3.2 \(\int x^5 (a+b \csc ^{-1}(c x)) \, dx\)

Optimal. Leaf size=89 \[ \frac {1}{6} x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {b x^5 \sqrt {1-\frac {1}{c^2 x^2}}}{30 c}+\frac {4 b x \sqrt {1-\frac {1}{c^2 x^2}}}{45 c^5}+\frac {2 b x^3 \sqrt {1-\frac {1}{c^2 x^2}}}{45 c^3} \]

[Out]

1/6*x^6*(a+b*arccsc(c*x))+4/45*b*x*(1-1/c^2/x^2)^(1/2)/c^5+2/45*b*x^3*(1-1/c^2/x^2)^(1/2)/c^3+1/30*b*x^5*(1-1/
c^2/x^2)^(1/2)/c

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Rubi [A]  time = 0.04, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5221, 271, 191} \[ \frac {1}{6} x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {b x^5 \sqrt {1-\frac {1}{c^2 x^2}}}{30 c}+\frac {2 b x^3 \sqrt {1-\frac {1}{c^2 x^2}}}{45 c^3}+\frac {4 b x \sqrt {1-\frac {1}{c^2 x^2}}}{45 c^5} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*ArcCsc[c*x]),x]

[Out]

(4*b*Sqrt[1 - 1/(c^2*x^2)]*x)/(45*c^5) + (2*b*Sqrt[1 - 1/(c^2*x^2)]*x^3)/(45*c^3) + (b*Sqrt[1 - 1/(c^2*x^2)]*x
^5)/(30*c) + (x^6*(a + b*ArcCsc[c*x]))/6

Rule 191

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]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(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]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 5221

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCsc[c*x]
))/(d*(m + 1)), x] + Dist[(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]

Rubi steps

\begin {align*} \int x^5 \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac {1}{6} x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {b \int \frac {x^4}{\sqrt {1-\frac {1}{c^2 x^2}}} \, dx}{6 c}\\ &=\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^5}{30 c}+\frac {1}{6} x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {(2 b) \int \frac {x^2}{\sqrt {1-\frac {1}{c^2 x^2}}} \, dx}{15 c^3}\\ &=\frac {2 b \sqrt {1-\frac {1}{c^2 x^2}} x^3}{45 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^5}{30 c}+\frac {1}{6} x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {(4 b) \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}}} \, dx}{45 c^5}\\ &=\frac {4 b \sqrt {1-\frac {1}{c^2 x^2}} x}{45 c^5}+\frac {2 b \sqrt {1-\frac {1}{c^2 x^2}} x^3}{45 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^5}{30 c}+\frac {1}{6} x^6 \left (a+b \csc ^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 72, normalized size = 0.81 \[ \frac {a x^6}{6}+b \sqrt {\frac {c^2 x^2-1}{c^2 x^2}} \left (\frac {4 x}{45 c^5}+\frac {2 x^3}{45 c^3}+\frac {x^5}{30 c}\right )+\frac {1}{6} b x^6 \csc ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*ArcCsc[c*x]),x]

[Out]

(a*x^6)/6 + b*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)]*((4*x)/(45*c^5) + (2*x^3)/(45*c^3) + x^5/(30*c)) + (b*x^6*ArcCsc[
c*x])/6

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fricas [A]  time = 1.39, size = 62, normalized size = 0.70 \[ \frac {15 \, b c^{6} x^{6} \operatorname {arccsc}\left (c x\right ) + 15 \, a c^{6} x^{6} + {\left (3 \, b c^{4} x^{4} + 4 \, b c^{2} x^{2} + 8 \, b\right )} \sqrt {c^{2} x^{2} - 1}}{90 \, c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arccsc(c*x)),x, algorithm="fricas")

[Out]

1/90*(15*b*c^6*x^6*arccsc(c*x) + 15*a*c^6*x^6 + (3*b*c^4*x^4 + 4*b*c^2*x^2 + 8*b)*sqrt(c^2*x^2 - 1))/c^6

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giac [B]  time = 0.21, size = 518, normalized size = 5.82 \[ \frac {1}{5760} \, {\left (\frac {15 \, b x^{6} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{6} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {15 \, a x^{6} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{6}}{c} + \frac {6 \, b x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}}{c^{2}} + \frac {90 \, b x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4} \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {90 \, a x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}}{c^{3}} + \frac {50 \, b x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}{c^{4}} + \frac {225 \, b x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{c^{5}} + \frac {225 \, a x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{5}} + \frac {300 \, b x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{6}} + \frac {300 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{7}} + \frac {300 \, a}{c^{7}} - \frac {300 \, b}{c^{8} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {225 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{9} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {225 \, a}{c^{9} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} - \frac {50 \, b}{c^{10} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {90 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{11} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}} + \frac {90 \, a}{c^{11} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}} - \frac {6 \, b}{c^{12} x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}} + \frac {15 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{13} x^{6} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{6}} + \frac {15 \, a}{c^{13} x^{6} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{6}}\right )} c \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arccsc(c*x)),x, algorithm="giac")

[Out]

1/5760*(15*b*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6*arcsin(1/(c*x))/c + 15*a*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6/c
+ 6*b*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5/c^2 + 90*b*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*arcsin(1/(c*x))/c^3 + 9
0*a*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4/c^3 + 50*b*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3/c^4 + 225*b*x^2*(sqrt(-1/
(c^2*x^2) + 1) + 1)^2*arcsin(1/(c*x))/c^5 + 225*a*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2/c^5 + 300*b*x*(sqrt(-1/(c
^2*x^2) + 1) + 1)/c^6 + 300*b*arcsin(1/(c*x))/c^7 + 300*a/c^7 - 300*b/(c^8*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 2
25*b*arcsin(1/(c*x))/(c^9*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) + 225*a/(c^9*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2)
- 50*b/(c^10*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 90*b*arcsin(1/(c*x))/(c^11*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1
)^4) + 90*a/(c^11*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4) - 6*b/(c^12*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 15*b*a
rcsin(1/(c*x))/(c^13*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6) + 15*a/(c^13*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6))*c

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maple [A]  time = 0.05, size = 83, normalized size = 0.93 \[ \frac {\frac {c^{6} x^{6} a}{6}+b \left (\frac {c^{6} x^{6} \mathrm {arccsc}\left (c x \right )}{6}+\frac {\left (c^{2} x^{2}-1\right ) \left (3 c^{4} x^{4}+4 c^{2} x^{2}+8\right )}{90 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*arccsc(c*x)),x)

[Out]

1/c^6*(1/6*c^6*x^6*a+b*(1/6*c^6*x^6*arccsc(c*x)+1/90*(c^2*x^2-1)*(3*c^4*x^4+4*c^2*x^2+8)/((c^2*x^2-1)/c^2/x^2)
^(1/2)/c/x))

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maxima [A]  time = 0.35, size = 80, normalized size = 0.90 \[ \frac {1}{6} \, a x^{6} + \frac {1}{90} \, {\left (15 \, x^{6} \operatorname {arccsc}\left (c x\right ) + \frac {3 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 10 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arccsc(c*x)),x, algorithm="maxima")

[Out]

1/6*a*x^6 + 1/90*(15*x^6*arccsc(c*x) + (3*c^4*x^5*(-1/(c^2*x^2) + 1)^(5/2) + 10*c^2*x^3*(-1/(c^2*x^2) + 1)^(3/
2) + 15*x*sqrt(-1/(c^2*x^2) + 1))/c^5)*b

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^5\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b*asin(1/(c*x))),x)

[Out]

int(x^5*(a + b*asin(1/(c*x))), x)

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sympy [A]  time = 4.42, size = 153, normalized size = 1.72 \[ \frac {a x^{6}}{6} + \frac {b x^{6} \operatorname {acsc}{\left (c x \right )}}{6} + \frac {b \left (\begin {cases} \frac {x^{4} \sqrt {c^{2} x^{2} - 1}}{5 c} + \frac {4 x^{2} \sqrt {c^{2} x^{2} - 1}}{15 c^{3}} + \frac {8 \sqrt {c^{2} x^{2} - 1}}{15 c^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{4} \sqrt {- c^{2} x^{2} + 1}}{5 c} + \frac {4 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{15 c^{3}} + \frac {8 i \sqrt {- c^{2} x^{2} + 1}}{15 c^{5}} & \text {otherwise} \end {cases}\right )}{6 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(a+b*acsc(c*x)),x)

[Out]

a*x**6/6 + b*x**6*acsc(c*x)/6 + b*Piecewise((x**4*sqrt(c**2*x**2 - 1)/(5*c) + 4*x**2*sqrt(c**2*x**2 - 1)/(15*c
**3) + 8*sqrt(c**2*x**2 - 1)/(15*c**5), Abs(c**2*x**2) > 1), (I*x**4*sqrt(-c**2*x**2 + 1)/(5*c) + 4*I*x**2*sqr
t(-c**2*x**2 + 1)/(15*c**3) + 8*I*sqrt(-c**2*x**2 + 1)/(15*c**5), True))/(6*c)

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