\(\int \frac {\arctan (a x)}{(c+a^2 c x^2)^{5/2}} \, dx\) [245]

Optimal result

Integrand size = 19, antiderivative size = 101 \[ \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {1}{9 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2}{3 a c^2 \sqrt {c+a^2 c x^2}}+\frac {x \arctan (a x)}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \arctan (a x)}{3 c^2 \sqrt {c+a^2 c x^2}} \]

[Out]

1/9/a/c/(a^2*c*x^2+c)^(3/2)+1/3*x*arctan(a*x)/c/(a^2*c*x^2+c)^(3/2)+2/3/a/c^2/(a^2*c*x^2+c)^(1/2)+2/3*x*arctan
(a*x)/c^2/(a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5016, 5014} \[ \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {2 x \arctan (a x)}{3 c^2 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2}{3 a c^2 \sqrt {a^2 c x^2+c}}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}} \]

[In]

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

[Out]

1/(9*a*c*(c + a^2*c*x^2)^(3/2)) + 2/(3*a*c^2*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x])/(3*c*(c + a^2*c*x^2)^(3/2)
) + (2*x*ArcTan[a*x])/(3*c^2*Sqrt[c + a^2*c*x^2])

Rule 5014

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

Rule 5016

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{9 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {x \arctan (a x)}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c} \\ & = \frac {1}{9 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2}{3 a c^2 \sqrt {c+a^2 c x^2}}+\frac {x \arctan (a x)}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \arctan (a x)}{3 c^2 \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.62 \[ \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {\sqrt {c+a^2 c x^2} \left (7+6 a^2 x^2+\left (9 a x+6 a^3 x^3\right ) \arctan (a x)\right )}{9 a c^3 \left (1+a^2 x^2\right )^2} \]

[In]

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

[Out]

(Sqrt[c + a^2*c*x^2]*(7 + 6*a^2*x^2 + (9*a*x + 6*a^3*x^3)*ArcTan[a*x]))/(9*a*c^3*(1 + a^2*x^2)^2)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.50 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.38

method	result	size
default	\(-\frac {\left (i+3 \arctan \left (a x \right )\right ) \left (a^{3} x^{3}-3 i a^{2} x^{2}-3 a x +i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{72 \left (a^{2} x^{2}+1\right )^{2} c^{3} a}+\frac {3 \left (\arctan \left (a x \right )+i\right ) \left (a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 a \,c^{3} \left (a^{2} x^{2}+1\right )}+\frac {3 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a x +i\right ) \left (\arctan \left (a x \right )-i\right )}{8 a \,c^{3} \left (a^{2} x^{2}+1\right )}-\frac {\left (-i+3 \arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a^{3} x^{3}+3 i a^{2} x^{2}-3 a x -i\right )}{72 \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right ) c^{3} a}\)	\(240\)

[In]

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

[Out]

-1/72*(I+3*arctan(a*x))*(a^3*x^3-3*I*a^2*x^2-3*a*x+I)*(c*(a*x-I)*(I+a*x))^(1/2)/(a^2*x^2+1)^2/c^3/a+3/8*(arcta
n(a*x)+I)*(a*x-I)*(c*(a*x-I)*(I+a*x))^(1/2)/a/c^3/(a^2*x^2+1)+3/8*(c*(a*x-I)*(I+a*x))^(1/2)*(I+a*x)*(arctan(a*
x)-I)/a/c^3/(a^2*x^2+1)-1/72*(-I+3*arctan(a*x))*(c*(a*x-I)*(I+a*x))^(1/2)*(a^3*x^3+3*I*a^2*x^2-3*a*x-I)/(a^4*x
^4+2*a^2*x^2+1)/c^3/a

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.71 \[ \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {\sqrt {a^{2} c x^{2} + c} {\left (6 \, a^{2} x^{2} + 3 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arctan \left (a x\right ) + 7\right )}}{9 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \]

[In]

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

[Out]

1/9*sqrt(a^2*c*x^2 + c)*(6*a^2*x^2 + 3*(2*a^3*x^3 + 3*a*x)*arctan(a*x) + 7)/(a^5*c^3*x^4 + 2*a^3*c^3*x^2 + a*c
^3)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.85 \[ \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {1}{9} \, a {\left (\frac {6}{\sqrt {a^{2} c x^{2} + c} a^{2} c^{2}} + \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{2} c}\right )} + \frac {1}{3} \, {\left (\frac {2 \, x}{\sqrt {a^{2} c x^{2} + c} c^{2}} + \frac {x}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c}\right )} \arctan \left (a x\right ) \]

[In]

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

[Out]

1/9*a*(6/(sqrt(a^2*c*x^2 + c)*a^2*c^2) + 1/((a^2*c*x^2 + c)^(3/2)*a^2*c)) + 1/3*(2*x/(sqrt(a^2*c*x^2 + c)*c^2)
+ x/((a^2*c*x^2 + c)^(3/2)*c))*arctan(a*x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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