Integrand size = 12, antiderivative size = 64 \[ \int x^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2}{6 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {b \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{6 c^3} \] Output:
1/6*b*(1-1/c^2/x^2)^(1/2)*x^2/c+1/3*x^3*(a+b*arccsc(c*x))+1/6*b*arctanh((1 -1/c^2/x^2)^(1/2))/c^3
Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.33 \[ \int x^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {a x^3}{3}+\frac {b x^2 \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}}{6 c}+\frac {1}{3} b x^3 \csc ^{-1}(c x)+\frac {b \log \left (x \left (1+\sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}\right )\right )}{6 c^3} \] Input:
Integrate[x^2*(a + b*ArcCsc[c*x]),x]
Output:
(a*x^3)/3 + (b*x^2*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])/(6*c) + (b*x^3*ArcCsc[c *x])/3 + (b*Log[x*(1 + Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])])/(6*c^3)
Time = 0.22 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5744, 798, 52, 73, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx\) |
\(\Big \downarrow \) 5744 |
\(\displaystyle \frac {b \int \frac {x}{\sqrt {1-\frac {1}{c^2 x^2}}}dx}{3 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )-\frac {b \int \frac {x^4}{\sqrt {1-\frac {1}{c^2 x^2}}}d\frac {1}{x^2}}{6 c}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )-\frac {b \left (\frac {\int \frac {x^2}{\sqrt {1-\frac {1}{c^2 x^2}}}d\frac {1}{x^2}}{2 c^2}-x^2 \sqrt {1-\frac {1}{c^2 x^2}}\right )}{6 c}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )-\frac {b \left (x^2 \left (-\sqrt {1-\frac {1}{c^2 x^2}}\right )-\int \frac {1}{c^2-\frac {c^2}{x^4}}d\sqrt {1-\frac {1}{c^2 x^2}}\right )}{6 c}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )-\frac {b \left (x^2 \left (-\sqrt {1-\frac {1}{c^2 x^2}}\right )-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^2}\right )}{6 c}\) |
Input:
Int[x^2*(a + b*ArcCsc[c*x]),x]
Output:
(x^3*(a + b*ArcCsc[c*x]))/3 - (b*(-(Sqrt[1 - 1/(c^2*x^2)]*x^2) - ArcTanh[S qrt[1 - 1/(c^2*x^2)]]/c^2))/(6*c)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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]]
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Sim p[(d*x)^(m + 1)*((a + b*ArcCsc[c*x])/(d*(m + 1))), x] + Simp[b*(d/(c*(m + 1 ))) Int[(d*x)^(m - 1)/Sqrt[1 - 1/(c^2*x^2)], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.47
method | result | size |
parts | \(\frac {a \,x^{3}}{3}+\frac {b \left (\frac {c^{3} x^{3} \operatorname {arccsc}\left (c x \right )}{3}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (c x \sqrt {c^{2} x^{2}-1}+\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}\) | \(94\) |
derivativedivides | \(\frac {\frac {a \,c^{3} x^{3}}{3}+b \left (\frac {c^{3} x^{3} \operatorname {arccsc}\left (c x \right )}{3}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (c x \sqrt {c^{2} x^{2}-1}+\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}\) | \(98\) |
default | \(\frac {\frac {a \,c^{3} x^{3}}{3}+b \left (\frac {c^{3} x^{3} \operatorname {arccsc}\left (c x \right )}{3}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (c x \sqrt {c^{2} x^{2}-1}+\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}\) | \(98\) |
Input:
int(x^2*(a+b*arccsc(c*x)),x,method=_RETURNVERBOSE)
Output:
1/3*a*x^3+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)
Time = 0.14 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.47 \[ \int x^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {2 \, a c^{3} x^{3} - 4 \, b c^{3} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + \sqrt {c^{2} x^{2} - 1} b c x + 2 \, {\left (b c^{3} x^{3} - b c^{3}\right )} \operatorname {arccsc}\left (c x\right ) - b \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{6 \, c^{3}} \] Input:
integrate(x^2*(a+b*arccsc(c*x)),x, algorithm="fricas")
Output:
1/6*(2*a*c^3*x^3 - 4*b*c^3*arctan(-c*x + sqrt(c^2*x^2 - 1)) + sqrt(c^2*x^2 - 1)*b*c*x + 2*(b*c^3*x^3 - b*c^3)*arccsc(c*x) - b*log(-c*x + sqrt(c^2*x^ 2 - 1)))/c^3
Time = 2.31 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.67 \[ \int x^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {a x^{3}}{3} + \frac {b x^{3} \operatorname {acsc}{\left (c x \right )}}{3} + \frac {b \left (\begin {cases} \frac {x \sqrt {c^{2} x^{2} - 1}}{2 c} + \frac {\operatorname {acosh}{\left (c x \right )}}{2 c^{2}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{3}}{2 \sqrt {- c^{2} x^{2} + 1}} + \frac {i x}{2 c \sqrt {- c^{2} x^{2} + 1}} - \frac {i \operatorname {asin}{\left (c x \right )}}{2 c^{2}} & \text {otherwise} \end {cases}\right )}{3 c} \] Input:
integrate(x**2*(a+b*acsc(c*x)),x)
Output:
a*x**3/3 + b*x**3*acsc(c*x)/3 + b*Piecewise((x*sqrt(c**2*x**2 - 1)/(2*c) + acosh(c*x)/(2*c**2), Abs(c**2*x**2) > 1), (-I*c*x**3/(2*sqrt(-c**2*x**2 + 1)) + I*x/(2*c*sqrt(-c**2*x**2 + 1)) - I*asin(c*x)/(2*c**2), True))/(3*c)
Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.52 \[ \int x^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{3} \, a x^{3} + \frac {1}{12} \, {\left (4 \, x^{3} \operatorname {arccsc}\left (c x\right ) + \frac {\frac {2 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b \] Input:
integrate(x^2*(a+b*arccsc(c*x)),x, algorithm="maxima")
Output:
1/3*a*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
Leaf count of result is larger than twice the leaf count of optimal. 310 vs. \(2 (54) = 108\).
Time = 0.29 (sec) , antiderivative size = 310, normalized size of antiderivative = 4.84 \[ \int x^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{24} \, {\left (\frac {b x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {a x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}{c} + \frac {b x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{2}} + \frac {3 \, b x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {3 \, a x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{3}} + \frac {4 \, b \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} - \frac {4 \, b \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{4}} + \frac {3 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{5} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {3 \, a}{c^{5} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} - \frac {b}{c^{6} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {b \arcsin \left (\frac {1}{c x}\right )}{c^{7} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {a}{c^{7} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}\right )} c \] Input:
integrate(x^2*(a+b*arccsc(c*x)),x, algorithm="giac")
Output:
1/24*(b*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3*arcsin(1/(c*x))/c + a*x^3*(sqrt (-1/(c^2*x^2) + 1) + 1)^3/c + b*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2/c^2 + 3 *b*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*arcsin(1/(c*x))/c^3 + 3*a*x*(sqrt(-1/(c^ 2*x^2) + 1) + 1)/c^3 + 4*b*log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^4 - 4*b*log(1 /(abs(c)*abs(x)))/c^4 + 3*b*arcsin(1/(c*x))/(c^5*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3*a/(c^5*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) - b/(c^6*x^2*(sqrt(-1/(c ^2*x^2) + 1) + 1)^2) + b*arcsin(1/(c*x))/(c^7*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + a/(c^7*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3))*c
Timed out. \[ \int x^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int x^2\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \] Input:
int(x^2*(a + b*asin(1/(c*x))),x)
Output:
int(x^2*(a + b*asin(1/(c*x))), x)
\[ \int x^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\left (\int \mathit {acsc} \left (c x \right ) x^{2}d x \right ) b +\frac {a \,x^{3}}{3} \] Input:
int(x^2*(a+b*acsc(c*x)),x)
Output:
(3*int(acsc(c*x)*x**2,x)*b + a*x**3)/3