3.109 \(\int x^4 (a+b \tan ^{-1}(c x^3)) \, dx\)

Optimal. Leaf size=117 \[ \frac {1}{5} x^5 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac {b \log \left (c^{2/3} x^2+1\right )}{10 c^{5/3}}-\frac {\sqrt {3} b \tan ^{-1}\left (\frac {1-2 c^{2/3} x^2}{\sqrt {3}}\right )}{10 c^{5/3}}-\frac {b \log \left (c^{4/3} x^4-c^{2/3} x^2+1\right )}{20 c^{5/3}}-\frac {3 b x^2}{10 c} \]

[Out]

-3/10*b*x^2/c+1/5*x^5*(a+b*arctan(c*x^3))+1/10*b*ln(1+c^(2/3)*x^2)/c^(5/3)-1/20*b*ln(1-c^(2/3)*x^2+c^(4/3)*x^4
)/c^(5/3)-1/10*b*arctan(1/3*(1-2*c^(2/3)*x^2)*3^(1/2))*3^(1/2)/c^(5/3)

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Rubi [A]  time = 0.10, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5033, 275, 321, 200, 31, 634, 617, 204, 628} \[ \frac {1}{5} x^5 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac {b \log \left (c^{2/3} x^2+1\right )}{10 c^{5/3}}-\frac {b \log \left (c^{4/3} x^4-c^{2/3} x^2+1\right )}{20 c^{5/3}}-\frac {\sqrt {3} b \tan ^{-1}\left (\frac {1-2 c^{2/3} x^2}{\sqrt {3}}\right )}{10 c^{5/3}}-\frac {3 b x^2}{10 c} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*ArcTan[c*x^3]),x]

[Out]

(-3*b*x^2)/(10*c) + (x^5*(a + b*ArcTan[c*x^3]))/5 - (Sqrt[3]*b*ArcTan[(1 - 2*c^(2/3)*x^2)/Sqrt[3]])/(10*c^(5/3
)) + (b*Log[1 + c^(2/3)*x^2])/(10*c^(5/3)) - (b*Log[1 - c^(2/3)*x^2 + c^(4/3)*x^4])/(20*c^(5/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[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 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

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^4 \left (a+b \tan ^{-1}\left (c x^3\right )\right ) \, dx &=\frac {1}{5} x^5 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac {1}{5} (3 b c) \int \frac {x^7}{1+c^2 x^6} \, dx\\ &=\frac {1}{5} x^5 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac {1}{10} (3 b c) \operatorname {Subst}\left (\int \frac {x^3}{1+c^2 x^3} \, dx,x,x^2\right )\\ &=-\frac {3 b x^2}{10 c}+\frac {1}{5} x^5 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{1+c^2 x^3} \, dx,x,x^2\right )}{10 c}\\ &=-\frac {3 b x^2}{10 c}+\frac {1}{5} x^5 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac {b \operatorname {Subst}\left (\int \frac {1}{1+c^{2/3} x} \, dx,x,x^2\right )}{10 c}+\frac {b \operatorname {Subst}\left (\int \frac {2-c^{2/3} x}{1-c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )}{10 c}\\ &=-\frac {3 b x^2}{10 c}+\frac {1}{5} x^5 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac {b \log \left (1+c^{2/3} x^2\right )}{10 c^{5/3}}-\frac {b \operatorname {Subst}\left (\int \frac {-c^{2/3}+2 c^{4/3} x}{1-c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )}{20 c^{5/3}}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{1-c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )}{20 c}\\ &=-\frac {3 b x^2}{10 c}+\frac {1}{5} x^5 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac {b \log \left (1+c^{2/3} x^2\right )}{10 c^{5/3}}-\frac {b \log \left (1-c^{2/3} x^2+c^{4/3} x^4\right )}{20 c^{5/3}}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 c^{2/3} x^2\right )}{10 c^{5/3}}\\ &=-\frac {3 b x^2}{10 c}+\frac {1}{5} x^5 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac {\sqrt {3} b \tan ^{-1}\left (\frac {1-2 c^{2/3} x^2}{\sqrt {3}}\right )}{10 c^{5/3}}+\frac {b \log \left (1+c^{2/3} x^2\right )}{10 c^{5/3}}-\frac {b \log \left (1-c^{2/3} x^2+c^{4/3} x^4\right )}{20 c^{5/3}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 185, normalized size = 1.58 \[ \frac {a x^5}{5}+\frac {b \log \left (c^{2/3} x^2+1\right )}{10 c^{5/3}}-\frac {b \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )}{20 c^{5/3}}-\frac {b \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )}{20 c^{5/3}}-\frac {\sqrt {3} b \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{10 c^{5/3}}-\frac {\sqrt {3} b \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt {3}\right )}{10 c^{5/3}}-\frac {3 b x^2}{10 c}+\frac {1}{5} b x^5 \tan ^{-1}\left (c x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*ArcTan[c*x^3]),x]

[Out]

(-3*b*x^2)/(10*c) + (a*x^5)/5 + (b*x^5*ArcTan[c*x^3])/5 - (Sqrt[3]*b*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/(10*c^(5/3
)) - (Sqrt[3]*b*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(10*c^(5/3)) + (b*Log[1 + c^(2/3)*x^2])/(10*c^(5/3)) - (b*Log[1
- Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(20*c^(5/3)) - (b*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(20*c^(5/3))

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fricas [A]  time = 0.45, size = 137, normalized size = 1.17 \[ \frac {4 \, b c^{3} x^{5} \arctan \left (c x^{3}\right ) + 4 \, a c^{3} x^{5} - 6 \, b c^{2} x^{2} + 2 \, \sqrt {3} b {\left (c^{2}\right )}^{\frac {1}{6}} c \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (c^{2}\right )}^{\frac {2}{3}} x^{2} - {\left (c^{2}\right )}^{\frac {1}{3}}\right )} {\left (c^{2}\right )}^{\frac {1}{6}}}{3 \, c}\right ) - b {\left (c^{2}\right )}^{\frac {2}{3}} \log \left (c^{2} x^{4} - {\left (c^{2}\right )}^{\frac {2}{3}} x^{2} + {\left (c^{2}\right )}^{\frac {1}{3}}\right ) + 2 \, b {\left (c^{2}\right )}^{\frac {2}{3}} \log \left (c^{2} x^{2} + {\left (c^{2}\right )}^{\frac {2}{3}}\right )}{20 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arctan(c*x^3)),x, algorithm="fricas")

[Out]

1/20*(4*b*c^3*x^5*arctan(c*x^3) + 4*a*c^3*x^5 - 6*b*c^2*x^2 + 2*sqrt(3)*b*(c^2)^(1/6)*c*arctan(1/3*sqrt(3)*(2*
(c^2)^(2/3)*x^2 - (c^2)^(1/3))*(c^2)^(1/6)/c) - b*(c^2)^(2/3)*log(c^2*x^4 - (c^2)^(2/3)*x^2 + (c^2)^(1/3)) + 2
*b*(c^2)^(2/3)*log(c^2*x^2 + (c^2)^(2/3)))/c^3

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giac [A]  time = 3.95, size = 119, normalized size = 1.02 \[ \frac {1}{20} \, b c^{9} {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )} {\left | c \right |}^{\frac {2}{3}}\right )}{c^{10} {\left | c \right |}^{\frac {2}{3}}} - \frac {\log \left (x^{4} - \frac {x^{2}}{{\left | c \right |}^{\frac {2}{3}}} + \frac {1}{{\left | c \right |}^{\frac {4}{3}}}\right )}{c^{10} {\left | c \right |}^{\frac {2}{3}}} + \frac {2 \, \log \left (x^{2} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{c^{10} {\left | c \right |}^{\frac {2}{3}}}\right )} + \frac {2 \, b c x^{5} \arctan \left (c x^{3}\right ) + 2 \, a c x^{5} - 3 \, b x^{2}}{10 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arctan(c*x^3)),x, algorithm="giac")

[Out]

1/20*b*c^9*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 - 1/abs(c)^(2/3))*abs(c)^(2/3))/(c^10*abs(c)^(2/3)) - log(x^4
- x^2/abs(c)^(2/3) + 1/abs(c)^(4/3))/(c^10*abs(c)^(2/3)) + 2*log(x^2 + 1/abs(c)^(2/3))/(c^10*abs(c)^(2/3))) +
1/10*(2*b*c*x^5*arctan(c*x^3) + 2*a*c*x^5 - 3*b*x^2)/c

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maple [A]  time = 0.03, size = 113, normalized size = 0.97 \[ \frac {a \,x^{5}}{5}+\frac {b \,x^{5} \arctan \left (c \,x^{3}\right )}{5}-\frac {3 b \,x^{2}}{10 c}+\frac {b \ln \left (x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{10 c^{3} \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x^{4}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}} x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}\right )}{20 c^{3} \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{10 c^{3} \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*arctan(c*x^3)),x)

[Out]

1/5*a*x^5+1/5*b*x^5*arctan(c*x^3)-3/10*b*x^2/c+1/10*b/c^3/(1/c^2)^(2/3)*ln(x^2+(1/c^2)^(1/3))-1/20*b/c^3/(1/c^
2)^(2/3)*ln(x^4-(1/c^2)^(1/3)*x^2+(1/c^2)^(2/3))+1/10*b/c^3/(1/c^2)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/c^2
)^(1/3)*x^2-1))

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maxima [A]  time = 0.42, size = 106, normalized size = 0.91 \[ \frac {1}{5} \, a x^{5} + \frac {1}{20} \, {\left (4 \, x^{5} \arctan \left (c x^{3}\right ) - c {\left (\frac {6 \, x^{2}}{c^{2}} - \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {4}{3}} x^{2} - c^{\frac {2}{3}}\right )}}{3 \, c^{\frac {2}{3}}}\right )}{c^{\frac {8}{3}}} + \frac {\log \left (c^{\frac {4}{3}} x^{4} - c^{\frac {2}{3}} x^{2} + 1\right )}{c^{\frac {8}{3}}} - \frac {2 \, \log \left (\frac {c^{\frac {2}{3}} x^{2} + 1}{c^{\frac {2}{3}}}\right )}{c^{\frac {8}{3}}}\right )}\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arctan(c*x^3)),x, algorithm="maxima")

[Out]

1/5*a*x^5 + 1/20*(4*x^5*arctan(c*x^3) - c*(6*x^2/c^2 - 2*sqrt(3)*arctan(1/3*sqrt(3)*(2*c^(4/3)*x^2 - c^(2/3))/
c^(2/3))/c^(8/3) + log(c^(4/3)*x^4 - c^(2/3)*x^2 + 1)/c^(8/3) - 2*log((c^(2/3)*x^2 + 1)/c^(2/3))/c^(8/3)))*b

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mupad [B]  time = 1.94, size = 106, normalized size = 0.91 \[ \frac {a\,x^5}{5}+\frac {b\,\ln \left (c^{2/3}\,x^2+1\right )}{10\,c^{5/3}}-\frac {3\,b\,x^2}{10\,c}-\frac {\ln \left (1-2\,c^{2/3}\,x^2+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{20\,c^{5/3}}-\frac {\ln \left (2\,c^{2/3}\,x^2-1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{20\,c^{5/3}}+\frac {b\,x^5\,\mathrm {atan}\left (c\,x^3\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b*atan(c*x^3)),x)

[Out]

(a*x^5)/5 + (b*log(c^(2/3)*x^2 + 1))/(10*c^(5/3)) - (3*b*x^2)/(10*c) - (log(3^(1/2)*1i - 2*c^(2/3)*x^2 + 1)*(b
+ 3^(1/2)*b*1i))/(20*c^(5/3)) - (log(3^(1/2)*1i + 2*c^(2/3)*x^2 - 1)*(b - 3^(1/2)*b*1i))/(20*c^(5/3)) + (b*x^
5*atan(c*x^3))/5

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sympy [A]  time = 71.60, size = 359, normalized size = 3.07 \[ \begin {cases} \frac {a x^{5}}{5} - \frac {\sqrt [3]{-1} b c^{3} \left (\frac {1}{c^{2}}\right )^{\frac {7}{3}} \log {\left (x - \sqrt [6]{-1} \sqrt [6]{\frac {1}{c^{2}}} \right )}}{5} + \frac {3 \sqrt [3]{-1} b c^{3} \left (\frac {1}{c^{2}}\right )^{\frac {7}{3}} \log {\left (4 x^{2} - 4 \sqrt [6]{-1} x \sqrt [6]{\frac {1}{c^{2}}} + 4 \sqrt [3]{-1} \sqrt [3]{\frac {1}{c^{2}}} \right )}}{20} - \frac {\sqrt [3]{-1} b c^{3} \left (\frac {1}{c^{2}}\right )^{\frac {7}{3}} \log {\left (4 x^{2} + 4 \sqrt [6]{-1} x \sqrt [6]{\frac {1}{c^{2}}} + 4 \sqrt [3]{-1} \sqrt [3]{\frac {1}{c^{2}}} \right )}}{20} + \frac {\sqrt [3]{-1} \sqrt {3} b c^{3} \left (\frac {1}{c^{2}}\right )^{\frac {7}{3}} \operatorname {atan}{\left (\frac {2 \left (-1\right )^{\frac {5}{6}} \sqrt {3} x}{3 \sqrt [6]{\frac {1}{c^{2}}}} - \frac {\sqrt {3}}{3} \right )}}{10} - \frac {\sqrt [3]{-1} \sqrt {3} b c^{3} \left (\frac {1}{c^{2}}\right )^{\frac {7}{3}} \operatorname {atan}{\left (\frac {2 \left (-1\right )^{\frac {5}{6}} \sqrt {3} x}{3 \sqrt [6]{\frac {1}{c^{2}}}} + \frac {\sqrt {3}}{3} \right )}}{10} + \frac {\left (-1\right )^{\frac {5}{6}} b c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {11}{6}} \operatorname {atan}{\left (c x^{3} \right )}}{5} + \frac {b x^{5} \operatorname {atan}{\left (c x^{3} \right )}}{5} - \frac {3 b x^{2}}{10 c} & \text {for}\: c \neq 0 \\\frac {a x^{5}}{5} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*atan(c*x**3)),x)

[Out]

Piecewise((a*x**5/5 - (-1)**(1/3)*b*c**3*(c**(-2))**(7/3)*log(x - (-1)**(1/6)*(c**(-2))**(1/6))/5 + 3*(-1)**(1
/3)*b*c**3*(c**(-2))**(7/3)*log(4*x**2 - 4*(-1)**(1/6)*x*(c**(-2))**(1/6) + 4*(-1)**(1/3)*(c**(-2))**(1/3))/20
- (-1)**(1/3)*b*c**3*(c**(-2))**(7/3)*log(4*x**2 + 4*(-1)**(1/6)*x*(c**(-2))**(1/6) + 4*(-1)**(1/3)*(c**(-2))
**(1/3))/20 + (-1)**(1/3)*sqrt(3)*b*c**3*(c**(-2))**(7/3)*atan(2*(-1)**(5/6)*sqrt(3)*x/(3*(c**(-2))**(1/6)) -
sqrt(3)/3)/10 - (-1)**(1/3)*sqrt(3)*b*c**3*(c**(-2))**(7/3)*atan(2*(-1)**(5/6)*sqrt(3)*x/(3*(c**(-2))**(1/6))
+ sqrt(3)/3)/10 + (-1)**(5/6)*b*c**2*(c**(-2))**(11/6)*atan(c*x**3)/5 + b*x**5*atan(c*x**3)/5 - 3*b*x**2/(10*c
), Ne(c, 0)), (a*x**5/5, True))

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