\(\int (c+d x^2) \cot ^{-1}(a x) \, dx\) [56]

Optimal result

Integrand size = 12, antiderivative size = 58 \[ \int \left (c+d x^2\right ) \cot ^{-1}(a x) \, dx=\frac {d x^2}{6 a}+c x \cot ^{-1}(a x)+\frac {1}{3} d x^3 \cot ^{-1}(a x)+\frac {\left (3 a^2 c-d\right ) \log \left (1+a^2 x^2\right )}{6 a^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5033, 1607, 455, 45} \[ \int \left (c+d x^2\right ) \cot ^{-1}(a x) \, dx=\frac {\left (3 a^2 c-d\right ) \log \left (a^2 x^2+1\right )}{6 a^3}+c x \cot ^{-1}(a x)+\frac {1}{3} d x^3 \cot ^{-1}(a x)+\frac {d x^2}{6 a} \]

[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 45

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])

Rule 455

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 1607

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 5033

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])

Rubi steps \begin{align*} \text {integral}& = 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 \text {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 \text {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] (verified)

Time = 0.01 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.16 \[ \int \left (c+d x^2\right ) \cot ^{-1}(a x) \, dx=\frac {d x^2}{6 a}+c x \cot ^{-1}(a x)+\frac {1}{3} d x^3 \cot ^{-1}(a x)+\frac {c \log \left (1+a^2 x^2\right )}{2 a}-\frac {d \log \left (1+a^2 x^2\right )}{6 a^3} \]

[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)

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.98

method	result	size
parts	\(\frac {d \,x^{3} \operatorname {arccot}\left (a x \right )}{3}+c x \,\operatorname {arccot}\left (a x \right )+\frac {a \left (\frac {d \,x^{2}}{2 a^{2}}+\frac {\left (3 a^{2} c -d \right ) \ln \left (a^{2} x^{2}+1\right )}{2 a^{4}}\right )}{3}\)	\(57\)
derivativedivides	\(\frac {\operatorname {arccot}\left (a x \right ) c a x +\frac {a \,\operatorname {arccot}\left (a x \right ) d \,x^{3}}{3}+\frac {\frac {a^{2} d \,x^{2}}{2}+\frac {\left (3 a^{2} c -d \right ) \ln \left (a^{2} x^{2}+1\right )}{2}}{3 a^{2}}}{a}\)	\(62\)
default	\(\frac {\operatorname {arccot}\left (a x \right ) c a x +\frac {a \,\operatorname {arccot}\left (a x \right ) d \,x^{3}}{3}+\frac {\frac {a^{2} d \,x^{2}}{2}+\frac {\left (3 a^{2} c -d \right ) \ln \left (a^{2} x^{2}+1\right )}{2}}{3 a^{2}}}{a}\)	\(62\)
parallelrisch	\(\frac {2 x^{3} \operatorname {arccot}\left (a x \right ) a^{3} d +6 x \,\operatorname {arccot}\left (a x \right ) a^{3} c +a^{2} d \,x^{2}+3 \ln \left (a^{2} x^{2}+1\right ) a^{2} c -\ln \left (a^{2} x^{2}+1\right ) d}{6 a^{3}}\)	\(68\)
risch	\(\frac {i \left (d \,x^{3}+3 c x \right ) \ln \left (i a x +1\right )}{6}-\frac {i d \,x^{3} \ln \left (-i a x +1\right )}{6}+\frac {\pi d \,x^{3}}{6}-\frac {i c x \ln \left (-i a x +1\right )}{2}+\frac {\pi c x}{2}+\frac {d \,x^{2}}{6 a}+\frac {\ln \left (-a^{2} x^{2}-1\right ) c}{2 a}-\frac {\ln \left (-a^{2} x^{2}-1\right ) d}{6 a^{3}}\)	\(106\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.98 \[ \int \left (c+d x^2\right ) \cot ^{-1}(a x) \, dx=\frac {a^{2} d x^{2} + 2 \, {\left (a^{3} d x^{3} + 3 \, a^{3} c x\right )} \operatorname {arccot}\left (a x\right ) + {\left (3 \, a^{2} c - d\right )} \log \left (a^{2} x^{2} + 1\right )}{6 \, a^{3}} \]

[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

Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.26 \[ \int \left (c+d x^2\right ) \cot ^{-1}(a x) \, dx=\begin {cases} c x \operatorname {acot}{\left (a x \right )} + \frac {d x^{3} \operatorname {acot}{\left (a x \right )}}{3} + \frac {c \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{2 a} + \frac {d x^{2}}{6 a} - \frac {d \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{6 a^{3}} & \text {for}\: a \neq 0 \\\frac {\pi \left (c x + \frac {d x^{3}}{3}\right )}{2} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.91 \[ \int \left (c+d x^2\right ) \cot ^{-1}(a x) \, dx=\frac {1}{6} \, a {\left (\frac {d x^{2}}{a^{2}} + \frac {{\left (3 \, a^{2} c - d\right )} \log \left (a^{2} x^{2} + 1\right )}{a^{4}}\right )} + \frac {1}{3} \, {\left (d x^{3} + 3 \, c x\right )} \operatorname {arccot}\left (a x\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.71 \[ \int \left (c+d x^2\right ) \cot ^{-1}(a x) \, dx=\frac {1}{6} \, {\left (\frac {2 \, {\left (d + \frac {3 \, c}{x^{2}}\right )} x^{3} \arctan \left (\frac {1}{a x}\right )}{a} + \frac {{\left (d + \frac {3 \, c}{x^{2}} - \frac {d}{a^{2} x^{2}}\right )} x^{2}}{a^{2}} + \frac {{\left (3 \, a^{2} c - d\right )} \log \left (\frac {1}{a^{2} x^{2}} + 1\right )}{a^{4}} - \frac {{\left (3 \, a^{2} c - d\right )} \log \left (\frac {1}{a^{2} x^{2}}\right )}{a^{4}}\right )} a \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.83 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.07 \[ \int \left (c+d x^2\right ) \cot ^{-1}(a x) \, dx=\frac {d\,x^3\,\mathrm {acot}\left (a\,x\right )}{3}-\frac {\frac {d\,\ln \left (a^2\,x^2+1\right )}{6}-a^2\,\left (\frac {c\,\ln \left (a^2\,x^2+1\right )}{2}+\frac {d\,x^2}{6}\right )}{a^3}+c\,x\,\mathrm {acot}\left (a\,x\right ) \]

[In]

int(acot(a*x)*(c + d*x^2),x)

[Out]

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