3.1.35 \(\int \sqrt {a \tanh ^3(x)} \, dx\) [35]

Optimal. Leaf size=63 \[ -2 \coth (x) \sqrt {a \tanh ^3(x)}+\frac {\text {ArcTan}\left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}+\frac {\tanh ^{-1}\left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)} \]

[Out]

-2*coth(x)*(a*tanh(x)^3)^(1/2)+arctan(tanh(x)^(1/2))*(a*tanh(x)^3)^(1/2)/tanh(x)^(3/2)+arctanh(tanh(x)^(1/2))*
(a*tanh(x)^3)^(1/2)/tanh(x)^(3/2)

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Rubi [A]
time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {3739, 3554, 3557, 335, 218, 212, 209} \begin {gather*} \frac {\sqrt {a \tanh ^3(x)} \text {ArcTan}\left (\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}+\frac {\tanh ^{-1}\left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}-2 \coth (x) \sqrt {a \tanh ^3(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*Tanh[x]^3],x]

[Out]

-2*Coth[x]*Sqrt[a*Tanh[x]^3] + (ArcTan[Sqrt[Tanh[x]]]*Sqrt[a*Tanh[x]^3])/Tanh[x]^(3/2) + (ArcTanh[Sqrt[Tanh[x]
]]*Sqrt[a*Tanh[x]^3])/Tanh[x]^(3/2)

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

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 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3557

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rule 3739

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])), Int[ActivateTrig[u]
*(Tan[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rubi steps

\begin {align*} \int \sqrt {a \tanh ^3(x)} \, dx &=\frac {\sqrt {a \tanh ^3(x)} \int \tanh ^{\frac {3}{2}}(x) \, dx}{\tanh ^{\frac {3}{2}}(x)}\\ &=-2 \coth (x) \sqrt {a \tanh ^3(x)}+\frac {\sqrt {a \tanh ^3(x)} \int \frac {1}{\sqrt {\tanh (x)}} \, dx}{\tanh ^{\frac {3}{2}}(x)}\\ &=-2 \coth (x) \sqrt {a \tanh ^3(x)}-\frac {\sqrt {a \tanh ^3(x)} \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (-1+x^2\right )} \, dx,x,\tanh (x)\right )}{\tanh ^{\frac {3}{2}}(x)}\\ &=-2 \coth (x) \sqrt {a \tanh ^3(x)}-\frac {\left (2 \sqrt {a \tanh ^3(x)}\right ) \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}\\ &=-2 \coth (x) \sqrt {a \tanh ^3(x)}+\frac {\sqrt {a \tanh ^3(x)} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}+\frac {\sqrt {a \tanh ^3(x)} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}\\ &=-2 \coth (x) \sqrt {a \tanh ^3(x)}+\frac {\tan ^{-1}\left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}+\frac {\tanh ^{-1}\left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 40, normalized size = 0.63 \begin {gather*} \frac {\left (\text {ArcTan}\left (\sqrt {\tanh (x)}\right )+\tanh ^{-1}\left (\sqrt {\tanh (x)}\right )-2 \sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*Tanh[x]^3],x]

[Out]

((ArcTan[Sqrt[Tanh[x]]] + ArcTanh[Sqrt[Tanh[x]]] - 2*Sqrt[Tanh[x]])*Sqrt[a*Tanh[x]^3])/Tanh[x]^(3/2)

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Maple [A]
time = 1.15, size = 59, normalized size = 0.94

method	result	size
derivativedivides	\(\frac {\sqrt {a \left (\tanh ^{3}\left (x \right )\right )}\, \left (-2 \sqrt {a \tanh \left (x \right )}+\sqrt {a}\, \arctan \left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right )+\sqrt {a}\, \arctanh \left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right )\right )}{\tanh \left (x \right ) \sqrt {a \tanh \left (x \right )}}\)	\(59\)
default	\(\frac {\sqrt {a \left (\tanh ^{3}\left (x \right )\right )}\, \left (-2 \sqrt {a \tanh \left (x \right )}+\sqrt {a}\, \arctan \left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right )+\sqrt {a}\, \arctanh \left (\frac {\sqrt {a \tanh \left (x \right )}}{\sqrt {a}}\right )\right )}{\tanh \left (x \right ) \sqrt {a \tanh \left (x \right )}}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*tanh(x)^3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

(a*tanh(x)^3)^(1/2)/tanh(x)/(a*tanh(x))^(1/2)*(-2*(a*tanh(x))^(1/2)+a^(1/2)*arctan((a*tanh(x))^(1/2)/a^(1/2))+
a^(1/2)*arctanh((a*tanh(x))^(1/2)/a^(1/2)))

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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((a*tanh(x)^3)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*tanh(x)^3), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (49) = 98\).
time = 0.40, size = 376, normalized size = 5.97 \begin {gather*} \left [-\frac {1}{2} \, \sqrt {-a} \arctan \left (\frac {{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )} \sqrt {-a} \sqrt {\frac {a \sinh \left (x\right )}{\cosh \left (x\right )}}}{a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} - a}\right ) + \frac {1}{4} \, \sqrt {-a} \log \left (-\frac {a \cosh \left (x\right )^{4} + 4 \, a \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, a \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, a \cosh \left (x\right ) \sinh \left (x\right )^{3} + a \sinh \left (x\right )^{4} + 2 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \sqrt {-a} \sqrt {\frac {a \sinh \left (x\right )}{\cosh \left (x\right )}} - 2 \, a}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}}\right ) - 2 \, \sqrt {\frac {a \sinh \left (x\right )}{\cosh \left (x\right )}}, -\frac {1}{2} \, \sqrt {a} \arctan \left (\frac {\sqrt {a} \sqrt {\frac {a \sinh \left (x\right )}{\cosh \left (x\right )}}}{a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} - a}\right ) + \frac {1}{4} \, \sqrt {a} \log \left (2 \, a \cosh \left (x\right )^{4} + 8 \, a \cosh \left (x\right )^{3} \sinh \left (x\right ) + 12 \, a \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 8 \, a \cosh \left (x\right ) \sinh \left (x\right )^{3} + 2 \, a \sinh \left (x\right )^{4} + 2 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + {\left (6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + 2 \, {\left (2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {a} \sqrt {\frac {a \sinh \left (x\right )}{\cosh \left (x\right )}} - a\right ) - 2 \, \sqrt {\frac {a \sinh \left (x\right )}{\cosh \left (x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*tanh(x)^3)^(1/2),x, algorithm="fricas")

[Out]

[-1/2*sqrt(-a)*arctan((cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)*sqrt(-a)*sqrt(a*sinh(x)/cosh(x))/(a*cosh(x)^
2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)^2 - a)) + 1/4*sqrt(-a)*log(-(a*cosh(x)^4 + 4*a*cosh(x)^3*sinh(x) + 6*a*cos
h(x)^2*sinh(x)^2 + 4*a*cosh(x)*sinh(x)^3 + a*sinh(x)^4 + 2*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqr
t(-a)*sqrt(a*sinh(x)/cosh(x)) - 2*a)/(cosh(x)^4 + 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 + 4*cosh(x)*sinh
(x)^3 + sinh(x)^4)) - 2*sqrt(a*sinh(x)/cosh(x)), -1/2*sqrt(a)*arctan(sqrt(a)*sqrt(a*sinh(x)/cosh(x))/(a*cosh(x
)^2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)^2 - a)) + 1/4*sqrt(a)*log(2*a*cosh(x)^4 + 8*a*cosh(x)^3*sinh(x) + 12*a*c
osh(x)^2*sinh(x)^2 + 8*a*cosh(x)*sinh(x)^3 + 2*a*sinh(x)^4 + 2*(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 +
(6*cosh(x)^2 + 1)*sinh(x)^2 + cosh(x)^2 + 2*(2*cosh(x)^3 + cosh(x))*sinh(x))*sqrt(a)*sqrt(a*sinh(x)/cosh(x)) -
a) - 2*sqrt(a*sinh(x)/cosh(x))]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a \tanh ^{3}{\left (x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*tanh(x)**3)**(1/2),x)

[Out]

Integral(sqrt(a*tanh(x)**3), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (49) = 98\).
time = 0.43, size = 115, normalized size = 1.83 \begin {gather*} \sqrt {a} \arctan \left (-\frac {\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}}{\sqrt {a}}\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - \frac {1}{2} \, \sqrt {a} \log \left ({\left | -\sqrt {a} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} - a} \right |}\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - \frac {4 \, a \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right )}{\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a} + \sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*tanh(x)^3)^(1/2),x, algorithm="giac")

[Out]

sqrt(a)*arctan(-(sqrt(a)*e^(2*x) - sqrt(a*e^(4*x) - a))/sqrt(a))*sgn(e^(4*x) - 1) - 1/2*sqrt(a)*log(abs(-sqrt(
a)*e^(2*x) + sqrt(a*e^(4*x) - a)))*sgn(e^(4*x) - 1) - 4*a*sgn(e^(4*x) - 1)/(sqrt(a)*e^(2*x) - sqrt(a*e^(4*x) -
a) + sqrt(a))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \sqrt {a\,{\mathrm {tanh}\left (x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*tanh(x)^3)^(1/2),x)

[Out]

int((a*tanh(x)^3)^(1/2), x)

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