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

Optimal. Leaf size=37 \[ -\frac{a+b \tan ^{-1}(c x)}{2 x^2}-\frac{1}{2} b c^2 \tan ^{-1}(c x)-\frac{b c}{2 x} \]

[Out]

-(b*c)/(2*x) - (b*c^2*ArcTan[c*x])/2 - (a + b*ArcTan[c*x])/(2*x^2)

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Rubi [A]  time = 0.0195559, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4852, 325, 203} \[ -\frac{a+b \tan ^{-1}(c x)}{2 x^2}-\frac{1}{2} b c^2 \tan ^{-1}(c x)-\frac{b c}{2 x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTan[c*x])/x^3,x]

[Out]

-(b*c)/(2*x) - (b*c^2*ArcTan[c*x])/2 - (a + b*ArcTan[c*x])/(2*x^2)

Rule 4852

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

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 203

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

Rubi steps

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

Mathematica [C]  time = 0.0028617, size = 46, normalized size = 1.24 \[ -\frac{b c \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-c^2 x^2\right )}{2 x}-\frac{a}{2 x^2}-\frac{b \tan ^{-1}(c x)}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTan[c*x])/x^3,x]

[Out]

-a/(2*x^2) - (b*ArcTan[c*x])/(2*x^2) - (b*c*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)])/(2*x)

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Maple [A]  time = 0.007, size = 35, normalized size = 1. \begin{align*} -{\frac{a}{2\,{x}^{2}}}-{\frac{b\arctan \left ( cx \right ) }{2\,{x}^{2}}}-{\frac{b{c}^{2}\arctan \left ( cx \right ) }{2}}-{\frac{bc}{2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a/x^2-1/2*b/x^2*arctan(c*x)-1/2*b*c^2*arctan(c*x)-1/2*b*c/x

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Maxima [A]  time = 1.46763, size = 42, normalized size = 1.14 \begin{align*} -\frac{1}{2} \,{\left ({\left (c \arctan \left (c x\right ) + \frac{1}{x}\right )} c + \frac{\arctan \left (c x\right )}{x^{2}}\right )} b - \frac{a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.56519, size = 70, normalized size = 1.89 \begin{align*} -\frac{b c x +{\left (b c^{2} x^{2} + b\right )} \arctan \left (c x\right ) + a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.744, size = 37, normalized size = 1. \begin{align*} - \frac{a}{2 x^{2}} - \frac{b c^{2} \operatorname{atan}{\left (c x \right )}}{2} - \frac{b c}{2 x} - \frac{b \operatorname{atan}{\left (c x \right )}}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/(2*x**2) - b*c**2*atan(c*x)/2 - b*c/(2*x) - b*atan(c*x)/(2*x**2)

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Giac [A]  time = 1.29837, size = 74, normalized size = 2. \begin{align*} \frac{b c^{2} i x^{2} \log \left (c i x + 1\right ) - b c^{2} i x^{2} \log \left (-c i x + 1\right ) - 2 \, b c x - 2 \, b \arctan \left (c x\right ) - 2 \, a}{4 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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