3.98 \(\int x^5 (a+b \tan ^{-1}(c x^3)) \, dx\)

Optimal. Leaf size=43 \[ \frac {1}{6} x^6 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac {b \tan ^{-1}\left (c x^3\right )}{6 c^2}-\frac {b x^3}{6 c} \]

[Out]

-1/6*b*x^3/c+1/6*b*arctan(c*x^3)/c^2+1/6*x^6*(a+b*arctan(c*x^3))

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5033, 275, 321, 203} \[ \frac {1}{6} x^6 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac {b \tan ^{-1}\left (c x^3\right )}{6 c^2}-\frac {b x^3}{6 c} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*ArcTan[c*x^3]),x]

[Out]

-(b*x^3)/(6*c) + (b*ArcTan[c*x^3])/(6*c^2) + (x^6*(a + b*ArcTan[c*x^3]))/6

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 5033

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTan
[c*x^n]))/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[(x^(n - 1)*(d*x)^(m + 1))/(1 + c^2*x^(2*n)), x], x]
/; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int x^5 \left (a+b \tan ^{-1}\left (c x^3\right )\right ) \, dx &=\frac {1}{6} x^6 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac {1}{2} (b c) \int \frac {x^8}{1+c^2 x^6} \, dx\\ &=\frac {1}{6} x^6 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac {1}{6} (b c) \operatorname {Subst}\left (\int \frac {x^2}{1+c^2 x^2} \, dx,x,x^3\right )\\ &=-\frac {b x^3}{6 c}+\frac {1}{6} x^6 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac {b \operatorname {Subst}\left (\int \frac {1}{1+c^2 x^2} \, dx,x,x^3\right )}{6 c}\\ &=-\frac {b x^3}{6 c}+\frac {b \tan ^{-1}\left (c x^3\right )}{6 c^2}+\frac {1}{6} x^6 \left (a+b \tan ^{-1}\left (c x^3\right )\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 48, normalized size = 1.12 \[ \frac {a x^6}{6}+\frac {b \tan ^{-1}\left (c x^3\right )}{6 c^2}-\frac {b x^3}{6 c}+\frac {1}{6} b x^6 \tan ^{-1}\left (c x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*ArcTan[c*x^3]),x]

[Out]

-1/6*(b*x^3)/c + (a*x^6)/6 + (b*ArcTan[c*x^3])/(6*c^2) + (b*x^6*ArcTan[c*x^3])/6

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 38, normalized size = 0.88 \[ \frac {a c^{2} x^{6} - b c x^{3} + {\left (b c^{2} x^{6} + b\right )} \arctan \left (c x^{3}\right )}{6 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arctan(c*x^3)),x, algorithm="fricas")

[Out]

1/6*(a*c^2*x^6 - b*c*x^3 + (b*c^2*x^6 + b)*arctan(c*x^3))/c^2

________________________________________________________________________________________

giac [A]  time = 0.17, size = 43, normalized size = 1.00 \[ \frac {a c x^{6} + \frac {{\left (c^{2} x^{6} \arctan \left (c x^{3}\right ) - c x^{3} + \arctan \left (c x^{3}\right )\right )} b}{c}}{6 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arctan(c*x^3)),x, algorithm="giac")

[Out]

1/6*(a*c*x^6 + (c^2*x^6*arctan(c*x^3) - c*x^3 + arctan(c*x^3))*b/c)/c

________________________________________________________________________________________

maple [A]  time = 0.03, size = 41, normalized size = 0.95 \[ \frac {x^{6} a}{6}+\frac {b \,x^{6} \arctan \left (c \,x^{3}\right )}{6}-\frac {b \,x^{3}}{6 c}+\frac {b \arctan \left (c \,x^{3}\right )}{6 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*arctan(c*x^3)),x)

[Out]

1/6*x^6*a+1/6*b*x^6*arctan(c*x^3)-1/6*b*x^3/c+1/6*b*arctan(c*x^3)/c^2

________________________________________________________________________________________

maxima [A]  time = 0.41, size = 43, normalized size = 1.00 \[ \frac {1}{6} \, a x^{6} + \frac {1}{6} \, {\left (x^{6} \arctan \left (c x^{3}\right ) - c {\left (\frac {x^{3}}{c^{2}} - \frac {\arctan \left (c x^{3}\right )}{c^{3}}\right )}\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arctan(c*x^3)),x, algorithm="maxima")

[Out]

1/6*a*x^6 + 1/6*(x^6*arctan(c*x^3) - c*(x^3/c^2 - arctan(c*x^3)/c^3))*b

________________________________________________________________________________________

mupad [B]  time = 0.40, size = 40, normalized size = 0.93 \[ \frac {a\,x^6}{6}-\frac {b\,x^3}{6\,c}+\frac {b\,\mathrm {atan}\left (c\,x^3\right )}{6\,c^2}+\frac {b\,x^6\,\mathrm {atan}\left (c\,x^3\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b*atan(c*x^3)),x)

[Out]

(a*x^6)/6 - (b*x^3)/(6*c) + (b*atan(c*x^3))/(6*c^2) + (b*x^6*atan(c*x^3))/6

________________________________________________________________________________________

sympy [A]  time = 89.85, size = 48, normalized size = 1.12 \[ \begin {cases} \frac {a x^{6}}{6} + \frac {b x^{6} \operatorname {atan}{\left (c x^{3} \right )}}{6} - \frac {b x^{3}}{6 c} + \frac {b \operatorname {atan}{\left (c x^{3} \right )}}{6 c^{2}} & \text {for}\: c \neq 0 \\\frac {a x^{6}}{6} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(a+b*atan(c*x**3)),x)

[Out]

Piecewise((a*x**6/6 + b*x**6*atan(c*x**3)/6 - b*x**3/(6*c) + b*atan(c*x**3)/(6*c**2), Ne(c, 0)), (a*x**6/6, Tr
ue))

________________________________________________________________________________________