3.2.32 \(\int x^3 (a+b \text {ArcTan}(\frac {c}{x})) \, dx\) [132]

Optimal. Leaf size=50 \[ -\frac {1}{4} b c^3 x+\frac {1}{12} b c x^3+\frac {1}{4} x^4 \left (a+b \text {ArcTan}\left (\frac {c}{x}\right )\right )+\frac {1}{4} b c^4 \text {ArcTan}\left (\frac {x}{c}\right ) \]

[Out]

-1/4*b*c^3*x+1/12*b*c*x^3+1/4*x^4*(a+b*arctan(c/x))+1/4*b*c^4*arctan(x/c)

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Rubi [A]
time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4946, 269, 308, 209} \begin {gather*} \frac {1}{4} x^4 \left (a+b \text {ArcTan}\left (\frac {c}{x}\right )\right )+\frac {1}{4} b c^4 \text {ArcTan}\left (\frac {x}{c}\right )-\frac {1}{4} b c^3 x+\frac {1}{12} b c x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*ArcTan[c/x]),x]

[Out]

-1/4*(b*c^3*x) + (b*c*x^3)/12 + (x^4*(a + b*ArcTan[c/x]))/4 + (b*c^4*ArcTan[x/c])/4

Rule 209

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

Rule 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

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^3 \left (a+b \tan ^{-1}\left (\frac {c}{x}\right )\right ) \, dx &=\frac {1}{4} x^4 \left (a+b \tan ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{4} (b c) \int \frac {x^2}{1+\frac {c^2}{x^2}} \, dx\\ &=\frac {1}{4} x^4 \left (a+b \tan ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{4} (b c) \int \frac {x^4}{c^2+x^2} \, dx\\ &=\frac {1}{4} x^4 \left (a+b \tan ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{4} (b c) \int \left (-c^2+x^2+\frac {c^4}{c^2+x^2}\right ) \, dx\\ &=-\frac {1}{4} b c^3 x+\frac {1}{12} b c x^3+\frac {1}{4} x^4 \left (a+b \tan ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{4} \left (b c^5\right ) \int \frac {1}{c^2+x^2} \, dx\\ &=-\frac {1}{4} b c^3 x+\frac {1}{12} b c x^3+\frac {1}{4} x^4 \left (a+b \tan ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{4} b c^4 \tan ^{-1}\left (\frac {x}{c}\right )\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 55, normalized size = 1.10 \begin {gather*} -\frac {1}{4} b c^3 x+\frac {1}{12} b c x^3+\frac {a x^4}{4}-\frac {1}{4} b c^4 \text {ArcTan}\left (\frac {c}{x}\right )+\frac {1}{4} b x^4 \text {ArcTan}\left (\frac {c}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*ArcTan[c/x]),x]

[Out]

-1/4*(b*c^3*x) + (b*c*x^3)/12 + (a*x^4)/4 - (b*c^4*ArcTan[c/x])/4 + (b*x^4*ArcTan[c/x])/4

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Maple [A]
time = 0.11, size = 56, normalized size = 1.12

method	result	size
derivativedivides	\(-c^{4} \left (-\frac {a \,x^{4}}{4 c^{4}}-\frac {b \,x^{4} \arctan \left (\frac {c}{x}\right )}{4 c^{4}}+\frac {b \arctan \left (\frac {c}{x}\right )}{4}-\frac {b \,x^{3}}{12 c^{3}}+\frac {b x}{4 c}\right )\)	\(56\)
default	\(-c^{4} \left (-\frac {a \,x^{4}}{4 c^{4}}-\frac {b \,x^{4} \arctan \left (\frac {c}{x}\right )}{4 c^{4}}+\frac {b \arctan \left (\frac {c}{x}\right )}{4}-\frac {b \,x^{3}}{12 c^{3}}+\frac {b x}{4 c}\right )\)	\(56\)
risch	\(\text {Expression too large to display}\)	\(697\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*arctan(c/x)),x,method=_RETURNVERBOSE)

[Out]

-c^4*(-1/4*a/c^4*x^4-1/4*b/c^4*x^4*arctan(c/x)+1/4*b*arctan(c/x)-1/12*b*x^3/c^3+1/4*b*x/c)

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Maxima [A]
time = 0.47, size = 45, normalized size = 0.90 \begin {gather*} \frac {1}{4} \, a x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \arctan \left (\frac {c}{x}\right ) + {\left (3 \, c^{3} \arctan \left (\frac {x}{c}\right ) - 3 \, c^{2} x + x^{3}\right )} c\right )} b \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*x^4 + 1/12*(3*x^4*arctan(c/x) + (3*c^3*arctan(x/c) - 3*c^2*x + x^3)*c)*b

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Fricas [A]
time = 1.87, size = 41, normalized size = 0.82 \begin {gather*} -\frac {1}{4} \, b c^{3} x + \frac {1}{12} \, b c x^{3} + \frac {1}{4} \, a x^{4} - \frac {1}{4} \, {\left (b c^{4} - b x^{4}\right )} \arctan \left (\frac {c}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*b*c^3*x + 1/12*b*c*x^3 + 1/4*a*x^4 - 1/4*(b*c^4 - b*x^4)*arctan(c/x)

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Sympy [A]
time = 0.15, size = 46, normalized size = 0.92 \begin {gather*} \frac {a x^{4}}{4} - \frac {b c^{4} \operatorname {atan}{\left (\frac {c}{x} \right )}}{4} - \frac {b c^{3} x}{4} + \frac {b c x^{3}}{12} + \frac {b x^{4} \operatorname {atan}{\left (\frac {c}{x} \right )}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**4/4 - b*c**4*atan(c/x)/4 - b*c**3*x/4 + b*c*x**3/12 + b*x**4*atan(c/x)/4

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Giac [C] Result contains complex when optimal does not.
time = 0.45, size = 81, normalized size = 1.62 \begin {gather*} \frac {{\left (6 \, b c^{5} \arctan \left (\frac {c}{x}\right ) - \frac {3 i \, b c^{9} \log \left (\frac {i \, c}{x} - 1\right )}{x^{4}} + \frac {3 i \, b c^{9} \log \left (-\frac {i \, c}{x} - 1\right )}{x^{4}} + 6 \, a c^{5} - \frac {6 \, b c^{8}}{x^{3}} + \frac {2 \, b c^{6}}{x}\right )} x^{4}}{24 \, c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(6*b*c^5*arctan(c/x) - 3*I*b*c^9*log(I*c/x - 1)/x^4 + 3*I*b*c^9*log(-I*c/x - 1)/x^4 + 6*a*c^5 - 6*b*c^8/x
^3 + 2*b*c^6/x)*x^4/c^5

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Mupad [B]
time = 0.41, size = 45, normalized size = 0.90 \begin {gather*} \frac {a\,x^4}{4}-\frac {b\,c^4\,\mathrm {atan}\left (\frac {c}{x}\right )}{4}+\frac {b\,x^4\,\mathrm {atan}\left (\frac {c}{x}\right )}{4}+\frac {b\,c\,x^3}{12}-\frac {b\,c^3\,x}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*atan(c/x)),x)

[Out]

(a*x^4)/4 - (b*c^4*atan(c/x))/4 + (b*x^4*atan(c/x))/4 + (b*c*x^3)/12 - (b*c^3*x)/4

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