3.2.64 \(\int \frac {(c+a^2 c x^2)^2 \text {ArcTan}(a x)}{x^4} \, dx\) [164]

Optimal. Leaf size=85 \[ -\frac {a c^2}{6 x^2}-\frac {c^2 \text {ArcTan}(a x)}{3 x^3}-\frac {2 a^2 c^2 \text {ArcTan}(a x)}{x}+a^4 c^2 x \text {ArcTan}(a x)+\frac {5}{3} a^3 c^2 \log (x)-\frac {4}{3} a^3 c^2 \log \left (1+a^2 x^2\right ) \]

[Out]

-1/6*a*c^2/x^2-1/3*c^2*arctan(a*x)/x^3-2*a^2*c^2*arctan(a*x)/x+a^4*c^2*x*arctan(a*x)+5/3*a^3*c^2*ln(x)-4/3*a^3
*c^2*ln(a^2*x^2+1)

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Rubi [A]
time = 0.08, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {5068, 4930, 266, 4946, 272, 46, 36, 29, 31} \begin {gather*} a^4 c^2 x \text {ArcTan}(a x)+\frac {5}{3} a^3 c^2 \log (x)-\frac {2 a^2 c^2 \text {ArcTan}(a x)}{x}-\frac {4}{3} a^3 c^2 \log \left (a^2 x^2+1\right )-\frac {c^2 \text {ArcTan}(a x)}{3 x^3}-\frac {a c^2}{6 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((c + a^2*c*x^2)^2*ArcTan[a*x])/x^4,x]

[Out]

-1/6*(a*c^2)/x^2 - (c^2*ArcTan[a*x])/(3*x^3) - (2*a^2*c^2*ArcTan[a*x])/x + a^4*c^2*x*ArcTan[a*x] + (5*a^3*c^2*
Log[x])/3 - (4*a^3*c^2*Log[1 + a^2*x^2])/3

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

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

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 266

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

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 4930

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

Rule 5068

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Int[Ex
pandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e,
c^2*d] && IGtQ[p, 0] && IGtQ[q, 1] && (EqQ[p, 1] || IntegerQ[m])

Rubi steps

\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)}{x^4} \, dx &=\int \left (a^4 c^2 \tan ^{-1}(a x)+\frac {c^2 \tan ^{-1}(a x)}{x^4}+\frac {2 a^2 c^2 \tan ^{-1}(a x)}{x^2}\right ) \, dx\\ &=c^2 \int \frac {\tan ^{-1}(a x)}{x^4} \, dx+\left (2 a^2 c^2\right ) \int \frac {\tan ^{-1}(a x)}{x^2} \, dx+\left (a^4 c^2\right ) \int \tan ^{-1}(a x) \, dx\\ &=-\frac {c^2 \tan ^{-1}(a x)}{3 x^3}-\frac {2 a^2 c^2 \tan ^{-1}(a x)}{x}+a^4 c^2 x \tan ^{-1}(a x)+\frac {1}{3} \left (a c^2\right ) \int \frac {1}{x^3 \left (1+a^2 x^2\right )} \, dx+\left (2 a^3 c^2\right ) \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx-\left (a^5 c^2\right ) \int \frac {x}{1+a^2 x^2} \, dx\\ &=-\frac {c^2 \tan ^{-1}(a x)}{3 x^3}-\frac {2 a^2 c^2 \tan ^{-1}(a x)}{x}+a^4 c^2 x \tan ^{-1}(a x)-\frac {1}{2} a^3 c^2 \log \left (1+a^2 x^2\right )+\frac {1}{6} \left (a c^2\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (1+a^2 x\right )} \, dx,x,x^2\right )+\left (a^3 c^2\right ) \text {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {c^2 \tan ^{-1}(a x)}{3 x^3}-\frac {2 a^2 c^2 \tan ^{-1}(a x)}{x}+a^4 c^2 x \tan ^{-1}(a x)-\frac {1}{2} a^3 c^2 \log \left (1+a^2 x^2\right )+\frac {1}{6} \left (a c^2\right ) \text {Subst}\left (\int \left (\frac {1}{x^2}-\frac {a^2}{x}+\frac {a^4}{1+a^2 x}\right ) \, dx,x,x^2\right )+\left (a^3 c^2\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\left (a^5 c^2\right ) \text {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac {a c^2}{6 x^2}-\frac {c^2 \tan ^{-1}(a x)}{3 x^3}-\frac {2 a^2 c^2 \tan ^{-1}(a x)}{x}+a^4 c^2 x \tan ^{-1}(a x)+\frac {5}{3} a^3 c^2 \log (x)-\frac {4}{3} a^3 c^2 \log \left (1+a^2 x^2\right )\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 68, normalized size = 0.80 \begin {gather*} \frac {c^2 \left (2 \left (-1-6 a^2 x^2+3 a^4 x^4\right ) \text {ArcTan}(a x)+a x \left (-1+10 a^2 x^2 \log (x)-8 a^2 x^2 \log \left (1+a^2 x^2\right )\right )\right )}{6 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((c + a^2*c*x^2)^2*ArcTan[a*x])/x^4,x]

[Out]

(c^2*(2*(-1 - 6*a^2*x^2 + 3*a^4*x^4)*ArcTan[a*x] + a*x*(-1 + 10*a^2*x^2*Log[x] - 8*a^2*x^2*Log[1 + a^2*x^2])))
/(6*x^3)

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Maple [A]
time = 0.13, size = 78, normalized size = 0.92

method	result	size
derivativedivides	\(a^{3} \left (a \,c^{2} x \arctan \left (a x \right )-\frac {2 c^{2} \arctan \left (a x \right )}{a x}-\frac {c^{2} \arctan \left (a x \right )}{3 a^{3} x^{3}}-\frac {c^{2} \left (4 \ln \left (a^{2} x^{2}+1\right )+\frac {1}{2 a^{2} x^{2}}-5 \ln \left (a x \right )\right )}{3}\right )\)	\(78\)
default	\(a^{3} \left (a \,c^{2} x \arctan \left (a x \right )-\frac {2 c^{2} \arctan \left (a x \right )}{a x}-\frac {c^{2} \arctan \left (a x \right )}{3 a^{3} x^{3}}-\frac {c^{2} \left (4 \ln \left (a^{2} x^{2}+1\right )+\frac {1}{2 a^{2} x^{2}}-5 \ln \left (a x \right )\right )}{3}\right )\)	\(78\)
risch	\(-\frac {i c^{2} \left (3 a^{4} x^{4}-6 a^{2} x^{2}-1\right ) \ln \left (i a x +1\right )}{6 x^{3}}+\frac {i c^{2} \left (3 x^{4} \ln \left (-i a x +1\right ) a^{4}-10 i \ln \left (x \right ) a^{3} x^{3}+8 i \ln \left (-9 a^{2} x^{2}-9\right ) a^{3} x^{3}-6 a^{2} x^{2} \ln \left (-i a x +1\right )+i a x -\ln \left (-i a x +1\right )\right )}{6 x^{3}}\)	\(125\)
meijerg	\(\frac {a^{3} c^{2} \left (\frac {4 a^{2} x^{2} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-2 \ln \left (a^{2} x^{2}+1\right )\right )}{4}+\frac {a^{3} c^{2} \left (-\frac {4 \arctan \left (\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-2 \ln \left (a^{2} x^{2}+1\right )+4 \ln \left (x \right )+4 \ln \left (a \right )\right )}{2}+\frac {a^{3} c^{2} \left (\frac {-\frac {4 a^{2} x^{2}}{9}+\frac {4}{3}}{a^{2} x^{2}}-\frac {4 \arctan \left (\sqrt {a^{2} x^{2}}\right )}{3 a^{2} x^{2} \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (a^{2} x^{2}+1\right )}{3}+\frac {4}{9}-\frac {4 \ln \left (x \right )}{3}-\frac {4 \ln \left (a \right )}{3}-\frac {2}{a^{2} x^{2}}\right )}{4}\)	\(183\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*c*x^2+c)^2*arctan(a*x)/x^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.25, size = 76, normalized size = 0.89 \begin {gather*} -\frac {1}{6} \, {\left (8 \, a^{2} c^{2} \log \left (a^{2} x^{2} + 1\right ) - 10 \, a^{2} c^{2} \log \left (x\right ) + \frac {c^{2}}{x^{2}}\right )} a + \frac {1}{3} \, {\left (3 \, a^{4} c^{2} x - \frac {6 \, a^{2} c^{2} x^{2} + c^{2}}{x^{3}}\right )} \arctan \left (a x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^2*arctan(a*x)/x^4,x, algorithm="maxima")

[Out]

-1/6*(8*a^2*c^2*log(a^2*x^2 + 1) - 10*a^2*c^2*log(x) + c^2/x^2)*a + 1/3*(3*a^4*c^2*x - (6*a^2*c^2*x^2 + c^2)/x
^3)*arctan(a*x)

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Fricas [A]
time = 2.27, size = 80, normalized size = 0.94 \begin {gather*} -\frac {8 \, a^{3} c^{2} x^{3} \log \left (a^{2} x^{2} + 1\right ) - 10 \, a^{3} c^{2} x^{3} \log \left (x\right ) + a c^{2} x - 2 \, {\left (3 \, a^{4} c^{2} x^{4} - 6 \, a^{2} c^{2} x^{2} - c^{2}\right )} \arctan \left (a x\right )}{6 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^2*arctan(a*x)/x^4,x, algorithm="fricas")

[Out]

-1/6*(8*a^3*c^2*x^3*log(a^2*x^2 + 1) - 10*a^3*c^2*x^3*log(x) + a*c^2*x - 2*(3*a^4*c^2*x^4 - 6*a^2*c^2*x^2 - c^
2)*arctan(a*x))/x^3

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Sympy [A]
time = 0.45, size = 87, normalized size = 1.02 \begin {gather*} \begin {cases} a^{4} c^{2} x \operatorname {atan}{\left (a x \right )} + \frac {5 a^{3} c^{2} \log {\left (x \right )}}{3} - \frac {4 a^{3} c^{2} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{3} - \frac {2 a^{2} c^{2} \operatorname {atan}{\left (a x \right )}}{x} - \frac {a c^{2}}{6 x^{2}} - \frac {c^{2} \operatorname {atan}{\left (a x \right )}}{3 x^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2*c*x**2+c)**2*atan(a*x)/x**4,x)

[Out]

Piecewise((a**4*c**2*x*atan(a*x) + 5*a**3*c**2*log(x)/3 - 4*a**3*c**2*log(x**2 + a**(-2))/3 - 2*a**2*c**2*atan
(a*x)/x - a*c**2/(6*x**2) - c**2*atan(a*x)/(3*x**3), Ne(a, 0)), (0, 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((a^2*c*x^2+c)^2*arctan(a*x)/x^4,x, algorithm="giac")

[Out]

sage0*x

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Mupad [B]
time = 0.47, size = 78, normalized size = 0.92 \begin {gather*} \frac {c^2\,\left (10\,a^3\,\ln \left (x\right )-8\,a^3\,\ln \left (a^2\,x^2+1\right )\right )}{6}-\frac {\frac {c^2\,\mathrm {atan}\left (a\,x\right )}{3}+\frac {a\,c^2\,x}{6}+2\,a^2\,c^2\,x^2\,\mathrm {atan}\left (a\,x\right )}{x^3}+a^4\,c^2\,x\,\mathrm {atan}\left (a\,x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((atan(a*x)*(c + a^2*c*x^2)^2)/x^4,x)

[Out]

(c^2*(10*a^3*log(x) - 8*a^3*log(a^2*x^2 + 1)))/6 - ((c^2*atan(a*x))/3 + (a*c^2*x)/6 + 2*a^2*c^2*x^2*atan(a*x))
/x^3 + a^4*c^2*x*atan(a*x)

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