3.1.2 \(\int x^4 (a+b \text {ArcTan}(c x)) \, dx\) [2]

Optimal. Leaf size=56 \[ \frac {b x^2}{10 c^3}-\frac {b x^4}{20 c}+\frac {1}{5} x^5 (a+b \text {ArcTan}(c x))-\frac {b \log \left (1+c^2 x^2\right )}{10 c^5} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 56, 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 = {4946, 272, 45} \begin {gather*} \frac {1}{5} x^5 (a+b \text {ArcTan}(c x))+\frac {b x^2}{10 c^3}-\frac {b \log \left (c^2 x^2+1\right )}{10 c^5}-\frac {b x^4}{20 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*ArcTan[c*x]),x]

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[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]]

Rule 4946

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^
n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))),
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 61, normalized size = 1.09 \begin {gather*} \frac {b x^2}{10 c^3}-\frac {b x^4}{20 c}+\frac {a x^5}{5}+\frac {1}{5} b x^5 \text {ArcTan}(c x)-\frac {b \log \left (1+c^2 x^2\right )}{10 c^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*ArcTan[c*x]),x]

[Out]

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

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Maple [A]
time = 0.05, size = 59, normalized size = 1.05

method	result	size
derivativedivides	\(\frac {\frac {c^{5} x^{5} a}{5}+\frac {c^{5} x^{5} b \arctan \left (c x \right )}{5}-\frac {c^{4} x^{4} b}{20}+\frac {b \,c^{2} x^{2}}{10}-\frac {b \ln \left (c^{2} x^{2}+1\right )}{10}}{c^{5}}\)	\(59\)
default	\(\frac {\frac {c^{5} x^{5} a}{5}+\frac {c^{5} x^{5} b \arctan \left (c x \right )}{5}-\frac {c^{4} x^{4} b}{20}+\frac {b \,c^{2} x^{2}}{10}-\frac {b \ln \left (c^{2} x^{2}+1\right )}{10}}{c^{5}}\)	\(59\)
risch	\(-\frac {i x^{5} b \ln \left (i c x +1\right )}{10}+\frac {i x^{5} b \ln \left (-i c x +1\right )}{10}+\frac {a \,x^{5}}{5}-\frac {b \,x^{4}}{20 c}+\frac {b \,x^{2}}{10 c^{3}}-\frac {b \ln \left (-c^{2} x^{2}-1\right )}{10 c^{5}}\)	\(73\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*arctan(c*x)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 56, normalized size = 1.00 \begin {gather*} \frac {1}{5} \, a x^{5} + \frac {1}{20} \, {\left (4 \, x^{5} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 2.03, size = 59, normalized size = 1.05 \begin {gather*} \frac {4 \, b c^{5} x^{5} \arctan \left (c x\right ) + 4 \, a c^{5} x^{5} - b c^{4} x^{4} + 2 \, b c^{2} x^{2} - 2 \, b \log \left (c^{2} x^{2} + 1\right )}{20 \, c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.26, size = 60, normalized size = 1.07 \begin {gather*} \begin {cases} \frac {a x^{5}}{5} + \frac {b x^{5} \operatorname {atan}{\left (c x \right )}}{5} - \frac {b x^{4}}{20 c} + \frac {b x^{2}}{10 c^{3}} - \frac {b \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{10 c^{5}} & \text {for}\: c \neq 0 \\\frac {a x^{5}}{5} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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Mupad [B]
time = 0.42, size = 54, normalized size = 0.96 \begin {gather*} \frac {a\,x^5}{5}-\frac {\frac {b\,\ln \left (c^2\,x^2+1\right )}{10}-\frac {b\,c^2\,x^2}{10}+\frac {b\,c^4\,x^4}{20}}{c^5}+\frac {b\,x^5\,\mathrm {atan}\left (c\,x\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b*atan(c*x)),x)

[Out]

(a*x^5)/5 - ((b*log(c^2*x^2 + 1))/10 - (b*c^2*x^2)/10 + (b*c^4*x^4)/20)/c^5 + (b*x^5*atan(c*x))/5

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