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

Optimal. Leaf size=39 \[ -\frac{a+b \tan ^{-1}\left (c x^3\right )}{3 x^3}-\frac{1}{6} b c \log \left (c^2 x^6+1\right )+b c \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0245742, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {5033, 266, 36, 29, 31} \[ -\frac{a+b \tan ^{-1}\left (c x^3\right )}{3 x^3}-\frac{1}{6} b c \log \left (c^2 x^6+1\right )+b c \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTan[c*x^3])/x^4,x]

[Out]

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

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]

Rule 266

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 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 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]

Rubi steps

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

Mathematica [A]  time = 0.0067643, size = 44, normalized size = 1.13 \[ -\frac{a}{3 x^3}-\frac{1}{6} b c \log \left (c^2 x^6+1\right )-\frac{b \tan ^{-1}\left (c x^3\right )}{3 x^3}+b c \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.026, size = 39, normalized size = 1. \begin{align*} -{\frac{a}{3\,{x}^{3}}}-{\frac{b\arctan \left ( c{x}^{3} \right ) }{3\,{x}^{3}}}-{\frac{bc\ln \left ({c}^{2}{x}^{6}+1 \right ) }{6}}+bc\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.993088, size = 55, normalized size = 1.41 \begin{align*} -\frac{1}{6} \,{\left (c{\left (\log \left (c^{2} x^{6} + 1\right ) - \log \left (x^{6}\right )\right )} + \frac{2 \, \arctan \left (c x^{3}\right )}{x^{3}}\right )} b - \frac{a}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 3.27881, size = 111, normalized size = 2.85 \begin{align*} -\frac{b c x^{3} \log \left (c^{2} x^{6} + 1\right ) - 6 \, b c x^{3} \log \left (x\right ) + 2 \, b \arctan \left (c x^{3}\right ) + 2 \, a}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 155.503, size = 505, normalized size = 12.95 \begin{align*} \begin{cases} - \frac{a - \infty i b}{3 x^{3}} & \text{for}\: c = - \frac{i}{x^{3}} \\- \frac{a + \infty i b}{3 x^{3}} & \text{for}\: c = \frac{i}{x^{3}} \\- \frac{a}{3 x^{3}} & \text{for}\: c = 0 \\- \frac{a x^{6}}{3 x^{9} + \frac{3 x^{3}}{c^{2}}} - \frac{a}{3 c^{2} x^{9} + 3 x^{3}} + \frac{i b c^{28} x^{9} \left (\frac{1}{c^{2}}\right )^{\frac{29}{2}} \operatorname{atan}{\left (c x^{3} \right )}}{\frac{3 x^{9}}{c^{2}} + \frac{3 x^{3}}{c^{4}}} + \frac{i b c^{26} x^{3} \left (\frac{1}{c^{2}}\right )^{\frac{29}{2}} \operatorname{atan}{\left (c x^{3} \right )}}{\frac{3 x^{9}}{c^{2}} + \frac{3 x^{3}}{c^{4}}} + \frac{3 b x^{9} \log{\left (x \right )}}{\frac{3 x^{9}}{c} + \frac{3 x^{3}}{c^{3}}} - \frac{b x^{9} \log{\left (x - \sqrt [6]{-1} \sqrt [6]{\frac{1}{c^{2}}} \right )}}{\frac{3 x^{9}}{c} + \frac{3 x^{3}}{c^{3}}} - \frac{b x^{9} \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 )}}{\frac{3 x^{9}}{c} + \frac{3 x^{3}}{c^{3}}} + \frac{2 b x^{9} \log{\left (2 \right )}}{\frac{3 x^{9}}{c} + \frac{3 x^{3}}{c^{3}}} - \frac{b x^{6} \operatorname{atan}{\left (c x^{3} \right )}}{3 x^{9} + \frac{3 x^{3}}{c^{2}}} + \frac{3 b x^{3} \log{\left (x \right )}}{3 c x^{9} + \frac{3 x^{3}}{c}} - \frac{b x^{3} \log{\left (x - \sqrt [6]{-1} \sqrt [6]{\frac{1}{c^{2}}} \right )}}{3 c x^{9} + \frac{3 x^{3}}{c}} - \frac{b x^{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 )}}{3 c x^{9} + \frac{3 x^{3}}{c}} + \frac{2 b x^{3} \log{\left (2 \right )}}{3 c x^{9} + \frac{3 x^{3}}{c}} - \frac{b \operatorname{atan}{\left (c x^{3} \right )}}{3 c^{2} x^{9} + 3 x^{3}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-(a - oo*I*b)/(3*x**3), Eq(c, -I/x**3)), (-(a + oo*I*b)/(3*x**3), Eq(c, I/x**3)), (-a/(3*x**3), Eq(
c, 0)), (-a*x**6/(3*x**9 + 3*x**3/c**2) - a/(3*c**2*x**9 + 3*x**3) + I*b*c**28*x**9*(c**(-2))**(29/2)*atan(c*x
**3)/(3*x**9/c**2 + 3*x**3/c**4) + I*b*c**26*x**3*(c**(-2))**(29/2)*atan(c*x**3)/(3*x**9/c**2 + 3*x**3/c**4) +
3*b*x**9*log(x)/(3*x**9/c + 3*x**3/c**3) - b*x**9*log(x - (-1)**(1/6)*(c**(-2))**(1/6))/(3*x**9/c + 3*x**3/c*
*3) - b*x**9*log(4*x**2 + 4*(-1)**(1/6)*x*(c**(-2))**(1/6) + 4*(-1)**(1/3)*(c**(-2))**(1/3))/(3*x**9/c + 3*x**
3/c**3) + 2*b*x**9*log(2)/(3*x**9/c + 3*x**3/c**3) - b*x**6*atan(c*x**3)/(3*x**9 + 3*x**3/c**2) + 3*b*x**3*log
(x)/(3*c*x**9 + 3*x**3/c) - b*x**3*log(x - (-1)**(1/6)*(c**(-2))**(1/6))/(3*c*x**9 + 3*x**3/c) - b*x**3*log(4*
x**2 + 4*(-1)**(1/6)*x*(c**(-2))**(1/6) + 4*(-1)**(1/3)*(c**(-2))**(1/3))/(3*c*x**9 + 3*x**3/c) + 2*b*x**3*log
(2)/(3*c*x**9 + 3*x**3/c) - b*atan(c*x**3)/(3*c**2*x**9 + 3*x**3), True))

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Giac [A]  time = 1.11099, size = 81, normalized size = 2.08 \begin{align*} -\frac{b c^{3} x^{3} \log \left (c^{2} x^{6} + 1\right ) - 2 \, b c^{3} x^{3} \log \left (c x^{3}\right ) + 2 \, b c^{2} \arctan \left (c x^{3}\right ) + 2 \, a c^{2}}{6 \, c^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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