3.11 \(\int x^6 \tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}) \, dx\)

Optimal. Leaf size=124 \[ -\frac{d^2 \left (d+e x^2\right )^{3/2}}{7 (-e)^{7/2}}+\frac{d^3 \sqrt{d+e x^2}}{7 (-e)^{7/2}}-\frac{\left (d+e x^2\right )^{7/2}}{49 (-e)^{7/2}}+\frac{3 d \left (d+e x^2\right )^{5/2}}{35 (-e)^{7/2}}+\frac{1}{7} x^7 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \]

[Out]

(d^3*Sqrt[d + e*x^2])/(7*(-e)^(7/2)) - (d^2*(d + e*x^2)^(3/2))/(7*(-e)^(7/2)) + (3*d*(d + e*x^2)^(5/2))/(35*(-
e)^(7/2)) - (d + e*x^2)^(7/2)/(49*(-e)^(7/2)) + (x^7*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/7

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Rubi [A]  time = 0.0670544, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {5151, 266, 43} \[ -\frac{d^2 \left (d+e x^2\right )^{3/2}}{7 (-e)^{7/2}}+\frac{d^3 \sqrt{d+e x^2}}{7 (-e)^{7/2}}-\frac{\left (d+e x^2\right )^{7/2}}{49 (-e)^{7/2}}+\frac{3 d \left (d+e x^2\right )^{5/2}}{35 (-e)^{7/2}}+\frac{1}{7} x^7 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^6*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]],x]

[Out]

(d^3*Sqrt[d + e*x^2])/(7*(-e)^(7/2)) - (d^2*(d + e*x^2)^(3/2))/(7*(-e)^(7/2)) + (3*d*(d + e*x^2)^(5/2))/(35*(-
e)^(7/2)) - (d + e*x^2)^(7/2)/(49*(-e)^(7/2)) + (x^7*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/7

Rule 5151

Int[ArcTan[((c_.)*(x_))/Sqrt[(a_.) + (b_.)*(x_)^2]]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*ArcTa
n[(c*x)/Sqrt[a + b*x^2]])/(d*(m + 1)), x] - Dist[c/(d*(m + 1)), Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x], x] /; F
reeQ[{a, b, c, d, m}, x] && EqQ[b + c^2, 0] && NeQ[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])

Rubi steps

\begin{align*} \int x^6 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \, dx &=\frac{1}{7} x^7 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )-\frac{1}{7} \sqrt{-e} \int \frac{x^7}{\sqrt{d+e x^2}} \, dx\\ &=\frac{1}{7} x^7 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )-\frac{1}{14} \sqrt{-e} \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{d+e x}} \, dx,x,x^2\right )\\ &=\frac{1}{7} x^7 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )-\frac{1}{14} \sqrt{-e} \operatorname{Subst}\left (\int \left (-\frac{d^3}{e^3 \sqrt{d+e x}}+\frac{3 d^2 \sqrt{d+e x}}{e^3}-\frac{3 d (d+e x)^{3/2}}{e^3}+\frac{(d+e x)^{5/2}}{e^3}\right ) \, dx,x,x^2\right )\\ &=\frac{d^3 \sqrt{d+e x^2}}{7 (-e)^{7/2}}-\frac{d^2 \left (d+e x^2\right )^{3/2}}{7 (-e)^{7/2}}+\frac{3 d \left (d+e x^2\right )^{5/2}}{35 (-e)^{7/2}}-\frac{\left (d+e x^2\right )^{7/2}}{49 (-e)^{7/2}}+\frac{1}{7} x^7 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.104445, size = 83, normalized size = 0.67 \[ \frac{\sqrt{d+e x^2} \left (-8 d^2 e x^2+16 d^3+6 d e^2 x^4-5 e^3 x^6\right )}{245 (-e)^{7/2}}+\frac{1}{7} x^7 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^6*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]],x]

[Out]

(Sqrt[d + e*x^2]*(16*d^3 - 8*d^2*e*x^2 + 6*d*e^2*x^4 - 5*e^3*x^6))/(245*(-e)^(7/2)) + (x^7*ArcTan[(Sqrt[-e]*x)
/Sqrt[d + e*x^2]])/7

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Maple [B]  time = 0.046, size = 230, normalized size = 1.9 \begin{align*}{\frac{{x}^{7}}{7}\arctan \left ({x\sqrt{-e}{\frac{1}{\sqrt{e{x}^{2}+d}}}} \right ) }+{\frac{{x}^{8}}{63\,d}\sqrt{-e}\sqrt{e{x}^{2}+d}}-{\frac{8\,{x}^{6}}{441\,e}\sqrt{-e}\sqrt{e{x}^{2}+d}}+{\frac{16\,d{x}^{4}}{735\,{e}^{2}}\sqrt{-e}\sqrt{e{x}^{2}+d}}-{\frac{64\,{d}^{2}{x}^{2}}{2205\,{e}^{3}}\sqrt{-e}\sqrt{e{x}^{2}+d}}+{\frac{128\,{d}^{3}}{2205\,{e}^{4}}\sqrt{-e}\sqrt{e{x}^{2}+d}}-{\frac{{x}^{6}}{63\,de}\sqrt{-e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{2\,{x}^{4}}{147\,{e}^{2}}\sqrt{-e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-{\frac{8\,d{x}^{2}}{735\,{e}^{3}}\sqrt{-e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{16\,{d}^{2}}{2205\,{e}^{4}}\sqrt{-e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2)),x)

[Out]

1/7*x^7*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))+1/63*(-e)^(1/2)/d*x^8*(e*x^2+d)^(1/2)-8/441*(-e)^(1/2)/e*x^6*(e*x
^2+d)^(1/2)+16/735*(-e)^(1/2)/e^2*d*x^4*(e*x^2+d)^(1/2)-64/2205*(-e)^(1/2)/e^3*d^2*x^2*(e*x^2+d)^(1/2)+128/220
5*(-e)^(1/2)/e^4*d^3*(e*x^2+d)^(1/2)-1/63*(-e)^(1/2)/d*x^6*(e*x^2+d)^(3/2)/e+2/147*(-e)^(1/2)/e^2*x^4*(e*x^2+d
)^(3/2)-8/735*(-e)^(1/2)*d/e^3*x^2*(e*x^2+d)^(3/2)+16/2205*(-e)^(1/2)*d^2/e^4*(e*x^2+d)^(3/2)

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Maxima [A]  time = 1.00712, size = 225, normalized size = 1.81 \begin{align*} \frac{1}{7} \, x^{7} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) - \frac{{\left (35 \,{\left (e x^{2} + d\right )}^{\frac{9}{2}} - 135 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} d + 189 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{3}\right )} \sqrt{-e}}{2205 \, d e^{4}} + \frac{{\left (35 \,{\left (e x^{2} + d\right )}^{\frac{9}{2}} - 180 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} d + 378 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{e x^{2} + d} d^{4}\right )} \sqrt{-e}}{2205 \, d e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2)),x, algorithm="maxima")

[Out]

1/7*x^7*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) - 1/2205*(35*(e*x^2 + d)^(9/2) - 135*(e*x^2 + d)^(7/2)*d + 189*(e*x
^2 + d)^(5/2)*d^2 - 105*(e*x^2 + d)^(3/2)*d^3)*sqrt(-e)/(d*e^4) + 1/2205*(35*(e*x^2 + d)^(9/2) - 180*(e*x^2 +
d)^(7/2)*d + 378*(e*x^2 + d)^(5/2)*d^2 - 420*(e*x^2 + d)^(3/2)*d^3 + 315*sqrt(e*x^2 + d)*d^4)*sqrt(-e)/(d*e^4)

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Fricas [A]  time = 2.3296, size = 182, normalized size = 1.47 \begin{align*} \frac{35 \, e^{4} x^{7} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) -{\left (5 \, e^{3} x^{6} - 6 \, d e^{2} x^{4} + 8 \, d^{2} e x^{2} - 16 \, d^{3}\right )} \sqrt{e x^{2} + d} \sqrt{-e}}{245 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2)),x, algorithm="fricas")

[Out]

1/245*(35*e^4*x^7*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) - (5*e^3*x^6 - 6*d*e^2*x^4 + 8*d^2*e*x^2 - 16*d^3)*sqrt(e
*x^2 + d)*sqrt(-e))/e^4

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Sympy [A]  time = 15.8235, size = 124, normalized size = 1. \begin{align*} \begin{cases} \frac{16 i d^{3} \sqrt{d + e x^{2}}}{245 e^{\frac{7}{2}}} - \frac{8 i d^{2} x^{2} \sqrt{d + e x^{2}}}{245 e^{\frac{5}{2}}} + \frac{6 i d x^{4} \sqrt{d + e x^{2}}}{245 e^{\frac{3}{2}}} + \frac{i x^{7} \operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{7} - \frac{i x^{6} \sqrt{d + e x^{2}}}{49 \sqrt{e}} & \text{for}\: e \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*atan(x*(-e)**(1/2)/(e*x**2+d)**(1/2)),x)

[Out]

Piecewise((16*I*d**3*sqrt(d + e*x**2)/(245*e**(7/2)) - 8*I*d**2*x**2*sqrt(d + e*x**2)/(245*e**(5/2)) + 6*I*d*x
**4*sqrt(d + e*x**2)/(245*e**(3/2)) + I*x**7*atanh(sqrt(e)*x/sqrt(d + e*x**2))/7 - I*x**6*sqrt(d + e*x**2)/(49
*sqrt(e)), Ne(e, 0)), (0, True))

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Giac [A]  time = 1.18974, size = 185, normalized size = 1.49 \begin{align*} \frac{1}{7} \, x^{7} \arctan \left (\frac{x \sqrt{-e}}{\sqrt{x^{2} e + d}}\right ) + \frac{1}{245} \,{\left (35 \, \sqrt{-x^{2} e^{2} - d e} d^{3} e^{3} + 35 \,{\left (-x^{2} e^{2} - d e\right )}^{\frac{3}{2}} d^{2} e^{2} + 21 \,{\left (x^{2} e^{2} + d e\right )}^{2} \sqrt{-x^{2} e^{2} - d e} d e - 5 \,{\left (x^{2} e^{2} + d e\right )}^{3} \sqrt{-x^{2} e^{2} - d e}\right )} e^{\left (-7\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2)),x, algorithm="giac")

[Out]

1/7*x^7*arctan(x*sqrt(-e)/sqrt(x^2*e + d)) + 1/245*(35*sqrt(-x^2*e^2 - d*e)*d^3*e^3 + 35*(-x^2*e^2 - d*e)^(3/2
)*d^2*e^2 + 21*(x^2*e^2 + d*e)^2*sqrt(-x^2*e^2 - d*e)*d*e - 5*(x^2*e^2 + d*e)^3*sqrt(-x^2*e^2 - d*e))*e^(-7)