3.1.21 \(\int \frac {\tanh ^{-1}(\frac {\sqrt {e} x}{\sqrt {d+e x^2}})}{x^{11/2}} \, dx\) [21]

Optimal. Leaf size=173 \[ -\frac {4 \sqrt {e} \sqrt {d+e x^2}}{63 d x^{7/2}}+\frac {20 e^{3/2} \sqrt {d+e x^2}}{189 d^2 x^{3/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{9 x^{9/2}}+\frac {10 e^{9/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{189 d^{9/4} \sqrt {d+e x^2}} \]

[Out]

-2/9*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^(9/2)+20/189*e^(3/2)*(e*x^2+d)^(1/2)/d^2/x^(3/2)-4/63*e^(1/2)*(e*x^2
+d)^(1/2)/d/x^(7/2)+10/189*e^(9/4)*(cos(2*arctan(e^(1/4)*x^(1/2)/d^(1/4)))^2)^(1/2)/cos(2*arctan(e^(1/4)*x^(1/
2)/d^(1/4)))*EllipticF(sin(2*arctan(e^(1/4)*x^(1/2)/d^(1/4))),1/2*2^(1/2))*(d^(1/2)+x*e^(1/2))*((e*x^2+d)/(d^(
1/2)+x*e^(1/2))^2)^(1/2)/d^(9/4)/(e*x^2+d)^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6356, 331, 335, 226} \begin {gather*} \frac {10 e^{9/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{189 d^{9/4} \sqrt {d+e x^2}}+\frac {20 e^{3/2} \sqrt {d+e x^2}}{189 d^2 x^{3/2}}-\frac {4 \sqrt {e} \sqrt {d+e x^2}}{63 d x^{7/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{9 x^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/x^(11/2),x]

[Out]

(-4*Sqrt[e]*Sqrt[d + e*x^2])/(63*d*x^(7/2)) + (20*e^(3/2)*Sqrt[d + e*x^2])/(189*d^2*x^(3/2)) - (2*ArcTanh[(Sqr
t[e]*x)/Sqrt[d + e*x^2]])/(9*x^(9/2)) + (10*e^(9/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*
x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(189*d^(9/4)*Sqrt[d + e*x^2])

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

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 \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^{11/2}} \, dx &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{9 x^{9/2}}+\frac {1}{9} \left (2 \sqrt {e}\right ) \int \frac {1}{x^{9/2} \sqrt {d+e x^2}} \, dx\\ &=-\frac {4 \sqrt {e} \sqrt {d+e x^2}}{63 d x^{7/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{9 x^{9/2}}-\frac {\left (10 e^{3/2}\right ) \int \frac {1}{x^{5/2} \sqrt {d+e x^2}} \, dx}{63 d}\\ &=-\frac {4 \sqrt {e} \sqrt {d+e x^2}}{63 d x^{7/2}}+\frac {20 e^{3/2} \sqrt {d+e x^2}}{189 d^2 x^{3/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{9 x^{9/2}}+\frac {\left (10 e^{5/2}\right ) \int \frac {1}{\sqrt {x} \sqrt {d+e x^2}} \, dx}{189 d^2}\\ &=-\frac {4 \sqrt {e} \sqrt {d+e x^2}}{63 d x^{7/2}}+\frac {20 e^{3/2} \sqrt {d+e x^2}}{189 d^2 x^{3/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{9 x^{9/2}}+\frac {\left (20 e^{5/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {d+e x^4}} \, dx,x,\sqrt {x}\right )}{189 d^2}\\ &=-\frac {4 \sqrt {e} \sqrt {d+e x^2}}{63 d x^{7/2}}+\frac {20 e^{3/2} \sqrt {d+e x^2}}{189 d^2 x^{3/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{9 x^{9/2}}+\frac {10 e^{9/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{189 d^{9/4} \sqrt {d+e x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.20, size = 154, normalized size = 0.89 \begin {gather*} \frac {4 \sqrt {e} x \sqrt {d+e x^2} \left (-3 d+5 e x^2\right )-42 d^2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{189 d^2 x^{9/2}}+\frac {20 \sqrt {\frac {i \sqrt {d}}{\sqrt {e}}} e^3 \sqrt {1+\frac {d}{e x^2}} x F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {d}}{\sqrt {e}}}}{\sqrt {x}}\right )\right |-1\right )}{189 d^{5/2} \sqrt {d+e x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/x^(11/2),x]

[Out]

(4*Sqrt[e]*x*Sqrt[d + e*x^2]*(-3*d + 5*e*x^2) - 42*d^2*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(189*d^2*x^(9/2))
+ (20*Sqrt[(I*Sqrt[d])/Sqrt[e]]*e^3*Sqrt[1 + d/(e*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[d])/Sqrt[e]]/Sqrt[
x]], -1])/(189*d^(5/2)*Sqrt[d + e*x^2])

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Maple [F]
time = 0.29, size = 0, normalized size = 0.00 \[\int \frac {\arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{x^{\frac {11}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^(11/2),x)

[Out]

int(arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^(11/2),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^(11/2),x, algorithm="maxima")

[Out]

2*d*integrate(-1/9*x*e^(1/2*log(x^2*e + d) + 1/2)/((x^4*e^2 + d*x^2*e)*x^(11/2) - (x^2*e + d)*e^(log(x^2*e + d
) + 11/2*log(x))), x) - 1/9*log(x*e^(1/2) + sqrt(x^2*e + d))/x^(9/2) + 1/9*log(-x*e^(1/2) + sqrt(x^2*e + d))/x
^(9/2)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.10, size = 260, normalized size = 1.50 \begin {gather*} -\frac {21 \, d^{2} \sqrt {x} \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 ) - 4 \, {\left (5 \, x^{3} \cosh \left (\frac {1}{2}\right )^{3} + 15 \, x^{3} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right )^{2} + 5 \, x^{3} \sinh \left (\frac {1}{2}\right )^{3} - 3 \, d x \cosh \left (\frac {1}{2}\right ) + 3 \, {\left (5 \, x^{3} \cosh \left (\frac {1}{2}\right )^{2} - d x\right )} \sinh \left (\frac {1}{2}\right )\right )} \sqrt {x} \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 )}} - 20 \, {\left (x^{5} \cosh \left (\frac {1}{2}\right )^{4} + 4 \, x^{5} \cosh \left (\frac {1}{2}\right )^{3} \sinh \left (\frac {1}{2}\right ) + 6 \, x^{5} \cosh \left (\frac {1}{2}\right )^{2} \sinh \left (\frac {1}{2}\right )^{2} + 4 \, x^{5} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right )^{3} + x^{5} \sinh \left (\frac {1}{2}\right )^{4}\right )} {\rm weierstrassPInverse}\left (-\frac {4 \, d}{\cosh \left (\frac {1}{2}\right )^{2} + 2 \, \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right ) + \sinh \left (\frac {1}{2}\right )^{2}}, 0, x\right )}{189 \, d^{2} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^(11/2),x, algorithm="fricas")

[Out]

-1/189*(21*d^2*sqrt(x)*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) - 4*(5*x^3*c
osh(1/2)^3 + 15*x^3*cosh(1/2)*sinh(1/2)^2 + 5*x^3*sinh(1/2)^3 - 3*d*x*cosh(1/2) + 3*(5*x^3*cosh(1/2)^2 - d*x)*
sinh(1/2))*sqrt(x)*sqrt(((x^2 + d)*cosh(1/2) + (x^2 - d)*sinh(1/2))/(cosh(1/2) - sinh(1/2))) - 20*(x^5*cosh(1/
2)^4 + 4*x^5*cosh(1/2)^3*sinh(1/2) + 6*x^5*cosh(1/2)^2*sinh(1/2)^2 + 4*x^5*cosh(1/2)*sinh(1/2)^3 + x^5*sinh(1/
2)^4)*weierstrassPInverse(-4*d/(cosh(1/2)^2 + 2*cosh(1/2)*sinh(1/2) + sinh(1/2)^2), 0, x))/(d^2*x^5)

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Sympy [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(atanh(x*e**(1/2)/(e*x**2+d)**(1/2))/x**(11/2),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^(11/2),x, algorithm="giac")

[Out]

integrate(arctanh(sqrt(e)*x/sqrt(e*x^2 + d))/x^(11/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right )}{x^{11/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh((e^(1/2)*x)/(d + e*x^2)^(1/2))/x^(11/2),x)

[Out]

int(atanh((e^(1/2)*x)/(d + e*x^2)^(1/2))/x^(11/2), x)

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