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

Optimal. Leaf size=109 \[ \frac{\left (15 a^4 c^2-10 a^2 c d+3 d^2\right ) \log \left (a^2 x^2+1\right )}{30 a^5}+\frac{d x^2 \left (10 a^2 c-3 d\right )}{30 a^3}+c^2 x \cot ^{-1}(a x)+\frac{2}{3} c d x^3 \cot ^{-1}(a x)+\frac{d^2 x^4}{20 a}+\frac{1}{5} d^2 x^5 \cot ^{-1}(a x) \]

[Out]

((10*a^2*c - 3*d)*d*x^2)/(30*a^3) + (d^2*x^4)/(20*a) + c^2*x*ArcCot[a*x] + (2*c*d*x^3*ArcCot[a*x])/3 + (d^2*x^
5*ArcCot[a*x])/5 + ((15*a^4*c^2 - 10*a^2*c*d + 3*d^2)*Log[1 + a^2*x^2])/(30*a^5)

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Rubi [A]  time = 0.126527, antiderivative size = 109, 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 = {194, 4913, 1594, 1247, 698} \[ \frac{\left (15 a^4 c^2-10 a^2 c d+3 d^2\right ) \log \left (a^2 x^2+1\right )}{30 a^5}+\frac{d x^2 \left (10 a^2 c-3 d\right )}{30 a^3}+c^2 x \cot ^{-1}(a x)+\frac{2}{3} c d x^3 \cot ^{-1}(a x)+\frac{d^2 x^4}{20 a}+\frac{1}{5} d^2 x^5 \cot ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^2*ArcCot[a*x],x]

[Out]

((10*a^2*c - 3*d)*d*x^2)/(30*a^3) + (d^2*x^4)/(20*a) + c^2*x*ArcCot[a*x] + (2*c*d*x^3*ArcCot[a*x])/3 + (d^2*x^
5*ArcCot[a*x])/5 + ((15*a^4*c^2 - 10*a^2*c*d + 3*d^2)*Log[1 + a^2*x^2])/(30*a^5)

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

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 1594

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

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \left (c+d x^2\right )^2 \cot ^{-1}(a x) \, dx &=c^2 x \cot ^{-1}(a x)+\frac{2}{3} c d x^3 \cot ^{-1}(a x)+\frac{1}{5} d^2 x^5 \cot ^{-1}(a x)+a \int \frac{c^2 x+\frac{2}{3} c d x^3+\frac{d^2 x^5}{5}}{1+a^2 x^2} \, dx\\ &=c^2 x \cot ^{-1}(a x)+\frac{2}{3} c d x^3 \cot ^{-1}(a x)+\frac{1}{5} d^2 x^5 \cot ^{-1}(a x)+a \int \frac{x \left (c^2+\frac{2}{3} c d x^2+\frac{d^2 x^4}{5}\right )}{1+a^2 x^2} \, dx\\ &=c^2 x \cot ^{-1}(a x)+\frac{2}{3} c d x^3 \cot ^{-1}(a x)+\frac{1}{5} d^2 x^5 \cot ^{-1}(a x)+\frac{1}{2} a \operatorname{Subst}\left (\int \frac{c^2+\frac{2 c d x}{3}+\frac{d^2 x^2}{5}}{1+a^2 x} \, dx,x,x^2\right )\\ &=c^2 x \cot ^{-1}(a x)+\frac{2}{3} c d x^3 \cot ^{-1}(a x)+\frac{1}{5} d^2 x^5 \cot ^{-1}(a x)+\frac{1}{2} a \operatorname{Subst}\left (\int \left (\frac{\left (10 a^2 c-3 d\right ) d}{15 a^4}+\frac{d^2 x}{5 a^2}+\frac{15 a^4 c^2-10 a^2 c d+3 d^2}{15 a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{\left (10 a^2 c-3 d\right ) d x^2}{30 a^3}+\frac{d^2 x^4}{20 a}+c^2 x \cot ^{-1}(a x)+\frac{2}{3} c d x^3 \cot ^{-1}(a x)+\frac{1}{5} d^2 x^5 \cot ^{-1}(a x)+\frac{\left (15 a^4 c^2-10 a^2 c d+3 d^2\right ) \log \left (1+a^2 x^2\right )}{30 a^5}\\ \end{align*}

Mathematica [A]  time = 0.0659073, size = 97, normalized size = 0.89 \[ \frac{\left (30 a^4 c^2-20 a^2 c d+6 d^2\right ) \log \left (a^2 x^2+1\right )+4 a^5 x \cot ^{-1}(a x) \left (15 c^2+10 c d x^2+3 d^2 x^4\right )+a^2 d x^2 \left (a^2 \left (20 c+3 d x^2\right )-6 d\right )}{60 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)^2*ArcCot[a*x],x]

[Out]

(a^2*d*x^2*(-6*d + a^2*(20*c + 3*d*x^2)) + 4*a^5*x*(15*c^2 + 10*c*d*x^2 + 3*d^2*x^4)*ArcCot[a*x] + (30*a^4*c^2
- 20*a^2*c*d + 6*d^2)*Log[1 + a^2*x^2])/(60*a^5)

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Maple [A]  time = 0.042, size = 119, normalized size = 1.1 \begin{align*}{\frac{{d}^{2}{x}^{5}{\rm arccot} \left (ax\right )}{5}}+{\frac{2\,cd{x}^{3}{\rm arccot} \left (ax\right )}{3}}+{c}^{2}x{\rm arccot} \left (ax\right )+{\frac{dc{x}^{2}}{3\,a}}+{\frac{{d}^{2}{x}^{4}}{20\,a}}-{\frac{{d}^{2}{x}^{2}}{10\,{a}^{3}}}+{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ){c}^{2}}{2\,a}}-{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ) cd}{3\,{a}^{3}}}+{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ){d}^{2}}{10\,{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*d^2*x^5*arccot(a*x)+2/3*c*d*x^3*arccot(a*x)+c^2*x*arccot(a*x)+1/3/a*d*c*x^2+1/20*d^2*x^4/a-1/10/a^3*x^2*d^
2+1/2/a*ln(a^2*x^2+1)*c^2-1/3/a^3*ln(a^2*x^2+1)*c*d+1/10/a^5*ln(a^2*x^2+1)*d^2

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Maxima [A]  time = 1.01283, size = 139, normalized size = 1.28 \begin{align*} \frac{1}{60} \, a{\left (\frac{3 \, a^{2} d^{2} x^{4} + 2 \,{\left (10 \, a^{2} c d - 3 \, d^{2}\right )} x^{2}}{a^{4}} + \frac{2 \,{\left (15 \, a^{4} c^{2} - 10 \, a^{2} c d + 3 \, d^{2}\right )} \log \left (a^{2} x^{2} + 1\right )}{a^{6}}\right )} + \frac{1}{15} \,{\left (3 \, d^{2} x^{5} + 10 \, c d x^{3} + 15 \, c^{2} 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)^2*arccot(a*x),x, algorithm="maxima")

[Out]

1/60*a*((3*a^2*d^2*x^4 + 2*(10*a^2*c*d - 3*d^2)*x^2)/a^4 + 2*(15*a^4*c^2 - 10*a^2*c*d + 3*d^2)*log(a^2*x^2 + 1
)/a^6) + 1/15*(3*d^2*x^5 + 10*c*d*x^3 + 15*c^2*x)*arccot(a*x)

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Fricas [A]  time = 1.92299, size = 240, normalized size = 2.2 \begin{align*} \frac{3 \, a^{4} d^{2} x^{4} + 2 \,{\left (10 \, a^{4} c d - 3 \, a^{2} d^{2}\right )} x^{2} + 4 \,{\left (3 \, a^{5} d^{2} x^{5} + 10 \, a^{5} c d x^{3} + 15 \, a^{5} c^{2} x\right )} \operatorname{arccot}\left (a x\right ) + 2 \,{\left (15 \, a^{4} c^{2} - 10 \, a^{2} c d + 3 \, d^{2}\right )} \log \left (a^{2} x^{2} + 1\right )}{60 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(3*a^4*d^2*x^4 + 2*(10*a^4*c*d - 3*a^2*d^2)*x^2 + 4*(3*a^5*d^2*x^5 + 10*a^5*c*d*x^3 + 15*a^5*c^2*x)*arcco
t(a*x) + 2*(15*a^4*c^2 - 10*a^2*c*d + 3*d^2)*log(a^2*x^2 + 1))/a^5

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Sympy [A]  time = 1.93479, size = 151, normalized size = 1.39 \begin{align*} \begin{cases} c^{2} x \operatorname{acot}{\left (a x \right )} + \frac{2 c d x^{3} \operatorname{acot}{\left (a x \right )}}{3} + \frac{d^{2} x^{5} \operatorname{acot}{\left (a x \right )}}{5} + \frac{c^{2} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{2 a} + \frac{c d x^{2}}{3 a} + \frac{d^{2} x^{4}}{20 a} - \frac{c d \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{3 a^{3}} - \frac{d^{2} x^{2}}{10 a^{3}} + \frac{d^{2} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{10 a^{5}} & \text{for}\: a \neq 0 \\\frac{\pi \left (c^{2} x + \frac{2 c d x^{3}}{3} + \frac{d^{2} x^{5}}{5}\right )}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((c**2*x*acot(a*x) + 2*c*d*x**3*acot(a*x)/3 + d**2*x**5*acot(a*x)/5 + c**2*log(x**2 + a**(-2))/(2*a)
+ c*d*x**2/(3*a) + d**2*x**4/(20*a) - c*d*log(x**2 + a**(-2))/(3*a**3) - d**2*x**2/(10*a**3) + d**2*log(x**2 +
a**(-2))/(10*a**5), Ne(a, 0)), (pi*(c**2*x + 2*c*d*x**3/3 + d**2*x**5/5)/2, True))

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Giac [A]  time = 1.11553, size = 142, normalized size = 1.3 \begin{align*} \frac{1}{15} \,{\left (3 \, d^{2} x^{5} + 10 \, c d x^{3} + 15 \, c^{2} x\right )} \arctan \left (\frac{1}{a x}\right ) + \frac{3 \, a^{3} d^{2} x^{4} + 20 \, a^{3} c d x^{2} - 6 \, a d^{2} x^{2}}{60 \, a^{4}} + \frac{{\left (15 \, a^{4} c^{2} - 10 \, a^{2} c d + 3 \, d^{2}\right )} \log \left (a^{2} x^{2} + 1\right )}{30 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(3*d^2*x^5 + 10*c*d*x^3 + 15*c^2*x)*arctan(1/(a*x)) + 1/60*(3*a^3*d^2*x^4 + 20*a^3*c*d*x^2 - 6*a*d^2*x^2)
/a^4 + 1/30*(15*a^4*c^2 - 10*a^2*c*d + 3*d^2)*log(a^2*x^2 + 1)/a^5