\(\int \frac {x \arctan (a x)^3}{(c+a^2 c x^2)^{5/2}} \, dx\) [454]

Optimal result

Integrand size = 22, antiderivative size = 199 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {2 x}{27 a c \left (c+a^2 c x^2\right )^{3/2}}-\frac {40 x}{27 a c^2 \sqrt {c+a^2 c x^2}}+\frac {2 \arctan (a x)}{9 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {4 \arctan (a x)}{3 a^2 c^2 \sqrt {c+a^2 c x^2}}+\frac {x \arctan (a x)^2}{3 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \arctan (a x)^2}{3 a c^2 \sqrt {c+a^2 c x^2}}-\frac {\arctan (a x)^3}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}} \]

[Out]

-2/27*x/a/c/(a^2*c*x^2+c)^(3/2)+2/9*arctan(a*x)/a^2/c/(a^2*c*x^2+c)^(3/2)+1/3*x*arctan(a*x)^2/a/c/(a^2*c*x^2+c
)^(3/2)-1/3*arctan(a*x)^3/a^2/c/(a^2*c*x^2+c)^(3/2)-40/27*x/a/c^2/(a^2*c*x^2+c)^(1/2)+4/3*arctan(a*x)/a^2/c^2/
(a^2*c*x^2+c)^(1/2)+2/3*x*arctan(a*x)^2/a/c^2/(a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5050, 5020, 5018, 197, 198} \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {2 x \arctan (a x)^2}{3 a c^2 \sqrt {a^2 c x^2+c}}+\frac {4 \arctan (a x)}{3 a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^3}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x \arctan (a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \arctan (a x)}{9 a^2 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {40 x}{27 a c^2 \sqrt {a^2 c x^2+c}}-\frac {2 x}{27 a c \left (a^2 c x^2+c\right )^{3/2}} \]

[In]

Int[(x*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(5/2),x]

[Out]

(-2*x)/(27*a*c*(c + a^2*c*x^2)^(3/2)) - (40*x)/(27*a*c^2*Sqrt[c + a^2*c*x^2]) + (2*ArcTan[a*x])/(9*a^2*c*(c +
a^2*c*x^2)^(3/2)) + (4*ArcTan[a*x])/(3*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^2)/(3*a*c*(c + a^2*c*x^2)
^(3/2)) + (2*x*ArcTan[a*x]^2)/(3*a*c^2*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]^3/(3*a^2*c*(c + a^2*c*x^2)^(3/2))

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

Rule 5018

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[b*p*((a + b*ArcTan[
c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2])), x] + (-Dist[b^2*p*(p - 1), Int[(a + b*ArcTan[c*x])^(p - 2)/(d + e*x^2)^(
3/2), x], x] + Simp[x*((a + b*ArcTan[c*x])^p/(d*Sqrt[d + e*x^2])), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e,
c^2*d] && GtQ[p, 1]

Rule 5020

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Simp[b*p*(d + e*x^2)^(q +
1)*((a + b*ArcTan[c*x])^(p - 1)/(4*c*d*(q + 1)^2)), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q +
1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[b^2*p*((p - 1)/(4*(q + 1)^2)), Int[(d + e*x^2)^q*(a + b*ArcTan[c*x])^(
p - 2), x], x] - Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*d*(q + 1))), x]) /; FreeQ[{a, b, c, d, e
}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]

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 {\arctan (a x)^3}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{a} \\ & = \frac {2 \arctan (a x)}{9 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {x \arctan (a x)^2}{3 a c \left (c+a^2 c x^2\right )^{3/2}}-\frac {\arctan (a x)^3}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{9 a}+\frac {2 \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a c} \\ & = -\frac {2 x}{27 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 \arctan (a x)}{9 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {4 \arctan (a x)}{3 a^2 c^2 \sqrt {c+a^2 c x^2}}+\frac {x \arctan (a x)^2}{3 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \arctan (a x)^2}{3 a c^2 \sqrt {c+a^2 c x^2}}-\frac {\arctan (a x)^3}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {4 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{27 a c}-\frac {4 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a c} \\ & = -\frac {2 x}{27 a c \left (c+a^2 c x^2\right )^{3/2}}-\frac {40 x}{27 a c^2 \sqrt {c+a^2 c x^2}}+\frac {2 \arctan (a x)}{9 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {4 \arctan (a x)}{3 a^2 c^2 \sqrt {c+a^2 c x^2}}+\frac {x \arctan (a x)^2}{3 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \arctan (a x)^2}{3 a c^2 \sqrt {c+a^2 c x^2}}-\frac {\arctan (a x)^3}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.46 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {\sqrt {c+a^2 c x^2} \left (-2 a x \left (21+20 a^2 x^2\right )+6 \left (7+6 a^2 x^2\right ) \arctan (a x)+9 a x \left (3+2 a^2 x^2\right ) \arctan (a x)^2-9 \arctan (a x)^3\right )}{27 c^3 \left (a+a^3 x^2\right )^2} \]

[In]

Integrate[(x*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(5/2),x]

[Out]

(Sqrt[c + a^2*c*x^2]*(-2*a*x*(21 + 20*a^2*x^2) + 6*(7 + 6*a^2*x^2)*ArcTan[a*x] + 9*a*x*(3 + 2*a^2*x^2)*ArcTan[
a*x]^2 - 9*ArcTan[a*x]^3))/(27*c^3*(a + a^3*x^2)^2)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.46 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.57

method	result	size
default	\(\frac {\left (9 i \arctan \left (a x \right )^{2}+9 \arctan \left (a x \right )^{3}-2 i-6 \arctan \left (a x \right )\right ) \left (i a^{3} x^{3}+3 a^{2} x^{2}-3 i a x -1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{216 \left (a^{2} x^{2}+1\right )^{2} a^{2} c^{3}}-\frac {\left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )+3 i \arctan \left (a x \right )^{2}-6 i\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 c^{3} a^{2} \left (a^{2} x^{2}+1\right )}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )-3 i \arctan \left (a x \right )^{2}+6 i\right )}{8 c^{3} a^{2} \left (a^{2} x^{2}+1\right )}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a^{3} x^{3}-3 a^{2} x^{2}-3 i a x +1\right ) \left (-9 i \arctan \left (a x \right )^{2}+9 \arctan \left (a x \right )^{3}+2 i-6 \arctan \left (a x \right )\right )}{216 c^{3} a^{2} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}\)	\(312\)

[In]

int(x*arctan(a*x)^3/(a^2*c*x^2+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/216*(9*I*arctan(a*x)^2+9*arctan(a*x)^3-2*I-6*arctan(a*x))*(I*a^3*x^3+3*a^2*x^2-3*I*a*x-1)*(c*(a*x-I)*(I+a*x)
)^(1/2)/(a^2*x^2+1)^2/a^2/c^3-1/8*(arctan(a*x)^3-6*arctan(a*x)+3*I*arctan(a*x)^2-6*I)*(1+I*a*x)*(c*(a*x-I)*(I+
a*x))^(1/2)/c^3/a^2/(a^2*x^2+1)+1/8*(c*(a*x-I)*(I+a*x))^(1/2)*(I*a*x-1)*(arctan(a*x)^3-6*arctan(a*x)-3*I*arcta
n(a*x)^2+6*I)/c^3/a^2/(a^2*x^2+1)-1/216*(c*(a*x-I)*(I+a*x))^(1/2)*(I*a^3*x^3-3*a^2*x^2-3*I*a*x+1)*(-9*I*arctan
(a*x)^2+9*arctan(a*x)^3+2*I-6*arctan(a*x))/c^3/a^2/(a^4*x^4+2*a^2*x^2+1)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.52 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {{\left (40 \, a^{3} x^{3} - 9 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arctan \left (a x\right )^{2} + 9 \, \arctan \left (a x\right )^{3} + 42 \, a x - 6 \, {\left (6 \, a^{2} x^{2} + 7\right )} \arctan \left (a x\right )\right )} \sqrt {a^{2} c x^{2} + c}}{27 \, {\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )}} \]

[In]

integrate(x*arctan(a*x)^3/(a^2*c*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

-1/27*(40*a^3*x^3 - 9*(2*a^3*x^3 + 3*a*x)*arctan(a*x)^2 + 9*arctan(a*x)^3 + 42*a*x - 6*(6*a^2*x^2 + 7)*arctan(
a*x))*sqrt(a^2*c*x^2 + c)/(a^6*c^3*x^4 + 2*a^4*c^3*x^2 + a^2*c^3)

Sympy [F]

\[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x \operatorname {atan}^{3}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(x*atan(a*x)**3/(a**2*c*x**2+c)**(5/2),x)

[Out]

Integral(x*atan(a*x)**3/(c*(a**2*x**2 + 1))**(5/2), x)

Maxima [F]

\[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x*arctan(a*x)^3/(a^2*c*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

integrate(x*arctan(a*x)^3/(a^2*c*x^2 + c)^(5/2), x)

Giac [F]

\[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x*arctan(a*x)^3/(a^2*c*x^2+c)^(5/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x\,{\mathrm {atan}\left (a\,x\right )}^3}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

[In]

int((x*atan(a*x)^3)/(c + a^2*c*x^2)^(5/2),x)

[Out]

int((x*atan(a*x)^3)/(c + a^2*c*x^2)^(5/2), x)