3.452 \(\int \frac {x^3 \tan ^{-1}(a x)^3}{(c+a^2 c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=237 \[ -\frac {x^2 \tan ^{-1}(a x)^3}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 x^2 \tan ^{-1}(a x)}{9 a^2 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {2 x^3}{27 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x^3 \tan ^{-1}(a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac {2 \tan ^{-1}(a x)^3}{3 a^4 c^2 \sqrt {a^2 c x^2+c}}+\frac {40 \tan ^{-1}(a x)}{9 a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {40 x}{9 a^3 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 x \tan ^{-1}(a x)^2}{a^3 c^2 \sqrt {a^2 c x^2+c}} \]

[Out]

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

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Rubi [A]  time = 0.41, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4940, 4930, 4898, 191, 4938} \[ -\frac {40 x}{9 a^3 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 x \tan ^{-1}(a x)^2}{a^3 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 \tan ^{-1}(a x)^3}{3 a^4 c^2 \sqrt {a^2 c x^2+c}}+\frac {40 \tan ^{-1}(a x)}{9 a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 x^3}{27 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x^3 \tan ^{-1}(a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac {x^2 \tan ^{-1}(a x)^3}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 x^2 \tan ^{-1}(a x)}{9 a^2 c \left (a^2 c x^2+c\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 191

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 4898

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 4930

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]

Rule 4938

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

Rule 4940

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

Rubi steps

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

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Mathematica [A]  time = 0.14, size = 104, normalized size = 0.44 \[ \frac {\sqrt {a^2 c x^2+c} \left (-2 a x \left (61 a^2 x^2+60\right )-9 \left (3 a^2 x^2+2\right ) \tan ^{-1}(a x)^3+9 a x \left (7 a^2 x^2+6\right ) \tan ^{-1}(a x)^2+6 \left (21 a^2 x^2+20\right ) \tan ^{-1}(a x)\right )}{27 a^4 c^3 \left (a^2 x^2+1\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + a^2*c*x^2]*(-2*a*x*(60 + 61*a^2*x^2) + 6*(20 + 21*a^2*x^2)*ArcTan[a*x] + 9*a*x*(6 + 7*a^2*x^2)*ArcTa
n[a*x]^2 - 9*(2 + 3*a^2*x^2)*ArcTan[a*x]^3))/(27*a^4*c^3*(1 + a^2*x^2)^2)

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fricas [A]  time = 0.41, size = 113, normalized size = 0.48 \[ -\frac {{\left (122 \, a^{3} x^{3} + 9 \, {\left (3 \, a^{2} x^{2} + 2\right )} \arctan \left (a x\right )^{3} - 9 \, {\left (7 \, a^{3} x^{3} + 6 \, a x\right )} \arctan \left (a x\right )^{2} + 120 \, a x - 6 \, {\left (21 \, a^{2} x^{2} + 20\right )} \arctan \left (a x\right )\right )} \sqrt {a^{2} c x^{2} + c}}{27 \, {\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*(122*a^3*x^3 + 9*(3*a^2*x^2 + 2)*arctan(a*x)^3 - 9*(7*a^3*x^3 + 6*a*x)*arctan(a*x)^2 + 120*a*x - 6*(21*a
^2*x^2 + 20)*arctan(a*x))*sqrt(a^2*c*x^2 + c)/(a^8*c^3*x^4 + 2*a^6*c^3*x^2 + a^4*c^3)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [C]  time = 2.63, size = 312, normalized size = 1.32 \[ -\frac {\left (9 i \arctan \left (a x \right )^{2}+9 \arctan \left (a x \right )^{3}-2 i-6 \arctan \left (a x \right )\right ) \left (i x^{3} a^{3}+3 a^{2} x^{2}-3 i a x -1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{216 \left (a^{2} x^{2}+1\right )^{2} c^{3} a^{4}}-\frac {3 \left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )+3 i \arctan \left (a x \right )^{2}-6 i\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 a^{4} c^{3} \left (a^{2} x^{2}+1\right )}+\frac {3 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )-3 i \arctan \left (a x \right )^{2}+6 i\right )}{8 a^{4} c^{3} \left (a^{2} x^{2}+1\right )}+\frac {\left (-9 i \arctan \left (a x \right )^{2}+9 \arctan \left (a x \right )^{3}+2 i-6 \arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i x^{3} a^{3}-3 a^{2} x^{2}-3 i a x +1\right )}{216 \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right ) c^{3} a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^3}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \operatorname {atan}^{3}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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