3.56 \(\int (c+d x^2) \cot ^{-1}(a x) \, dx\)

Optimal. Leaf size=58 \[ \frac{\left (3 a^2 c-d\right ) \log \left (a^2 x^2+1\right )}{6 a^3}+c x \cot ^{-1}(a x)+\frac{d x^2}{6 a}+\frac{1}{3} d x^3 \cot ^{-1}(a x) \]

[Out]

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

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Rubi [A]  time = 0.061042, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4913, 1593, 444, 43} \[ \frac{\left (3 a^2 c-d\right ) \log \left (a^2 x^2+1\right )}{6 a^3}+c x \cot ^{-1}(a x)+\frac{d x^2}{6 a}+\frac{1}{3} d x^3 \cot ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)*ArcCot[a*x],x]

[Out]

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

Rule 4913

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^q, x]}, Dist[a + b*ArcCot[c*x], u, x] + Dist[b*c, Int[u/(1 + c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x]
&& (IntegerQ[q] || ILtQ[q + 1/2, 0])

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 43

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])

Rubi steps

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

Mathematica [A]  time = 0.0089964, size = 67, normalized size = 1.16 \[ \frac{c \log \left (a^2 x^2+1\right )}{2 a}-\frac{d \log \left (a^2 x^2+1\right )}{6 a^3}+c x \cot ^{-1}(a x)+\frac{d x^2}{6 a}+\frac{1}{3} d x^3 \cot ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)*ArcCot[a*x],x]

[Out]

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

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Maple [A]  time = 0.04, size = 60, normalized size = 1. \begin{align*}{\frac{d{x}^{3}{\rm arccot} \left (ax\right )}{3}}+cx{\rm arccot} \left (ax\right )+{\frac{d{x}^{2}}{6\,a}}+{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ) c}{2\,a}}-{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ) d}{6\,{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)*arccot(a*x),x)

[Out]

1/3*d*x^3*arccot(a*x)+c*x*arccot(a*x)+1/6*d*x^2/a+1/2/a*ln(a^2*x^2+1)*c-1/6/a^3*ln(a^2*x^2+1)*d

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Maxima [A]  time = 0.953438, size = 72, normalized size = 1.24 \begin{align*} \frac{1}{6} \, a{\left (\frac{d x^{2}}{a^{2}} + \frac{{\left (3 \, a^{2} c - d\right )} \log \left (a^{2} x^{2} + 1\right )}{a^{4}}\right )} + \frac{1}{3} \,{\left (d x^{3} + 3 \, c x\right )} \operatorname{arccot}\left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)*arccot(a*x),x, algorithm="maxima")

[Out]

1/6*a*(d*x^2/a^2 + (3*a^2*c - d)*log(a^2*x^2 + 1)/a^4) + 1/3*(d*x^3 + 3*c*x)*arccot(a*x)

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Fricas [A]  time = 2.14745, size = 127, normalized size = 2.19 \begin{align*} \frac{a^{2} d x^{2} + 2 \,{\left (a^{3} d x^{3} + 3 \, a^{3} c x\right )} \operatorname{arccot}\left (a x\right ) +{\left (3 \, a^{2} c - d\right )} \log \left (a^{2} x^{2} + 1\right )}{6 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)*arccot(a*x),x, algorithm="fricas")

[Out]

1/6*(a^2*d*x^2 + 2*(a^3*d*x^3 + 3*a^3*c*x)*arccot(a*x) + (3*a^2*c - d)*log(a^2*x^2 + 1))/a^3

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Sympy [A]  time = 0.836268, size = 73, normalized size = 1.26 \begin{align*} \begin{cases} c x \operatorname{acot}{\left (a x \right )} + \frac{d x^{3} \operatorname{acot}{\left (a x \right )}}{3} + \frac{c \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{2 a} + \frac{d x^{2}}{6 a} - \frac{d \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{6 a^{3}} & \text{for}\: a \neq 0 \\\frac{\pi \left (c x + \frac{d x^{3}}{3}\right )}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)*acot(a*x),x)

[Out]

Piecewise((c*x*acot(a*x) + d*x**3*acot(a*x)/3 + c*log(x**2 + a**(-2))/(2*a) + d*x**2/(6*a) - d*log(x**2 + a**(
-2))/(6*a**3), Ne(a, 0)), (pi*(c*x + d*x**3/3)/2, True))

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Giac [A]  time = 1.10448, size = 74, normalized size = 1.28 \begin{align*} \frac{d x^{2}}{6 \, a} + \frac{1}{3} \,{\left (d x^{3} + 3 \, c x\right )} \arctan \left (\frac{1}{a x}\right ) + \frac{{\left (3 \, a^{2} c - d\right )} \log \left (a^{2} x^{2} + 1\right )}{6 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)*arccot(a*x),x, algorithm="giac")

[Out]

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