3.54 \(\int (c+d x^2)^3 \cot ^{-1}(a x) \, dx\)

Optimal. Leaf size=168 \[ \frac{d x^2 \left (35 a^4 c^2-21 a^2 c d+5 d^2\right )}{70 a^5}+\frac{\left (-35 a^4 c^2 d+35 a^6 c^3+21 a^2 c d^2-5 d^3\right ) \log \left (a^2 x^2+1\right )}{70 a^7}+\frac{d^2 x^4 \left (21 a^2 c-5 d\right )}{140 a^3}+c^2 d x^3 \cot ^{-1}(a x)+c^3 x \cot ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac{d^3 x^6}{42 a}+\frac{1}{7} d^3 x^7 \cot ^{-1}(a x) \]

[Out]

(d*(35*a^4*c^2 - 21*a^2*c*d + 5*d^2)*x^2)/(70*a^5) + ((21*a^2*c - 5*d)*d^2*x^4)/(140*a^3) + (d^3*x^6)/(42*a) +
c^3*x*ArcCot[a*x] + c^2*d*x^3*ArcCot[a*x] + (3*c*d^2*x^5*ArcCot[a*x])/5 + (d^3*x^7*ArcCot[a*x])/7 + ((35*a^6*
c^3 - 35*a^4*c^2*d + 21*a^2*c*d^2 - 5*d^3)*Log[1 + a^2*x^2])/(70*a^7)

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Rubi [A]  time = 0.120067, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {194, 4913, 1810, 260} \[ \frac{d x^2 \left (35 a^4 c^2-21 a^2 c d+5 d^2\right )}{70 a^5}+\frac{\left (-35 a^4 c^2 d+35 a^6 c^3+21 a^2 c d^2-5 d^3\right ) \log \left (a^2 x^2+1\right )}{70 a^7}+\frac{d^2 x^4 \left (21 a^2 c-5 d\right )}{140 a^3}+c^2 d x^3 \cot ^{-1}(a x)+c^3 x \cot ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac{d^3 x^6}{42 a}+\frac{1}{7} d^3 x^7 \cot ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^3*ArcCot[a*x],x]

[Out]

(d*(35*a^4*c^2 - 21*a^2*c*d + 5*d^2)*x^2)/(70*a^5) + ((21*a^2*c - 5*d)*d^2*x^4)/(140*a^3) + (d^3*x^6)/(42*a) +
c^3*x*ArcCot[a*x] + c^2*d*x^3*ArcCot[a*x] + (3*c*d^2*x^5*ArcCot[a*x])/5 + (d^3*x^7*ArcCot[a*x])/7 + ((35*a^6*
c^3 - 35*a^4*c^2*d + 21*a^2*c*d^2 - 5*d^3)*Log[1 + a^2*x^2])/(70*a^7)

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 1810

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \left (c+d x^2\right )^3 \cot ^{-1}(a x) \, dx &=c^3 x \cot ^{-1}(a x)+c^2 d x^3 \cot ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cot ^{-1}(a x)+a \int \frac{c^3 x+c^2 d x^3+\frac{3}{5} c d^2 x^5+\frac{d^3 x^7}{7}}{1+a^2 x^2} \, dx\\ &=c^3 x \cot ^{-1}(a x)+c^2 d x^3 \cot ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cot ^{-1}(a x)+a \int \left (\frac{d \left (35 a^4 c^2-21 a^2 c d+5 d^2\right ) x}{35 a^6}+\frac{\left (21 a^2 c-5 d\right ) d^2 x^3}{35 a^4}+\frac{d^3 x^5}{7 a^2}+\frac{\left (35 a^6 c^3-35 a^4 c^2 d+21 a^2 c d^2-5 d^3\right ) x}{35 a^6 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=\frac{d \left (35 a^4 c^2-21 a^2 c d+5 d^2\right ) x^2}{70 a^5}+\frac{\left (21 a^2 c-5 d\right ) d^2 x^4}{140 a^3}+\frac{d^3 x^6}{42 a}+c^3 x \cot ^{-1}(a x)+c^2 d x^3 \cot ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cot ^{-1}(a x)+\frac{\left (35 a^6 c^3-35 a^4 c^2 d+21 a^2 c d^2-5 d^3\right ) \int \frac{x}{1+a^2 x^2} \, dx}{35 a^5}\\ &=\frac{d \left (35 a^4 c^2-21 a^2 c d+5 d^2\right ) x^2}{70 a^5}+\frac{\left (21 a^2 c-5 d\right ) d^2 x^4}{140 a^3}+\frac{d^3 x^6}{42 a}+c^3 x \cot ^{-1}(a x)+c^2 d x^3 \cot ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cot ^{-1}(a x)+\frac{\left (35 a^6 c^3-35 a^4 c^2 d+21 a^2 c d^2-5 d^3\right ) \log \left (1+a^2 x^2\right )}{70 a^7}\\ \end{align*}

Mathematica [A]  time = 0.103544, size = 149, normalized size = 0.89 \[ \frac{a^2 d x^2 \left (a^4 \left (210 c^2+63 c d x^2+10 d^2 x^4\right )-3 a^2 d \left (42 c+5 d x^2\right )+30 d^2\right )+6 \left (-35 a^4 c^2 d+35 a^6 c^3+21 a^2 c d^2-5 d^3\right ) \log \left (a^2 x^2+1\right )+12 a^7 x \cot ^{-1}(a x) \left (35 c^2 d x^2+35 c^3+21 c d^2 x^4+5 d^3 x^6\right )}{420 a^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)^3*ArcCot[a*x],x]

[Out]

(a^2*d*x^2*(30*d^2 - 3*a^2*d*(42*c + 5*d*x^2) + a^4*(210*c^2 + 63*c*d*x^2 + 10*d^2*x^4)) + 12*a^7*x*(35*c^3 +
35*c^2*d*x^2 + 21*c*d^2*x^4 + 5*d^3*x^6)*ArcCot[a*x] + 6*(35*a^6*c^3 - 35*a^4*c^2*d + 21*a^2*c*d^2 - 5*d^3)*Lo
g[1 + a^2*x^2])/(420*a^7)

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Maple [A]  time = 0.043, size = 191, normalized size = 1.1 \begin{align*}{\frac{{d}^{3}{x}^{7}{\rm arccot} \left (ax\right )}{7}}+{\frac{3\,c{d}^{2}{x}^{5}{\rm arccot} \left (ax\right )}{5}}+{c}^{2}d{x}^{3}{\rm arccot} \left (ax\right )+{c}^{3}x{\rm arccot} \left (ax\right )+{\frac{{c}^{2}d{x}^{2}}{2\,a}}+{\frac{3\,c{x}^{4}{d}^{2}}{20\,a}}+{\frac{{d}^{3}{x}^{6}}{42\,a}}-{\frac{3\,c{d}^{2}{x}^{2}}{10\,{a}^{3}}}-{\frac{{d}^{3}{x}^{4}}{28\,{a}^{3}}}+{\frac{{d}^{3}{x}^{2}}{14\,{a}^{5}}}+{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ){c}^{3}}{2\,a}}-{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ){c}^{2}d}{2\,{a}^{3}}}+{\frac{3\,\ln \left ({a}^{2}{x}^{2}+1 \right ) c{d}^{2}}{10\,{a}^{5}}}-{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ){d}^{3}}{14\,{a}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)^3*arccot(a*x),x)

[Out]

1/7*d^3*x^7*arccot(a*x)+3/5*c*d^2*x^5*arccot(a*x)+c^2*d*x^3*arccot(a*x)+c^3*x*arccot(a*x)+1/2*c^2*d*x^2/a+3/20
/a*x^4*c*d^2+1/42*d^3*x^6/a-3/10/a^3*c*d^2*x^2-1/28/a^3*d^3*x^4+1/14/a^5*d^3*x^2+1/2/a*ln(a^2*x^2+1)*c^3-1/2/a
^3*ln(a^2*x^2+1)*c^2*d+3/10/a^5*ln(a^2*x^2+1)*c*d^2-1/14/a^7*ln(a^2*x^2+1)*d^3

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Maxima [A]  time = 0.971695, size = 215, normalized size = 1.28 \begin{align*} \frac{1}{420} \, a{\left (\frac{10 \, a^{4} d^{3} x^{6} + 3 \,{\left (21 \, a^{4} c d^{2} - 5 \, a^{2} d^{3}\right )} x^{4} + 6 \,{\left (35 \, a^{4} c^{2} d - 21 \, a^{2} c d^{2} + 5 \, d^{3}\right )} x^{2}}{a^{6}} + \frac{6 \,{\left (35 \, a^{6} c^{3} - 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} - 5 \, d^{3}\right )} \log \left (a^{2} x^{2} + 1\right )}{a^{8}}\right )} + \frac{1}{35} \,{\left (5 \, d^{3} x^{7} + 21 \, c d^{2} x^{5} + 35 \, c^{2} d x^{3} + 35 \, c^{3} 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)^3*arccot(a*x),x, algorithm="maxima")

[Out]

1/420*a*((10*a^4*d^3*x^6 + 3*(21*a^4*c*d^2 - 5*a^2*d^3)*x^4 + 6*(35*a^4*c^2*d - 21*a^2*c*d^2 + 5*d^3)*x^2)/a^6
+ 6*(35*a^6*c^3 - 35*a^4*c^2*d + 21*a^2*c*d^2 - 5*d^3)*log(a^2*x^2 + 1)/a^8) + 1/35*(5*d^3*x^7 + 21*c*d^2*x^5
+ 35*c^2*d*x^3 + 35*c^3*x)*arccot(a*x)

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Fricas [A]  time = 1.83765, size = 366, normalized size = 2.18 \begin{align*} \frac{10 \, a^{6} d^{3} x^{6} + 3 \,{\left (21 \, a^{6} c d^{2} - 5 \, a^{4} d^{3}\right )} x^{4} + 6 \,{\left (35 \, a^{6} c^{2} d - 21 \, a^{4} c d^{2} + 5 \, a^{2} d^{3}\right )} x^{2} + 12 \,{\left (5 \, a^{7} d^{3} x^{7} + 21 \, a^{7} c d^{2} x^{5} + 35 \, a^{7} c^{2} d x^{3} + 35 \, a^{7} c^{3} x\right )} \operatorname{arccot}\left (a x\right ) + 6 \,{\left (35 \, a^{6} c^{3} - 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} - 5 \, d^{3}\right )} \log \left (a^{2} x^{2} + 1\right )}{420 \, a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3*arccot(a*x),x, algorithm="fricas")

[Out]

1/420*(10*a^6*d^3*x^6 + 3*(21*a^6*c*d^2 - 5*a^4*d^3)*x^4 + 6*(35*a^6*c^2*d - 21*a^4*c*d^2 + 5*a^2*d^3)*x^2 + 1
2*(5*a^7*d^3*x^7 + 21*a^7*c*d^2*x^5 + 35*a^7*c^2*d*x^3 + 35*a^7*c^3*x)*arccot(a*x) + 6*(35*a^6*c^3 - 35*a^4*c^
2*d + 21*a^2*c*d^2 - 5*d^3)*log(a^2*x^2 + 1))/a^7

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Sympy [A]  time = 3.53637, size = 243, normalized size = 1.45 \begin{align*} \begin{cases} c^{3} x \operatorname{acot}{\left (a x \right )} + c^{2} d x^{3} \operatorname{acot}{\left (a x \right )} + \frac{3 c d^{2} x^{5} \operatorname{acot}{\left (a x \right )}}{5} + \frac{d^{3} x^{7} \operatorname{acot}{\left (a x \right )}}{7} + \frac{c^{3} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{2 a} + \frac{c^{2} d x^{2}}{2 a} + \frac{3 c d^{2} x^{4}}{20 a} + \frac{d^{3} x^{6}}{42 a} - \frac{c^{2} d \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{2 a^{3}} - \frac{3 c d^{2} x^{2}}{10 a^{3}} - \frac{d^{3} x^{4}}{28 a^{3}} + \frac{3 c d^{2} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{10 a^{5}} + \frac{d^{3} x^{2}}{14 a^{5}} - \frac{d^{3} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{14 a^{7}} & \text{for}\: a \neq 0 \\\frac{\pi \left (c^{3} x + c^{2} d x^{3} + \frac{3 c d^{2} x^{5}}{5} + \frac{d^{3} x^{7}}{7}\right )}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)**3*acot(a*x),x)

[Out]

Piecewise((c**3*x*acot(a*x) + c**2*d*x**3*acot(a*x) + 3*c*d**2*x**5*acot(a*x)/5 + d**3*x**7*acot(a*x)/7 + c**3
*log(x**2 + a**(-2))/(2*a) + c**2*d*x**2/(2*a) + 3*c*d**2*x**4/(20*a) + d**3*x**6/(42*a) - c**2*d*log(x**2 + a
**(-2))/(2*a**3) - 3*c*d**2*x**2/(10*a**3) - d**3*x**4/(28*a**3) + 3*c*d**2*log(x**2 + a**(-2))/(10*a**5) + d*
*3*x**2/(14*a**5) - d**3*log(x**2 + a**(-2))/(14*a**7), Ne(a, 0)), (pi*(c**3*x + c**2*d*x**3 + 3*c*d**2*x**5/5
+ d**3*x**7/7)/2, True))

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Giac [A]  time = 1.13143, size = 221, normalized size = 1.32 \begin{align*} \frac{1}{35} \,{\left (5 \, d^{3} x^{7} + 21 \, c d^{2} x^{5} + 35 \, c^{2} d x^{3} + 35 \, c^{3} x\right )} \arctan \left (\frac{1}{a x}\right ) + \frac{10 \, a^{5} d^{3} x^{6} + 63 \, a^{5} c d^{2} x^{4} + 210 \, a^{5} c^{2} d x^{2} - 15 \, a^{3} d^{3} x^{4} - 126 \, a^{3} c d^{2} x^{2} + 30 \, a d^{3} x^{2}}{420 \, a^{6}} + \frac{{\left (35 \, a^{6} c^{3} - 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} - 5 \, d^{3}\right )} \log \left (a^{2} x^{2} + 1\right )}{70 \, a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3*arccot(a*x),x, algorithm="giac")

[Out]

1/35*(5*d^3*x^7 + 21*c*d^2*x^5 + 35*c^2*d*x^3 + 35*c^3*x)*arctan(1/(a*x)) + 1/420*(10*a^5*d^3*x^6 + 63*a^5*c*d
^2*x^4 + 210*a^5*c^2*d*x^2 - 15*a^3*d^3*x^4 - 126*a^3*c*d^2*x^2 + 30*a*d^3*x^2)/a^6 + 1/70*(35*a^6*c^3 - 35*a^
4*c^2*d + 21*a^2*c*d^2 - 5*d^3)*log(a^2*x^2 + 1)/a^7