3.1.9 \(\int x^6 \tanh ^{-1}(\frac {\sqrt {e} x}{\sqrt {d+e x^2}}) \, dx\) [9]

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

[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)

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Rubi [A]
time = 0.04, 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 = {6356, 272, 45} \begin {gather*} \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 ) \end {gather*}

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 45

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 272

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 6356

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} \text {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} \text {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*}

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Mathematica [A]
time = 0.04, size = 79, normalized size = 0.69 \begin {gather*} \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 ) \end {gather*}

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(223\) vs. \(2(84)=168\).
time = 0.01, size = 224, normalized size = 1.96

method	result	size
default	\(\frac {x^{7} \arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{7}+\frac {e^{\frac {3}{2}} \left (\frac {x^{8} \sqrt {e \,x^{2}+d}}{9 e}-\frac {8 d \left (\frac {x^{6} \sqrt {e \,x^{2}+d}}{7 e}-\frac {6 d \left (\frac {x^{4} \sqrt {e \,x^{2}+d}}{5 e}-\frac {4 d \left (\frac {x^{2} \sqrt {e \,x^{2}+d}}{3 e}-\frac {2 d \sqrt {e \,x^{2}+d}}{3 e^{2}}\right )}{5 e}\right )}{7 e}\right )}{9 e}\right )}{7 d}-\frac {\sqrt {e}\, \left (\frac {x^{6} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{9 e}-\frac {2 d \left (\frac {x^{4} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{7 e}-\frac {4 d \left (\frac {x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{5 e}-\frac {2 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{15 e^{2}}\right )}{7 e}\right )}{3 e}\right )}{7 d}\)	\(224\)

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,method=_RETURNVERBOSE)

[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))))

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Maxima [A]
time = 0.26, size = 162, normalized size = 1.42 \begin {gather*} \frac {1}{7} \, x^{7} \operatorname {artanh}\left (\frac {x e^{\frac {1}{2}}}{\sqrt {x^{2} e + d}}\right ) - \frac {{\left (35 \, {\left (x^{2} e + d\right )}^{\frac {9}{2}} - 135 \, {\left (x^{2} e + d\right )}^{\frac {7}{2}} d + 189 \, {\left (x^{2} e + d\right )}^{\frac {5}{2}} d^{2} - 105 \, {\left (x^{2} e + d\right )}^{\frac {3}{2}} d^{3}\right )} e^{\left (-\frac {7}{2}\right )}}{2205 \, d} + \frac {{\left (35 \, {\left (x^{2} e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x^{2} e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x^{2} e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x^{2} e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x^{2} e + d} d^{4}\right )} e^{\left (-\frac {7}{2}\right )}}{2205 \, d} \end {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (84) = 168\).
time = 0.36, size = 442, normalized size = 3.88 \begin {gather*} \frac {35 \, {\left (x^{7} \cosh \left (\frac {1}{2}\right )^{7} + 7 \, x^{7} \cosh \left (\frac {1}{2}\right )^{6} \sinh \left (\frac {1}{2}\right ) + 21 \, x^{7} \cosh \left (\frac {1}{2}\right )^{5} \sinh \left (\frac {1}{2}\right )^{2} + 35 \, x^{7} \cosh \left (\frac {1}{2}\right )^{4} \sinh \left (\frac {1}{2}\right )^{3} + 35 \, x^{7} \cosh \left (\frac {1}{2}\right )^{3} \sinh \left (\frac {1}{2}\right )^{4} + 21 \, x^{7} \cosh \left (\frac {1}{2}\right )^{2} \sinh \left (\frac {1}{2}\right )^{5} + 7 \, x^{7} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right )^{6} + x^{7} \sinh \left (\frac {1}{2}\right )^{7}\right )} \log \left (\frac {2 \, x^{2} \cosh \left (\frac {1}{2}\right )^{2} + 4 \, x^{2} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right ) + 2 \, x^{2} \sinh \left (\frac {1}{2}\right )^{2} + 2 \, {\left (x \cosh \left (\frac {1}{2}\right ) + x \sinh \left (\frac {1}{2}\right )\right )} \sqrt {\frac {{\left (x^{2} + d\right )} \cosh \left (\frac {1}{2}\right ) + {\left (x^{2} - d\right )} \sinh \left (\frac {1}{2}\right )}{\cosh \left (\frac {1}{2}\right ) - \sinh \left (\frac {1}{2}\right )}} + d}{d}\right ) - 2 \, {\left (5 \, x^{6} \cosh \left (\frac {1}{2}\right )^{6} + 30 \, x^{6} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right )^{5} + 5 \, x^{6} \sinh \left (\frac {1}{2}\right )^{6} - 6 \, d x^{4} \cosh \left (\frac {1}{2}\right )^{4} + 8 \, d^{2} x^{2} \cosh \left (\frac {1}{2}\right )^{2} + 3 \, {\left (25 \, x^{6} \cosh \left (\frac {1}{2}\right )^{2} - 2 \, d x^{4}\right )} \sinh \left (\frac {1}{2}\right )^{4} + 4 \, {\left (25 \, x^{6} \cosh \left (\frac {1}{2}\right )^{3} - 6 \, d x^{4} \cosh \left (\frac {1}{2}\right )\right )} \sinh \left (\frac {1}{2}\right )^{3} - 16 \, d^{3} + {\left (75 \, x^{6} \cosh \left (\frac {1}{2}\right )^{4} - 36 \, d x^{4} \cosh \left (\frac {1}{2}\right )^{2} + 8 \, d^{2} x^{2}\right )} \sinh \left (\frac {1}{2}\right )^{2} + 2 \, {\left (15 \, x^{6} \cosh \left (\frac {1}{2}\right )^{5} - 12 \, d x^{4} \cosh \left (\frac {1}{2}\right )^{3} + 8 \, d^{2} x^{2} \cosh \left (\frac {1}{2}\right )\right )} \sinh \left (\frac {1}{2}\right )\right )} \sqrt {\frac {{\left (x^{2} + d\right )} \cosh \left (\frac {1}{2}\right ) + {\left (x^{2} - d\right )} \sinh \left (\frac {1}{2}\right )}{\cosh \left (\frac {1}{2}\right ) - \sinh \left (\frac {1}{2}\right )}}}{490 \, {\left (\cosh \left (\frac {1}{2}\right )^{7} + 7 \, \cosh \left (\frac {1}{2}\right )^{6} \sinh \left (\frac {1}{2}\right ) + 21 \, \cosh \left (\frac {1}{2}\right )^{5} \sinh \left (\frac {1}{2}\right )^{2} + 35 \, \cosh \left (\frac {1}{2}\right )^{4} \sinh \left (\frac {1}{2}\right )^{3} + 35 \, \cosh \left (\frac {1}{2}\right )^{3} \sinh \left (\frac {1}{2}\right )^{4} + 21 \, \cosh \left (\frac {1}{2}\right )^{2} \sinh \left (\frac {1}{2}\right )^{5} + 7 \, \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right )^{6} + \sinh \left (\frac {1}{2}\right )^{7}\right )}} \end {gather*}

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*(x^7*cosh(1/2)^7 + 7*x^7*cosh(1/2)^6*sinh(1/2) + 21*x^7*cosh(1/2)^5*sinh(1/2)^2 + 35*x^7*cosh(1/2)^4
*sinh(1/2)^3 + 35*x^7*cosh(1/2)^3*sinh(1/2)^4 + 21*x^7*cosh(1/2)^2*sinh(1/2)^5 + 7*x^7*cosh(1/2)*sinh(1/2)^6 +
x^7*sinh(1/2)^7)*log((2*x^2*cosh(1/2)^2 + 4*x^2*cosh(1/2)*sinh(1/2) + 2*x^2*sinh(1/2)^2 + 2*(x*cosh(1/2) + x*
sinh(1/2))*sqrt(((x^2 + d)*cosh(1/2) + (x^2 - d)*sinh(1/2))/(cosh(1/2) - sinh(1/2))) + d)/d) - 2*(5*x^6*cosh(1
/2)^6 + 30*x^6*cosh(1/2)*sinh(1/2)^5 + 5*x^6*sinh(1/2)^6 - 6*d*x^4*cosh(1/2)^4 + 8*d^2*x^2*cosh(1/2)^2 + 3*(25
*x^6*cosh(1/2)^2 - 2*d*x^4)*sinh(1/2)^4 + 4*(25*x^6*cosh(1/2)^3 - 6*d*x^4*cosh(1/2))*sinh(1/2)^3 - 16*d^3 + (7
5*x^6*cosh(1/2)^4 - 36*d*x^4*cosh(1/2)^2 + 8*d^2*x^2)*sinh(1/2)^2 + 2*(15*x^6*cosh(1/2)^5 - 12*d*x^4*cosh(1/2)
^3 + 8*d^2*x^2*cosh(1/2))*sinh(1/2))*sqrt(((x^2 + d)*cosh(1/2) + (x^2 - d)*sinh(1/2))/(cosh(1/2) - sinh(1/2)))
)/(cosh(1/2)^7 + 7*cosh(1/2)^6*sinh(1/2) + 21*cosh(1/2)^5*sinh(1/2)^2 + 35*cosh(1/2)^4*sinh(1/2)^3 + 35*cosh(1
/2)^3*sinh(1/2)^4 + 21*cosh(1/2)^2*sinh(1/2)^5 + 7*cosh(1/2)*sinh(1/2)^6 + sinh(1/2)^7)

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Sympy [A]
time = 1.90, size = 116, normalized size = 1.02 \begin {gather*} \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} \end {gather*}

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))

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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]

Timed out

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

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)

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