3.51 \(\int (1+\tanh (x))^{5/2} \, dx\)

Optimal. Leaf size=45 \[ 4 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{\tanh (x)+1}}{\sqrt{2}}\right )-\frac{2}{3} (\tanh (x)+1)^{3/2}-4 \sqrt{\tanh (x)+1} \]

[Out]

4*Sqrt[2]*ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]] - 4*Sqrt[1 + Tanh[x]] - (2*(1 + Tanh[x])^(3/2))/3

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Rubi [A]  time = 0.0308033, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3478, 3480, 206} \[ 4 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{\tanh (x)+1}}{\sqrt{2}}\right )-\frac{2}{3} (\tanh (x)+1)^{3/2}-4 \sqrt{\tanh (x)+1} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Tanh[x])^(5/2),x]

[Out]

4*Sqrt[2]*ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]] - 4*Sqrt[1 + Tanh[x]] - (2*(1 + Tanh[x])^(3/2))/3

Rule 3478

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a + b*Tan[c + d*x])^(n - 1))/(d*(n - 1)
), x] + Dist[2*a, Int[(a + b*Tan[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && G
tQ[n, 1]

Rule 3480

Int[Sqrt[(a_) + (b_.)*tan[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[(-2*b)/d, Subst[Int[1/(2*a - x^2), x], x, Sq
rt[a + b*Tan[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0]

Rule 206

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

Rubi steps

\begin{align*} \int (1+\tanh (x))^{5/2} \, dx &=-\frac{2}{3} (1+\tanh (x))^{3/2}+2 \int (1+\tanh (x))^{3/2} \, dx\\ &=-4 \sqrt{1+\tanh (x)}-\frac{2}{3} (1+\tanh (x))^{3/2}+4 \int \sqrt{1+\tanh (x)} \, dx\\ &=-4 \sqrt{1+\tanh (x)}-\frac{2}{3} (1+\tanh (x))^{3/2}+8 \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{1+\tanh (x)}\right )\\ &=4 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1+\tanh (x)}}{\sqrt{2}}\right )-4 \sqrt{1+\tanh (x)}-\frac{2}{3} (1+\tanh (x))^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.164421, size = 39, normalized size = 0.87 \[ 4 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{\tanh (x)+1}}{\sqrt{2}}\right )-\frac{2}{3} \sqrt{\tanh (x)+1} (\tanh (x)+7) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Tanh[x])^(5/2),x]

[Out]

4*Sqrt[2]*ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]] - (2*Sqrt[1 + Tanh[x]]*(7 + Tanh[x]))/3

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Maple [A]  time = 0.013, size = 35, normalized size = 0.8 \begin{align*} 4\,{\it Artanh} \left ( 1/2\,\sqrt{1+\tanh \left ( x \right ) }\sqrt{2} \right ) \sqrt{2}-4\,\sqrt{1+\tanh \left ( x \right ) }-{\frac{2}{3} \left ( 1+\tanh \left ( x \right ) \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+tanh(x))^(5/2),x)

[Out]

4*arctanh(1/2*(1+tanh(x))^(1/2)*2^(1/2))*2^(1/2)-4*(1+tanh(x))^(1/2)-2/3*(1+tanh(x))^(3/2)

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Maxima [B]  time = 1.58942, size = 95, normalized size = 2.11 \begin{align*} -2 \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \frac{\sqrt{2}}{\sqrt{e^{\left (-2 \, x\right )} + 1}}}{\sqrt{2} + \frac{\sqrt{2}}{\sqrt{e^{\left (-2 \, x\right )} + 1}}}\right ) - \frac{4 \, \sqrt{2}}{\sqrt{e^{\left (-2 \, x\right )} + 1}} - \frac{4 \, \sqrt{2}}{3 \,{\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+tanh(x))^(5/2),x, algorithm="maxima")

[Out]

-2*sqrt(2)*log(-(sqrt(2) - sqrt(2)/sqrt(e^(-2*x) + 1))/(sqrt(2) + sqrt(2)/sqrt(e^(-2*x) + 1))) - 4*sqrt(2)/sqr
t(e^(-2*x) + 1) - 4/3*sqrt(2)/(e^(-2*x) + 1)^(3/2)

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Fricas [B]  time = 2.32679, size = 871, normalized size = 19.36 \begin{align*} -\frac{2 \,{\left (2 \, \sqrt{2}{\left (4 \, \sqrt{2} \cosh \left (x\right )^{3} + 12 \, \sqrt{2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + 4 \, \sqrt{2} \sinh \left (x\right )^{3} + 3 \,{\left (4 \, \sqrt{2} \cosh \left (x\right )^{2} + \sqrt{2}\right )} \sinh \left (x\right ) + 3 \, \sqrt{2} \cosh \left (x\right )\right )} \sqrt{\frac{\cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} - 3 \,{\left (\sqrt{2} \cosh \left (x\right )^{4} + 4 \, \sqrt{2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sqrt{2} \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \sqrt{2} \cosh \left (x\right )^{2} + \sqrt{2}\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt{2} \cosh \left (x\right )^{2} + 4 \,{\left (\sqrt{2} \cosh \left (x\right )^{3} + \sqrt{2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt{2}\right )} \log \left (-2 \, \sqrt{2} \sqrt{\frac{\cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - 2 \, \cosh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) \sinh \left (x\right ) - 2 \, \sinh \left (x\right )^{2} - 1\right )\right )}}{3 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+tanh(x))^(5/2),x, algorithm="fricas")

[Out]

-2/3*(2*sqrt(2)*(4*sqrt(2)*cosh(x)^3 + 12*sqrt(2)*cosh(x)*sinh(x)^2 + 4*sqrt(2)*sinh(x)^3 + 3*(4*sqrt(2)*cosh(
x)^2 + sqrt(2))*sinh(x) + 3*sqrt(2)*cosh(x))*sqrt(cosh(x)/(cosh(x) - sinh(x))) - 3*(sqrt(2)*cosh(x)^4 + 4*sqrt
(2)*cosh(x)*sinh(x)^3 + sqrt(2)*sinh(x)^4 + 2*(3*sqrt(2)*cosh(x)^2 + sqrt(2))*sinh(x)^2 + 2*sqrt(2)*cosh(x)^2
+ 4*(sqrt(2)*cosh(x)^3 + sqrt(2)*cosh(x))*sinh(x) + sqrt(2))*log(-2*sqrt(2)*sqrt(cosh(x)/(cosh(x) - sinh(x)))*
(cosh(x) + sinh(x)) - 2*cosh(x)^2 - 4*cosh(x)*sinh(x) - 2*sinh(x)^2 - 1))/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + s
inh(x)^4 + 2*(3*cosh(x)^2 + 1)*sinh(x)^2 + 2*cosh(x)^2 + 4*(cosh(x)^3 + cosh(x))*sinh(x) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\tanh{\left (x \right )} + 1\right )^{\frac{5}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+tanh(x))**(5/2),x)

[Out]

Integral((tanh(x) + 1)**(5/2), x)

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Giac [B]  time = 1.1742, size = 130, normalized size = 2.89 \begin{align*} \frac{2}{3} \, \sqrt{2}{\left (\frac{2 \,{\left (6 \,{\left (\sqrt{e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} - 9 \, \sqrt{e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 9 \, e^{\left (2 \, x\right )} + 4\right )}}{{\left (\sqrt{e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )} - 1\right )}^{3}} - 3 \, \log \left (-2 \, \sqrt{e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 2 \, e^{\left (2 \, x\right )} + 1\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+tanh(x))^(5/2),x, algorithm="giac")

[Out]

2/3*sqrt(2)*(2*(6*(sqrt(e^(4*x) + e^(2*x)) - e^(2*x))^2 - 9*sqrt(e^(4*x) + e^(2*x)) + 9*e^(2*x) + 4)/(sqrt(e^(
4*x) + e^(2*x)) - e^(2*x) - 1)^3 - 3*log(-2*sqrt(e^(4*x) + e^(2*x)) + 2*e^(2*x) + 1))