-\frac {b^2 \text {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 ) \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right )}{3 c^3}-\frac {\left (i b^2\right ) \text {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 ) \text {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 ) \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right )}{3 c^3}-\frac {i b^2 \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )}{3 c^3}+\frac {i b^2 \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )}{3 c^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.00 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.53 \[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\frac {1}{3} \left (a^2 x^3+2 a b x^3 \csc ^{-1}(c x)+\frac {a b \left (-c x+c^3 x^3-\sqrt {-1+c^2 x^2} \log \left (-c x+\sqrt {-1+c^2 x^2}\right )\right )}{c^4 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {i b^2 \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^2 \left (c x+c^3 x^3 \csc ^{-1}(c x)^2+\csc ^{-1}(c x) \left (c^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2-\log \left (1-e^{i \csc ^{-1}(c x)}\right )+\log \left (1+e^{i \csc ^{-1}(c x)}\right )\right )+i \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )\right )}{c^3}\right ) \]

[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]*Log[-(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

Maple [A] (verified)

Time = 1.44 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.93

method	result	size
parts	\(\frac {a^{2} x^{3}}{3}+\frac {b^{2} \left (\frac {\left (c^{2} x^{2} \operatorname {arccsc}\left (c x \right )^{2}+\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+1\right ) c x}{3}-\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}+\frac {i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}+\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}-\frac {i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}\right )}{c^{3}}+\frac {2 a b \left (\frac {c^{3} x^{3} \operatorname {arccsc}\left (c x \right )}{3}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (c x \sqrt {c^{2} x^{2}-1}+\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}\)	\(268\)
derivativedivides	\(\frac {\frac {c^{3} x^{3} a^{2}}{3}+b^{2} \left (\frac {\left (c^{2} x^{2} \operatorname {arccsc}\left (c x \right )^{2}+\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+1\right ) c x}{3}-\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}+\frac {i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}+\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}-\frac {i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}\right )+2 a b \left (\frac {c^{3} x^{3} \operatorname {arccsc}\left (c x \right )}{3}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (c x \sqrt {c^{2} x^{2}-1}+\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}\)	\(269\)
default	\(\frac {\frac {c^{3} x^{3} a^{2}}{3}+b^{2} \left (\frac {\left (c^{2} x^{2} \operatorname {arccsc}\left (c x \right )^{2}+\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+1\right ) c x}{3}-\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}+\frac {i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}+\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}-\frac {i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}\right )+2 a b \left (\frac {c^{3} x^{3} \operatorname {arccsc}\left (c x \right )}{3}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (c x \sqrt {c^{2} x^{2}-1}+\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}\)	\(269\)

[In]

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

[Out]

1/3*a^2*x^3+b^2/c^3*(1/3*(c^2*x^2*arccsc(c*x)^2+arccsc(c*x)*c*x*((c^2*x^2-1)/c^2/x^2)^(1/2)+1)*c*x-1/3*arccsc(
c*x)*ln(1-I/c/x-(1-1/c^2/x^2)^(1/2))+1/3*I*polylog(2,I/c/x+(1-1/c^2/x^2)^(1/2))+1/3*arccsc(c*x)*ln(1+I/c/x+(1-
1/c^2/x^2)^(1/2))-1/3*I*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2)))+2*a*b/c^3*(1/3*c^3*x^3*arccsc(c*x)+1/6*(c^2*x^2
-1)^(1/2)*(c*x*(c^2*x^2-1)^(1/2)+ln(c*x+(c^2*x^2-1)^(1/2)))/((c^2*x^2-1)/c^2/x^2)^(1/2)/c/x)

Fricas [F]

\[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\int { {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]

[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)

Sympy [F]

\[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\int x^{2} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{2}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\int { {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]

[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

Giac [F]

\[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\int { {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \]

[In]

int(x^2*(a + b*asin(1/(c*x)))^2,x)

[Out]

int(x^2*(a + b*asin(1/(c*x)))^2, x)