Integrand size = 12, antiderivative size = 64 \[ \int x^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right ) \] Output:
1/6*b*(1-1/c^2/x^2)^(1/2)*x/c^3+1/12*b*(1-1/c^2/x^2)^(1/2)*x^3/c+1/4*x^4*( a+b*arccsc(c*x))
Time = 0.07 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.97 \[ \int x^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {a x^4}{4}+b \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}} \left (\frac {x}{6 c^3}+\frac {x^3}{12 c}\right )+\frac {1}{4} b x^4 \csc ^{-1}(c x) \] Input:
Integrate[x^3*(a + b*ArcCsc[c*x]),x]
Output:
(a*x^4)/4 + b*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)]*(x/(6*c^3) + x^3/(12*c)) + (b *x^4*ArcCsc[c*x])/4
Time = 0.22 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5744, 803, 746}
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.
\(\displaystyle \int x^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx\) |
\(\Big \downarrow \) 5744 |
\(\displaystyle \frac {b \int \frac {x^2}{\sqrt {1-\frac {1}{c^2 x^2}}}dx}{4 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )\) |
\(\Big \downarrow \) 803 |
\(\displaystyle \frac {b \left (\frac {2 \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}}}dx}{3 c^2}+\frac {1}{3} x^3 \sqrt {1-\frac {1}{c^2 x^2}}\right )}{4 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )\) |
\(\Big \downarrow \) 746 |
\(\displaystyle \frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {b \left (\frac {2 x \sqrt {1-\frac {1}{c^2 x^2}}}{3 c^2}+\frac {1}{3} x^3 \sqrt {1-\frac {1}{c^2 x^2}}\right )}{4 c}\) |
Input:
Int[x^3*(a + b*ArcCsc[c*x]),x]
Output:
(b*((2*Sqrt[1 - 1/(c^2*x^2)]*x)/(3*c^2) + (Sqrt[1 - 1/(c^2*x^2)]*x^3)/3))/ (4*c) + (x^4*(a + b*ArcCsc[c*x]))/4
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1) /a), x] /; FreeQ[{a, b, n, p}, x] && EqQ[1/n + p + 1, 0]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*(( a + b*x^n)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*(m + 1 ))) Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x] && I LtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]
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 = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.09
method | result | size |
parts | \(\frac {a \,x^{4}}{4}+\frac {b \left (\frac {c^{4} x^{4} \operatorname {arccsc}\left (c x \right )}{4}+\frac {\left (c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+2\right )}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}\) | \(70\) |
derivativedivides | \(\frac {\frac {a \,c^{4} x^{4}}{4}+b \left (\frac {c^{4} x^{4} \operatorname {arccsc}\left (c x \right )}{4}+\frac {\left (c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+2\right )}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}\) | \(74\) |
default | \(\frac {\frac {a \,c^{4} x^{4}}{4}+b \left (\frac {c^{4} x^{4} \operatorname {arccsc}\left (c x \right )}{4}+\frac {\left (c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+2\right )}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}\) | \(74\) |
Input:
int(x^3*(a+b*arccsc(c*x)),x,method=_RETURNVERBOSE)
Output:
1/4*a*x^4+b/c^4*(1/4*c^4*x^4*arccsc(c*x)+1/12*(c^2*x^2-1)*(c^2*x^2+2)/((c^ 2*x^2-1)/c^2/x^2)^(1/2)/c/x)
Time = 0.12 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.81 \[ \int x^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {3 \, b c^{4} x^{4} \operatorname {arccsc}\left (c x\right ) + 3 \, a c^{4} x^{4} + {\left (b c^{2} x^{2} + 2 \, b\right )} \sqrt {c^{2} x^{2} - 1}}{12 \, c^{4}} \] Input:
integrate(x^3*(a+b*arccsc(c*x)),x, algorithm="fricas")
Output:
1/12*(3*b*c^4*x^4*arccsc(c*x) + 3*a*c^4*x^4 + (b*c^2*x^2 + 2*b)*sqrt(c^2*x ^2 - 1))/c^4
Time = 1.61 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.67 \[ \int x^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {a x^{4}}{4} + \frac {b x^{4} \operatorname {acsc}{\left (c x \right )}}{4} + \frac {b \left (\begin {cases} \frac {x^{2} \sqrt {c^{2} x^{2} - 1}}{3 c} + \frac {2 \sqrt {c^{2} x^{2} - 1}}{3 c^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {2 i \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} & \text {otherwise} \end {cases}\right )}{4 c} \] Input:
integrate(x**3*(a+b*acsc(c*x)),x)
Output:
a*x**4/4 + b*x**4*acsc(c*x)/4 + b*Piecewise((x**2*sqrt(c**2*x**2 - 1)/(3*c ) + 2*sqrt(c**2*x**2 - 1)/(3*c**3), Abs(c**2*x**2) > 1), (I*x**2*sqrt(-c** 2*x**2 + 1)/(3*c) + 2*I*sqrt(-c**2*x**2 + 1)/(3*c**3), True))/(4*c)
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.92 \[ \int x^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{4} \, a x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arccsc}\left (c x\right ) + \frac {c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 3 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b \] Input:
integrate(x^3*(a+b*arccsc(c*x)),x, algorithm="maxima")
Output:
1/4*a*x^4 + 1/12*(3*x^4*arccsc(c*x) + (c^2*x^3*(-1/(c^2*x^2) + 1)^(3/2) + 3*x*sqrt(-1/(c^2*x^2) + 1))/c^3)*b
Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (54) = 108\).
Time = 0.14 (sec) , antiderivative size = 352, normalized size of antiderivative = 5.50 \[ \int x^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{192} \, {\left (\frac {3 \, b x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {3 \, a x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}}{c} + \frac {2 \, b x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}{c^{2}} + \frac {12 \, b x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {12 \, a x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{3}} + \frac {18 \, b x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{4}} + \frac {18 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{5}} + \frac {18 \, a}{c^{5}} - \frac {18 \, b}{c^{6} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {12 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{7} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {12 \, a}{c^{7} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} - \frac {2 \, b}{c^{8} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {3 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{9} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}} + \frac {3 \, a}{c^{9} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}}\right )} c \] Input:
integrate(x^3*(a+b*arccsc(c*x)),x, algorithm="giac")
Output:
1/192*(3*b*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*arcsin(1/(c*x))/c + 3*a*x^4* (sqrt(-1/(c^2*x^2) + 1) + 1)^4/c + 2*b*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3/ c^2 + 12*b*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*arcsin(1/(c*x))/c^3 + 12*a*x ^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2/c^3 + 18*b*x*(sqrt(-1/(c^2*x^2) + 1) + 1 )/c^4 + 18*b*arcsin(1/(c*x))/c^5 + 18*a/c^5 - 18*b/(c^6*x*(sqrt(-1/(c^2*x^ 2) + 1) + 1)) + 12*b*arcsin(1/(c*x))/(c^7*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1) ^2) + 12*a/(c^7*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) - 2*b/(c^8*x^3*(sqrt(- 1/(c^2*x^2) + 1) + 1)^3) + 3*b*arcsin(1/(c*x))/(c^9*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4) + 3*a/(c^9*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4))*c
Timed out. \[ \int x^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int x^3\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \] Input:
int(x^3*(a + b*asin(1/(c*x))),x)
Output:
int(x^3*(a + b*asin(1/(c*x))), x)
\[ \int x^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\left (\int \mathit {acsc} \left (c x \right ) x^{3}d x \right ) b +\frac {a \,x^{4}}{4} \] Input:
int(x^3*(a+b*acsc(c*x)),x)
Output:
(4*int(acsc(c*x)*x**3,x)*b + a*x**4)/4