3.13 \(\int x^5 (a+b \tan ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=144 \[ \frac{b x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{a b x}{3 c^5}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{6 c^6}+\frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac{b^2 x^4}{60 c^2}-\frac{4 b^2 x^2}{45 c^4}+\frac{23 b^2 \log \left (c^2 x^2+1\right )}{90 c^6}-\frac{b^2 x \tan ^{-1}(c x)}{3 c^5} \]

[Out]

-(a*b*x)/(3*c^5) - (4*b^2*x^2)/(45*c^4) + (b^2*x^4)/(60*c^2) - (b^2*x*ArcTan[c*x])/(3*c^5) + (b*x^3*(a + b*Arc
Tan[c*x]))/(9*c^3) - (b*x^5*(a + b*ArcTan[c*x]))/(15*c) + (a + b*ArcTan[c*x])^2/(6*c^6) + (x^6*(a + b*ArcTan[c
*x])^2)/6 + (23*b^2*Log[1 + c^2*x^2])/(90*c^6)

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Rubi [A]  time = 0.310175, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4852, 4916, 266, 43, 4846, 260, 4884} \[ \frac{b x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{a b x}{3 c^5}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{6 c^6}+\frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac{b^2 x^4}{60 c^2}-\frac{4 b^2 x^2}{45 c^4}+\frac{23 b^2 \log \left (c^2 x^2+1\right )}{90 c^6}-\frac{b^2 x \tan ^{-1}(c x)}{3 c^5} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*ArcTan[c*x])^2,x]

[Out]

-(a*b*x)/(3*c^5) - (4*b^2*x^2)/(45*c^4) + (b^2*x^4)/(60*c^2) - (b^2*x*ArcTan[c*x])/(3*c^5) + (b*x^3*(a + b*Arc
Tan[c*x]))/(9*c^3) - (b*x^5*(a + b*ArcTan[c*x]))/(15*c) + (a + b*ArcTan[c*x])^2/(6*c^6) + (x^6*(a + b*ArcTan[c
*x])^2)/6 + (23*b^2*Log[1 + c^2*x^2])/(90*c^6)

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 4916

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2/
e, Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x])^p, x], x] - Dist[(d*f^2)/e, Int[((f*x)^(m - 2)*(a + b*ArcTan[c*x])^p)
/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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])

Rule 4846

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x])^p, x] - Dist[b*c*p, Int[
(x*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]

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]

Rule 4884

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan[c*x])^(p +
1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]

Rubi steps

\begin{align*} \int x^5 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{3} (b c) \int \frac{x^6 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=\frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b \int x^4 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c}+\frac{b \int \frac{x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c}\\ &=-\frac{b x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{15} b^2 \int \frac{x^5}{1+c^2 x^2} \, dx+\frac{b \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c^3}-\frac{b \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c^3}\\ &=\frac{b x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{b x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{30} b^2 \operatorname{Subst}\left (\int \frac{x^2}{1+c^2 x} \, dx,x,x^2\right )-\frac{b \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c^5}+\frac{b \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{3 c^5}-\frac{b^2 \int \frac{x^3}{1+c^2 x^2} \, dx}{9 c^2}\\ &=-\frac{a b x}{3 c^5}+\frac{b x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{b x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{6 c^6}+\frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{30} b^2 \operatorname{Subst}\left (\int \left (-\frac{1}{c^4}+\frac{x}{c^2}+\frac{1}{c^4 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{b^2 \int \tan ^{-1}(c x) \, dx}{3 c^5}-\frac{b^2 \operatorname{Subst}\left (\int \frac{x}{1+c^2 x} \, dx,x,x^2\right )}{18 c^2}\\ &=-\frac{a b x}{3 c^5}-\frac{b^2 x^2}{30 c^4}+\frac{b^2 x^4}{60 c^2}-\frac{b^2 x \tan ^{-1}(c x)}{3 c^5}+\frac{b x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{b x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{6 c^6}+\frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{b^2 \log \left (1+c^2 x^2\right )}{30 c^6}+\frac{b^2 \int \frac{x}{1+c^2 x^2} \, dx}{3 c^4}-\frac{b^2 \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{18 c^2}\\ &=-\frac{a b x}{3 c^5}-\frac{4 b^2 x^2}{45 c^4}+\frac{b^2 x^4}{60 c^2}-\frac{b^2 x \tan ^{-1}(c x)}{3 c^5}+\frac{b x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{b x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{6 c^6}+\frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{23 b^2 \log \left (1+c^2 x^2\right )}{90 c^6}\\ \end{align*}

Mathematica [A]  time = 0.129538, size = 138, normalized size = 0.96 \[ \frac{c x \left (30 a^2 c^5 x^5-4 a b \left (3 c^4 x^4-5 c^2 x^2+15\right )+b^2 c x \left (3 c^2 x^2-16\right )\right )+4 b \tan ^{-1}(c x) \left (15 a \left (c^6 x^6+1\right )+b c x \left (-3 c^4 x^4+5 c^2 x^2-15\right )\right )+46 b^2 \log \left (c^2 x^2+1\right )+30 b^2 \left (c^6 x^6+1\right ) \tan ^{-1}(c x)^2}{180 c^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*ArcTan[c*x])^2,x]

[Out]

(c*x*(30*a^2*c^5*x^5 + b^2*c*x*(-16 + 3*c^2*x^2) - 4*a*b*(15 - 5*c^2*x^2 + 3*c^4*x^4)) + 4*b*(b*c*x*(-15 + 5*c
^2*x^2 - 3*c^4*x^4) + 15*a*(1 + c^6*x^6))*ArcTan[c*x] + 30*b^2*(1 + c^6*x^6)*ArcTan[c*x]^2 + 46*b^2*Log[1 + c^
2*x^2])/(180*c^6)

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Maple [A]  time = 0.013, size = 171, normalized size = 1.2 \begin{align*}{\frac{{x}^{6}{a}^{2}}{6}}+{\frac{{b}^{2}{x}^{6} \left ( \arctan \left ( cx \right ) \right ) ^{2}}{6}}-{\frac{{b}^{2}\arctan \left ( cx \right ){x}^{5}}{15\,c}}+{\frac{{b}^{2}\arctan \left ( cx \right ){x}^{3}}{9\,{c}^{3}}}-{\frac{{b}^{2}x\arctan \left ( cx \right ) }{3\,{c}^{5}}}+{\frac{{b}^{2} \left ( \arctan \left ( cx \right ) \right ) ^{2}}{6\,{c}^{6}}}+{\frac{{b}^{2}{x}^{4}}{60\,{c}^{2}}}-{\frac{4\,{b}^{2}{x}^{2}}{45\,{c}^{4}}}+{\frac{23\,{b}^{2}\ln \left ({c}^{2}{x}^{2}+1 \right ) }{90\,{c}^{6}}}+{\frac{ab{x}^{6}\arctan \left ( cx \right ) }{3}}-{\frac{a{x}^{5}b}{15\,c}}+{\frac{ab{x}^{3}}{9\,{c}^{3}}}-{\frac{xab}{3\,{c}^{5}}}+{\frac{ab\arctan \left ( cx \right ) }{3\,{c}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*arctan(c*x))^2,x)

[Out]

1/6*x^6*a^2+1/6*b^2*x^6*arctan(c*x)^2-1/15/c*b^2*arctan(c*x)*x^5+1/9/c^3*b^2*arctan(c*x)*x^3-1/3*b^2*x*arctan(
c*x)/c^5+1/6/c^6*b^2*arctan(c*x)^2+1/60*b^2*x^4/c^2-4/45*b^2*x^2/c^4+23/90*b^2*ln(c^2*x^2+1)/c^6+1/3*a*b*x^6*a
rctan(c*x)-1/15/c*x^5*a*b+1/9*a*b*x^3/c^3-1/3*a*b*x/c^5+1/3/c^6*a*b*arctan(c*x)

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Maxima [A]  time = 1.50921, size = 220, normalized size = 1.53 \begin{align*} \frac{1}{6} \, b^{2} x^{6} \arctan \left (c x\right )^{2} + \frac{1}{6} \, a^{2} x^{6} + \frac{1}{45} \,{\left (15 \, x^{6} \arctan \left (c x\right ) - c{\left (\frac{3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac{15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} a b - \frac{1}{180} \,{\left (4 \, c{\left (\frac{3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac{15 \, \arctan \left (c x\right )}{c^{7}}\right )} \arctan \left (c x\right ) - \frac{3 \, c^{4} x^{4} - 16 \, c^{2} x^{2} - 30 \, \arctan \left (c x\right )^{2} + 46 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arctan(c*x))^2,x, algorithm="maxima")

[Out]

1/6*b^2*x^6*arctan(c*x)^2 + 1/6*a^2*x^6 + 1/45*(15*x^6*arctan(c*x) - c*((3*c^4*x^5 - 5*c^2*x^3 + 15*x)/c^6 - 1
5*arctan(c*x)/c^7))*a*b - 1/180*(4*c*((3*c^4*x^5 - 5*c^2*x^3 + 15*x)/c^6 - 15*arctan(c*x)/c^7)*arctan(c*x) - (
3*c^4*x^4 - 16*c^2*x^2 - 30*arctan(c*x)^2 + 46*log(c^2*x^2 + 1))/c^6)*b^2

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Fricas [A]  time = 2.54298, size = 351, normalized size = 2.44 \begin{align*} \frac{30 \, a^{2} c^{6} x^{6} - 12 \, a b c^{5} x^{5} + 3 \, b^{2} c^{4} x^{4} + 20 \, a b c^{3} x^{3} - 16 \, b^{2} c^{2} x^{2} - 60 \, a b c x + 30 \,{\left (b^{2} c^{6} x^{6} + b^{2}\right )} \arctan \left (c x\right )^{2} + 46 \, b^{2} \log \left (c^{2} x^{2} + 1\right ) + 4 \,{\left (15 \, a b c^{6} x^{6} - 3 \, b^{2} c^{5} x^{5} + 5 \, b^{2} c^{3} x^{3} - 15 \, b^{2} c x + 15 \, a b\right )} \arctan \left (c x\right )}{180 \, c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arctan(c*x))^2,x, algorithm="fricas")

[Out]

1/180*(30*a^2*c^6*x^6 - 12*a*b*c^5*x^5 + 3*b^2*c^4*x^4 + 20*a*b*c^3*x^3 - 16*b^2*c^2*x^2 - 60*a*b*c*x + 30*(b^
2*c^6*x^6 + b^2)*arctan(c*x)^2 + 46*b^2*log(c^2*x^2 + 1) + 4*(15*a*b*c^6*x^6 - 3*b^2*c^5*x^5 + 5*b^2*c^3*x^3 -
15*b^2*c*x + 15*a*b)*arctan(c*x))/c^6

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Sympy [A]  time = 3.60678, size = 199, normalized size = 1.38 \begin{align*} \begin{cases} \frac{a^{2} x^{6}}{6} + \frac{a b x^{6} \operatorname{atan}{\left (c x \right )}}{3} - \frac{a b x^{5}}{15 c} + \frac{a b x^{3}}{9 c^{3}} - \frac{a b x}{3 c^{5}} + \frac{a b \operatorname{atan}{\left (c x \right )}}{3 c^{6}} + \frac{b^{2} x^{6} \operatorname{atan}^{2}{\left (c x \right )}}{6} - \frac{b^{2} x^{5} \operatorname{atan}{\left (c x \right )}}{15 c} + \frac{b^{2} x^{4}}{60 c^{2}} + \frac{b^{2} x^{3} \operatorname{atan}{\left (c x \right )}}{9 c^{3}} - \frac{4 b^{2} x^{2}}{45 c^{4}} - \frac{b^{2} x \operatorname{atan}{\left (c x \right )}}{3 c^{5}} + \frac{23 b^{2} \log{\left (x^{2} + \frac{1}{c^{2}} \right )}}{90 c^{6}} + \frac{b^{2} \operatorname{atan}^{2}{\left (c x \right )}}{6 c^{6}} & \text{for}\: c \neq 0 \\\frac{a^{2} x^{6}}{6} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(a+b*atan(c*x))**2,x)

[Out]

Piecewise((a**2*x**6/6 + a*b*x**6*atan(c*x)/3 - a*b*x**5/(15*c) + a*b*x**3/(9*c**3) - a*b*x/(3*c**5) + a*b*ata
n(c*x)/(3*c**6) + b**2*x**6*atan(c*x)**2/6 - b**2*x**5*atan(c*x)/(15*c) + b**2*x**4/(60*c**2) + b**2*x**3*atan
(c*x)/(9*c**3) - 4*b**2*x**2/(45*c**4) - b**2*x*atan(c*x)/(3*c**5) + 23*b**2*log(x**2 + c**(-2))/(90*c**6) + b
**2*atan(c*x)**2/(6*c**6), Ne(c, 0)), (a**2*x**6/6, True))

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Giac [A]  time = 1.20348, size = 243, normalized size = 1.69 \begin{align*} \frac{30 \, b^{2} c^{6} x^{6} \arctan \left (c x\right )^{2} + 60 \, a b c^{6} x^{6} \arctan \left (c x\right ) + 30 \, a^{2} c^{6} x^{6} - 12 \, b^{2} c^{5} x^{5} \arctan \left (c x\right ) - 12 \, a b c^{5} x^{5} + 3 \, b^{2} c^{4} x^{4} + 20 \, b^{2} c^{3} x^{3} \arctan \left (c x\right ) + 20 \, a b c^{3} x^{3} - 16 \, b^{2} c^{2} x^{2} - 60 \, b^{2} c x \arctan \left (c x\right ) - 60 \, \pi a b \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) - 60 \, a b c x + 30 \, b^{2} \arctan \left (c x\right )^{2} + 60 \, a b \arctan \left (c x\right ) + 46 \, b^{2} \log \left (c^{2} x^{2} + 1\right )}{180 \, c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arctan(c*x))^2,x, algorithm="giac")

[Out]

1/180*(30*b^2*c^6*x^6*arctan(c*x)^2 + 60*a*b*c^6*x^6*arctan(c*x) + 30*a^2*c^6*x^6 - 12*b^2*c^5*x^5*arctan(c*x)
- 12*a*b*c^5*x^5 + 3*b^2*c^4*x^4 + 20*b^2*c^3*x^3*arctan(c*x) + 20*a*b*c^3*x^3 - 16*b^2*c^2*x^2 - 60*b^2*c*x*
arctan(c*x) - 60*pi*a*b*sgn(c)*sgn(x) - 60*a*b*c*x + 30*b^2*arctan(c*x)^2 + 60*a*b*arctan(c*x) + 46*b^2*log(c^
2*x^2 + 1))/c^6