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

Optimal. Leaf size=114 \[ \frac {d^3 \sqrt {d+e x^2}}{7 e^{7/2}}-\frac {d^2 \left (d+e x^2\right )^{3/2}}{7 e^{7/2}}-\frac {\left (d+e x^2\right )^{7/2}}{49 e^{7/2}}+\frac {3 d \left (d+e x^2\right )^{5/2}}{35 e^{7/2}}+\frac {1}{7} x^7 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \]

[Out]

-1/7*d^2*(e*x^2+d)^(3/2)/e^(7/2)+3/35*d*(e*x^2+d)^(5/2)/e^(7/2)-1/49*(e*x^2+d)^(7/2)/e^(7/2)+1/7*x^7*arctanh(x
*e^(1/2)/(e*x^2+d)^(1/2))+1/7*d^3*(e*x^2+d)^(1/2)/e^(7/2)

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6221, 266, 43} \[ -\frac {d^2 \left (d+e x^2\right )^{3/2}}{7 e^{7/2}}+\frac {d^3 \sqrt {d+e x^2}}{7 e^{7/2}}-\frac {\left (d+e x^2\right )^{7/2}}{49 e^{7/2}}+\frac {3 d \left (d+e x^2\right )^{5/2}}{35 e^{7/2}}+\frac {1}{7} x^7 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^6*ArcTanh[(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*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/7

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])

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 6221

Int[ArcTanh[((c_.)*(x_))/Sqrt[(a_.) + (b_.)*(x_)^2]]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*ArcT
anh[(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] /;
FreeQ[{a, b, c, d, m}, x] && EqQ[b, c^2] && NeQ[m, -1]

Rubi steps

\begin {align*} \int x^6 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx &=\frac {1}{7} x^7 \tanh ^{-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 \tanh ^{-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 \tanh ^{-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 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 79, normalized size = 0.69 \[ \frac {\sqrt {d+e x^2} \left (16 d^3-8 d^2 e x^2+6 d e^2 x^4-5 e^3 x^6\right )}{245 e^{7/2}}+\frac {1}{7} x^7 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^6*ArcTanh[(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*ArcTanh[(Sqrt[e]*x)/Sq
rt[d + e*x^2]])/7

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 88, normalized size = 0.77 \[ \frac {35 \, e^{4} x^{7} \log \left (\frac {2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x + d}{d}\right ) - 2 \, {\left (5 \, e^{3} x^{6} - 6 \, d e^{2} x^{4} + 8 \, d^{2} e x^{2} - 16 \, d^{3}\right )} \sqrt {e x^{2} + d} \sqrt {e}}{490 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/490*(35*e^4*x^7*log((2*e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(e)*x + d)/d) - 2*(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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 1/2*(x^7/7*ln((1+(sqrt(exp(1)*x^2+d))^-1
*exp(1/2)*x)/(1-(sqrt(exp(1)*x^2+d))^-1*exp(1/2)*x))-2*d*exp(1/2)*((2401/5*sqrt(d+x^2*exp(1))*(d+x^2*exp(1))^2
*exp(1)^16-9604/5*sqrt(d+x^2*exp(1))*(d+x^2*exp(1))^2*exp(1)^15*exp(1/2)^2+14406/5*sqrt(d+x^2*exp(1))*(d+x^2*e
xp(1))^2*exp(1)^14*exp(1/2)^4-9604/5*sqrt(d+x^2*exp(1))*(d+x^2*exp(1))^2*exp(1)^13*exp(1/2)^6+2401/5*sqrt(d+x^
2*exp(1))*(d+x^2*exp(1))^2*exp(1)^12*exp(1/2)^8-2401*sqrt(d+x^2*exp(1))*(d+x^2*exp(1))*d*exp(1)^16+26411/3*sqr
t(d+x^2*exp(1))*(d+x^2*exp(1))*d*exp(1)^15*exp(1/2)^2-12005*sqrt(d+x^2*exp(1))*(d+x^2*exp(1))*d*exp(1)^14*exp(
1/2)^4+7203*sqrt(d+x^2*exp(1))*(d+x^2*exp(1))*d*exp(1)^13*exp(1/2)^6-4802/3*sqrt(d+x^2*exp(1))*(d+x^2*exp(1))*
d*exp(1)^12*exp(1/2)^8+7203*sqrt(d+x^2*exp(1))*d^2*exp(1)^16-21609*sqrt(d+x^2*exp(1))*d^2*exp(1)^15*exp(1/2)^2
+24010*sqrt(d+x^2*exp(1))*d^2*exp(1)^14*exp(1/2)^4-12005*sqrt(d+x^2*exp(1))*d^2*exp(1)^13*exp(1/2)^6+2401*sqrt
(d+x^2*exp(1))*d^2*exp(1)^12*exp(1/2)^8)/(16807*exp(1)^20-84035*exp(1)^19*exp(1/2)^2+168070*exp(1)^18*exp(1/2)
^4-168070*exp(1)^17*exp(1/2)^6+84035*exp(1)^16*exp(1/2)^8-16807*exp(1)^15*exp(1/2)^10)-2*d^3/2/(7*exp(1)^3-21*
exp(1)^2*exp(1/2)^2+21*exp(1)*exp(1/2)^4-7*exp(1/2)^6)/sqrt(-d*exp(1/2)^2+d*exp(1))/exp(1/2)*atan((sqrt(d+x^2*
exp(1))*exp(1)-sqrt(d+x^2*exp(1))*exp(1/2)^2)/sqrt(-d*exp(1/2)^2+d*exp(1))/exp(1/2))))

________________________________________________________________________________________

maple [B]  time = 0.04, size = 224, normalized size = 1.96 \[ \frac {x^{7} \arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{7}+\frac {e^{\frac {3}{2}} \left (\frac {x^{8} \sqrt {e \,x^{2}+d}}{9 e}-\frac {8 d \left (\frac {x^{6} \sqrt {e \,x^{2}+d}}{7 e}-\frac {6 d \left (\frac {x^{4} \sqrt {e \,x^{2}+d}}{5 e}-\frac {4 d \left (\frac {x^{2} \sqrt {e \,x^{2}+d}}{3 e}-\frac {2 d \sqrt {e \,x^{2}+d}}{3 e^{2}}\right )}{5 e}\right )}{7 e}\right )}{9 e}\right )}{7 d}-\frac {\sqrt {e}\, \left (\frac {x^{6} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{9 e}-\frac {2 d \left (\frac {x^{4} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{7 e}-\frac {4 d \left (\frac {x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{5 e}-\frac {2 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{15 e^{2}}\right )}{7 e}\right )}{3 e}\right )}{7 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))+1/7*e^(3/2)/d*(1/9*x^8/e*(e*x^2+d)^(1/2)-8/9*d/e*(1/7*x^6/e*(e*x^2+
d)^(1/2)-6/7*d/e*(1/5*x^4/e*(e*x^2+d)^(1/2)-4/5*d/e*(1/3*x^2/e*(e*x^2+d)^(1/2)-2/3*d/e^2*(e*x^2+d)^(1/2)))))-1
/7*e^(1/2)/d*(1/9*x^6*(e*x^2+d)^(3/2)/e-2/3*d/e*(1/7*x^4*(e*x^2+d)^(3/2)/e-4/7*d/e*(1/5*x^2*(e*x^2+d)^(3/2)/e-
2/15*d/e^2*(e*x^2+d)^(3/2))))

________________________________________________________________________________________

maxima [A]  time = 0.34, size = 155, normalized size = 1.36 \[ \frac {1}{7} \, x^{7} \operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right ) - \frac {35 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} - 135 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d + 189 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{2} - 105 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{3}}{2205 \, d e^{\frac {7}{2}}} + \frac {35 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x^{2} + d} d^{4}}{2205 \, d e^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*arctanh(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)/(d*e^(7/2)) + 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)/(d*e^(7/2))

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^6\,\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 8.51, size = 116, normalized size = 1.02 \[ \begin {cases} \frac {16 d^{3} \sqrt {d + e x^{2}}}{245 e^{\frac {7}{2}}} - \frac {8 d^{2} x^{2} \sqrt {d + e x^{2}}}{245 e^{\frac {5}{2}}} + \frac {6 d x^{4} \sqrt {d + e x^{2}}}{245 e^{\frac {3}{2}}} + \frac {x^{7} \operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{7} - \frac {x^{6} \sqrt {d + e x^{2}}}{49 \sqrt {e}} & \text {for}\: e \neq 0 \\0 & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________