\(\int x (c+a^2 c x^2)^3 \arctan (a x) \, dx\) [167]

Optimal result

Integrand size = 18, antiderivative size = 74 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=-\frac {c^3 x}{8 a}-\frac {1}{8} a c^3 x^3-\frac {3}{40} a^3 c^3 x^5-\frac {1}{56} a^5 c^3 x^7+\frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)}{8 a^2} \]

[Out]

-1/8*c^3*x/a-1/8*a*c^3*x^3-3/40*a^3*c^3*x^5-1/56*a^5*c^3*x^7+1/8*c^3*(a^2*x^2+1)^4*arctan(a*x)/a^2

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5050, 200} \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=-\frac {1}{56} a^5 c^3 x^7-\frac {3}{40} a^3 c^3 x^5+\frac {c^3 \left (a^2 x^2+1\right )^4 \arctan (a x)}{8 a^2}-\frac {1}{8} a c^3 x^3-\frac {c^3 x}{8 a} \]

[In]

Int[x*(c + a^2*c*x^2)^3*ArcTan[a*x],x]

[Out]

-1/8*(c^3*x)/a - (a*c^3*x^3)/8 - (3*a^3*c^3*x^5)/40 - (a^5*c^3*x^7)/56 + (c^3*(1 + a^2*x^2)^4*ArcTan[a*x])/(8*
a^2)

Rule 200

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 5050

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

Rubi steps \begin{align*} \text {integral}& = \frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)}{8 a^2}-\frac {\int \left (c+a^2 c x^2\right )^3 \, dx}{8 a} \\ & = \frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)}{8 a^2}-\frac {\int \left (c^3+3 a^2 c^3 x^2+3 a^4 c^3 x^4+a^6 c^3 x^6\right ) \, dx}{8 a} \\ & = -\frac {c^3 x}{8 a}-\frac {1}{8} a c^3 x^3-\frac {3}{40} a^3 c^3 x^5-\frac {1}{56} a^5 c^3 x^7+\frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)}{8 a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.73 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=-\frac {c^3 x}{8 a}-\frac {1}{8} a c^3 x^3-\frac {3}{40} a^3 c^3 x^5-\frac {1}{56} a^5 c^3 x^7+\frac {c^3 \arctan (a x)}{8 a^2}+\frac {1}{2} c^3 x^2 \arctan (a x)+\frac {3}{4} a^2 c^3 x^4 \arctan (a x)+\frac {1}{2} a^4 c^3 x^6 \arctan (a x)+\frac {1}{8} a^6 c^3 x^8 \arctan (a x) \]

[In]

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

[Out]

-1/8*(c^3*x)/a - (a*c^3*x^3)/8 - (3*a^3*c^3*x^5)/40 - (a^5*c^3*x^7)/56 + (c^3*ArcTan[a*x])/(8*a^2) + (c^3*x^2*
ArcTan[a*x])/2 + (3*a^2*c^3*x^4*ArcTan[a*x])/4 + (a^4*c^3*x^6*ArcTan[a*x])/2 + (a^6*c^3*x^8*ArcTan[a*x])/8

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.41

method	result	size
parts	\(\frac {c^{3} \arctan \left (a x \right ) a^{6} x^{8}}{8}+\frac {c^{3} \arctan \left (a x \right ) a^{4} x^{6}}{2}+\frac {3 c^{3} \arctan \left (a x \right ) a^{2} x^{4}}{4}+\frac {c^{3} \arctan \left (a x \right ) x^{2}}{2}+\frac {c^{3} \arctan \left (a x \right )}{8 a^{2}}-\frac {c^{3} \left (\frac {1}{7} a^{6} x^{7}+\frac {3}{5} a^{4} x^{5}+a^{2} x^{3}+x \right )}{8 a}\)	\(104\)
derivativedivides	\(\frac {\frac {c^{3} \arctan \left (a x \right ) a^{8} x^{8}}{8}+\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )}{2}+\frac {3 a^{4} c^{3} x^{4} \arctan \left (a x \right )}{4}+\frac {a^{2} c^{3} x^{2} \arctan \left (a x \right )}{2}+\frac {c^{3} \arctan \left (a x \right )}{8}-\frac {c^{3} \left (\frac {1}{7} a^{7} x^{7}+\frac {3}{5} a^{5} x^{5}+a^{3} x^{3}+a x \right )}{8}}{a^{2}}\)	\(107\)
default	\(\frac {\frac {c^{3} \arctan \left (a x \right ) a^{8} x^{8}}{8}+\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )}{2}+\frac {3 a^{4} c^{3} x^{4} \arctan \left (a x \right )}{4}+\frac {a^{2} c^{3} x^{2} \arctan \left (a x \right )}{2}+\frac {c^{3} \arctan \left (a x \right )}{8}-\frac {c^{3} \left (\frac {1}{7} a^{7} x^{7}+\frac {3}{5} a^{5} x^{5}+a^{3} x^{3}+a x \right )}{8}}{a^{2}}\)	\(107\)
parallelrisch	\(\frac {35 c^{3} \arctan \left (a x \right ) a^{8} x^{8}-5 a^{7} c^{3} x^{7}+140 a^{6} c^{3} x^{6} \arctan \left (a x \right )-21 a^{5} c^{3} x^{5}+210 a^{4} c^{3} x^{4} \arctan \left (a x \right )-35 a^{3} c^{3} x^{3}+140 a^{2} c^{3} x^{2} \arctan \left (a x \right )-35 a \,c^{3} x +35 c^{3} \arctan \left (a x \right )}{280 a^{2}}\)	\(116\)
risch	\(-\frac {i c^{3} \left (a^{2} x^{2}+1\right )^{4} \ln \left (i a x +1\right )}{16 a^{2}}+\frac {i c^{3} a^{6} x^{8} \ln \left (-i a x +1\right )}{16}-\frac {a^{5} c^{3} x^{7}}{56}+\frac {i c^{3} a^{4} x^{6} \ln \left (-i a x +1\right )}{4}-\frac {3 a^{3} c^{3} x^{5}}{40}+\frac {3 i c^{3} a^{2} x^{4} \ln \left (-i a x +1\right )}{8}-\frac {a \,c^{3} x^{3}}{8}+\frac {i c^{3} x^{2} \ln \left (-i a x +1\right )}{4}-\frac {c^{3} x}{8 a}+\frac {i c^{3} \ln \left (a^{2} x^{2}+1\right )}{32 a^{2}}+\frac {c^{3} \arctan \left (a x \right )}{16 a^{2}}\)	\(178\)
meijerg	\(\frac {c^{3} \left (\frac {x a \left (-45 a^{6} x^{6}+63 a^{4} x^{4}-105 a^{2} x^{2}+315\right )}{630}-\frac {x a \left (-9 a^{8} x^{8}+9\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{18 \sqrt {a^{2} x^{2}}}\right )}{4 a^{2}}+\frac {3 c^{3} \left (-\frac {2 a x \left (21 a^{4} x^{4}-35 a^{2} x^{2}+105\right )}{315}+\frac {2 a x \left (7 a^{6} x^{6}+7\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{21 \sqrt {a^{2} x^{2}}}\right )}{4 a^{2}}+\frac {3 c^{3} \left (\frac {a x \left (-5 a^{2} x^{2}+15\right )}{15}-\frac {a x \left (-5 a^{4} x^{4}+5\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{5 \sqrt {a^{2} x^{2}}}\right )}{4 a^{2}}+\frac {c^{3} \left (-2 a x +\frac {2 \left (3 a^{2} x^{2}+3\right ) \arctan \left (a x \right )}{3}\right )}{4 a^{2}}\)	\(223\)

[In]

int(x*(a^2*c*x^2+c)^3*arctan(a*x),x,method=_RETURNVERBOSE)

[Out]

1/8*c^3*arctan(a*x)*a^6*x^8+1/2*c^3*arctan(a*x)*a^4*x^6+3/4*c^3*arctan(a*x)*a^2*x^4+1/2*c^3*arctan(a*x)*x^2+1/
8*c^3/a^2*arctan(a*x)-1/8*c^3/a*(1/7*a^6*x^7+3/5*a^4*x^5+a^2*x^3+x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.34 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=-\frac {5 \, a^{7} c^{3} x^{7} + 21 \, a^{5} c^{3} x^{5} + 35 \, a^{3} c^{3} x^{3} + 35 \, a c^{3} x - 35 \, {\left (a^{8} c^{3} x^{8} + 4 \, a^{6} c^{3} x^{6} + 6 \, a^{4} c^{3} x^{4} + 4 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )}{280 \, a^{2}} \]

[In]

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

[Out]

-1/280*(5*a^7*c^3*x^7 + 21*a^5*c^3*x^5 + 35*a^3*c^3*x^3 + 35*a*c^3*x - 35*(a^8*c^3*x^8 + 4*a^6*c^3*x^6 + 6*a^4
*c^3*x^4 + 4*a^2*c^3*x^2 + c^3)*arctan(a*x))/a^2

Sympy [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.68 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\begin {cases} \frac {a^{6} c^{3} x^{8} \operatorname {atan}{\left (a x \right )}}{8} - \frac {a^{5} c^{3} x^{7}}{56} + \frac {a^{4} c^{3} x^{6} \operatorname {atan}{\left (a x \right )}}{2} - \frac {3 a^{3} c^{3} x^{5}}{40} + \frac {3 a^{2} c^{3} x^{4} \operatorname {atan}{\left (a x \right )}}{4} - \frac {a c^{3} x^{3}}{8} + \frac {c^{3} x^{2} \operatorname {atan}{\left (a x \right )}}{2} - \frac {c^{3} x}{8 a} + \frac {c^{3} \operatorname {atan}{\left (a x \right )}}{8 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(x*(a**2*c*x**2+c)**3*atan(a*x),x)

[Out]

Piecewise((a**6*c**3*x**8*atan(a*x)/8 - a**5*c**3*x**7/56 + a**4*c**3*x**6*atan(a*x)/2 - 3*a**3*c**3*x**5/40 +
3*a**2*c**3*x**4*atan(a*x)/4 - a*c**3*x**3/8 + c**3*x**2*atan(a*x)/2 - c**3*x/(8*a) + c**3*atan(a*x)/(8*a**2)
, Ne(a, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.99 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\frac {{\left (a^{2} c x^{2} + c\right )}^{4} \arctan \left (a x\right )}{8 \, a^{2} c} - \frac {5 \, a^{6} c^{4} x^{7} + 21 \, a^{4} c^{4} x^{5} + 35 \, a^{2} c^{4} x^{3} + 35 \, c^{4} x}{280 \, a c} \]

[In]

integrate(x*(a^2*c*x^2+c)^3*arctan(a*x),x, algorithm="maxima")

[Out]

1/8*(a^2*c*x^2 + c)^4*arctan(a*x)/(a^2*c) - 1/280*(5*a^6*c^4*x^7 + 21*a^4*c^4*x^5 + 35*a^2*c^4*x^3 + 35*c^4*x)
/(a*c)

Giac [F]

\[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x \arctan \left (a x\right ) \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.35 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\mathrm {atan}\left (a\,x\right )\,\left (\frac {a^6\,c^3\,x^8}{8}+\frac {a^4\,c^3\,x^6}{2}+\frac {3\,a^2\,c^3\,x^4}{4}+\frac {c^3\,x^2}{2}\right )-\frac {c^3\,x}{8\,a}-\frac {a\,c^3\,x^3}{8}+\frac {c^3\,\mathrm {atan}\left (a\,x\right )}{8\,a^2}-\frac {3\,a^3\,c^3\,x^5}{40}-\frac {a^5\,c^3\,x^7}{56} \]

[In]

int(x*atan(a*x)*(c + a^2*c*x^2)^3,x)

[Out]

atan(a*x)*((c^3*x^2)/2 + (3*a^2*c^3*x^4)/4 + (a^4*c^3*x^6)/2 + (a^6*c^3*x^8)/8) - (c^3*x)/(8*a) - (a*c^3*x^3)/
8 + (c^3*atan(a*x))/(8*a^2) - (3*a^3*c^3*x^5)/40 - (a^5*c^3*x^7)/56