\(\int (1+\tanh ^2(x))^{3/2} \, dx\) [227]

Optimal result

Integrand size = 10, antiderivative size = 50 \[ \int \left (1+\tanh ^2(x)\right )^{3/2} \, dx=-\frac {5}{2} \text {arcsinh}(\tanh (x))+2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \tanh (x)}{\sqrt {1+\tanh ^2(x)}}\right )-\frac {1}{2} \tanh (x) \sqrt {1+\tanh ^2(x)} \]

[Out]

-5/2*arcsinh(tanh(x))+2*arctanh(2^(1/2)*tanh(x)/(1+tanh(x)^2)^(1/2))*2^(1/2)-1/2*(1+tanh(x)^2)^(1/2)*tanh(x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3742, 427, 537, 221, 385, 212} \[ \int \left (1+\tanh ^2(x)\right )^{3/2} \, dx=-\frac {5}{2} \text {arcsinh}(\tanh (x))+2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \tanh (x)}{\sqrt {\tanh ^2(x)+1}}\right )-\frac {1}{2} \tanh (x) \sqrt {\tanh ^2(x)+1} \]

[In]

Int[(1 + Tanh[x]^2)^(3/2),x]

[Out]

(-5*ArcSinh[Tanh[x]])/2 + 2*Sqrt[2]*ArcTanh[(Sqrt[2]*Tanh[x])/Sqrt[1 + Tanh[x]^2]] - (Tanh[x]*Sqrt[1 + Tanh[x]
^2])/2

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 221

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

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 427

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c
+ d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rule 537

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 3742

Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x]
, x]}, Dist[c*(ff/f), Subst[Int[(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ
[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\left (1+x^2\right )^{3/2}}{1-x^2} \, dx,x,\tanh (x)\right ) \\ & = -\frac {1}{2} \tanh (x) \sqrt {1+\tanh ^2(x)}-\frac {1}{2} \text {Subst}\left (\int \frac {-3-5 x^2}{\left (1-x^2\right ) \sqrt {1+x^2}} \, dx,x,\tanh (x)\right ) \\ & = -\frac {1}{2} \tanh (x) \sqrt {1+\tanh ^2(x)}-\frac {5}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\tanh (x)\right )+4 \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {1+x^2}} \, dx,x,\tanh (x)\right ) \\ & = -\frac {5}{2} \text {arcsinh}(\tanh (x))-\frac {1}{2} \tanh (x) \sqrt {1+\tanh ^2(x)}+4 \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\tanh (x)}{\sqrt {1+\tanh ^2(x)}}\right ) \\ & = -\frac {5}{2} \text {arcsinh}(\tanh (x))+2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \tanh (x)}{\sqrt {1+\tanh ^2(x)}}\right )-\frac {1}{2} \tanh (x) \sqrt {1+\tanh ^2(x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.48 \[ \int \left (1+\tanh ^2(x)\right )^{3/2} \, dx=-\frac {\left (-4 \sqrt {2} \text {arcsinh}\left (\sqrt {2} \sinh (x)\right ) \cosh ^3(x)+5 \text {arctanh}\left (\frac {\sinh (x)}{\sqrt {\cosh (2 x)}}\right ) \cosh ^3(x)+\cosh (x) \sqrt {\cosh (2 x)} \sinh (x)\right ) \left (1+\tanh ^2(x)\right )^{3/2}}{2 \cosh ^{\frac {3}{2}}(2 x)} \]

[In]

Integrate[(1 + Tanh[x]^2)^(3/2),x]

[Out]

-1/2*((-4*Sqrt[2]*ArcSinh[Sqrt[2]*Sinh[x]]*Cosh[x]^3 + 5*ArcTanh[Sinh[x]/Sqrt[Cosh[2*x]]]*Cosh[x]^3 + Cosh[x]*
Sqrt[Cosh[2*x]]*Sinh[x])*(1 + Tanh[x]^2)^(3/2))/Cosh[2*x]^(3/2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(157\) vs. \(2(38)=76\).

Time = 0.08 (sec) , antiderivative size = 158, normalized size of antiderivative = 3.16

method	result	size
derivativedivides	\(-\frac {\left (\left (\tanh \left (x \right )-1\right )^{2}+2 \tanh \left (x \right )\right )^{\frac {3}{2}}}{6}-\frac {\tanh \left (x \right ) \sqrt {\left (\tanh \left (x \right )-1\right )^{2}+2 \tanh \left (x \right )}}{4}-\frac {5 \,\operatorname {arcsinh}\left (\tanh \left (x \right )\right )}{2}-\sqrt {\left (\tanh \left (x \right )-1\right )^{2}+2 \tanh \left (x \right )}+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2+2 \tanh \left (x \right )\right ) \sqrt {2}}{4 \sqrt {\left (\tanh \left (x \right )-1\right )^{2}+2 \tanh \left (x \right )}}\right )+\frac {\left (\left (1+\tanh \left (x \right )\right )^{2}-2 \tanh \left (x \right )\right )^{\frac {3}{2}}}{6}-\frac {\tanh \left (x \right ) \sqrt {\left (1+\tanh \left (x \right )\right )^{2}-2 \tanh \left (x \right )}}{4}+\sqrt {\left (1+\tanh \left (x \right )\right )^{2}-2 \tanh \left (x \right )}-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2-2 \tanh \left (x \right )\right ) \sqrt {2}}{4 \sqrt {\left (1+\tanh \left (x \right )\right )^{2}-2 \tanh \left (x \right )}}\right )\)	\(158\)
default	\(-\frac {\left (\left (\tanh \left (x \right )-1\right )^{2}+2 \tanh \left (x \right )\right )^{\frac {3}{2}}}{6}-\frac {\tanh \left (x \right ) \sqrt {\left (\tanh \left (x \right )-1\right )^{2}+2 \tanh \left (x \right )}}{4}-\frac {5 \,\operatorname {arcsinh}\left (\tanh \left (x \right )\right )}{2}-\sqrt {\left (\tanh \left (x \right )-1\right )^{2}+2 \tanh \left (x \right )}+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2+2 \tanh \left (x \right )\right ) \sqrt {2}}{4 \sqrt {\left (\tanh \left (x \right )-1\right )^{2}+2 \tanh \left (x \right )}}\right )+\frac {\left (\left (1+\tanh \left (x \right )\right )^{2}-2 \tanh \left (x \right )\right )^{\frac {3}{2}}}{6}-\frac {\tanh \left (x \right ) \sqrt {\left (1+\tanh \left (x \right )\right )^{2}-2 \tanh \left (x \right )}}{4}+\sqrt {\left (1+\tanh \left (x \right )\right )^{2}-2 \tanh \left (x \right )}-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2-2 \tanh \left (x \right )\right ) \sqrt {2}}{4 \sqrt {\left (1+\tanh \left (x \right )\right )^{2}-2 \tanh \left (x \right )}}\right )\)	\(158\)

[In]

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

[Out]

-1/6*((tanh(x)-1)^2+2*tanh(x))^(3/2)-1/4*tanh(x)*((tanh(x)-1)^2+2*tanh(x))^(1/2)-5/2*arcsinh(tanh(x))-((tanh(x
)-1)^2+2*tanh(x))^(1/2)+2^(1/2)*arctanh(1/4*(2+2*tanh(x))*2^(1/2)/((tanh(x)-1)^2+2*tanh(x))^(1/2))+1/6*((1+tan
h(x))^2-2*tanh(x))^(3/2)-1/4*tanh(x)*((1+tanh(x))^2-2*tanh(x))^(1/2)+((1+tanh(x))^2-2*tanh(x))^(1/2)-2^(1/2)*a
rctanh(1/4*(2-2*tanh(x))*2^(1/2)/((1+tanh(x))^2-2*tanh(x))^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1027 vs. \(2 (38) = 76\).

Time = 0.27 (sec) , antiderivative size = 1027, normalized size of antiderivative = 20.54 \[ \int \left (1+\tanh ^2(x)\right )^{3/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/4*(2*(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*(cosh(x)
^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + (28*cosh(x)^2 - 3)*sinh(x)^6 - 3*cosh(x)^6 + 2*(28*cosh(x)^3 - 9*cosh(x
))*sinh(x)^5 + 5*(14*cosh(x)^4 - 9*cosh(x)^2 + 1)*sinh(x)^4 + 5*cosh(x)^4 + 4*(14*cosh(x)^5 - 15*cosh(x)^3 + 5
*cosh(x))*sinh(x)^3 + (28*cosh(x)^6 - 45*cosh(x)^4 + 30*cosh(x)^2 - 4)*sinh(x)^2 - 4*cosh(x)^2 + 2*(4*cosh(x)^
7 - 9*cosh(x)^5 + 10*cosh(x)^3 - 4*cosh(x))*sinh(x) + (sqrt(2)*cosh(x)^6 + 6*sqrt(2)*cosh(x)*sinh(x)^5 + sqrt(
2)*sinh(x)^6 + 3*(5*sqrt(2)*cosh(x)^2 - sqrt(2))*sinh(x)^4 - 3*sqrt(2)*cosh(x)^4 + 4*(5*sqrt(2)*cosh(x)^3 - 3*
sqrt(2)*cosh(x))*sinh(x)^3 + (15*sqrt(2)*cosh(x)^4 - 18*sqrt(2)*cosh(x)^2 + 4*sqrt(2))*sinh(x)^2 + 4*sqrt(2)*c
osh(x)^2 + 2*(3*sqrt(2)*cosh(x)^5 - 6*sqrt(2)*cosh(x)^3 + 4*sqrt(2)*cosh(x))*sinh(x) - 4*sqrt(2))*sqrt((cosh(x
)^2 + sinh(x)^2)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4)/(cosh(x)^6 + 6*cosh(x)^5*sinh(x) + 15*cosh(
x)^4*sinh(x)^2 + 20*cosh(x)^3*sinh(x)^3 + 15*cosh(x)^2*sinh(x)^4 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6)) + 2*(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*(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(
2)*cosh(x)^2 + 2*sqrt(2)*cosh(x)*sinh(x) + sqrt(2)*sinh(x)^2 + sqrt(2))*sqrt((cosh(x)^2 + sinh(x)^2)/(cosh(x)^
2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 1)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)) - 5*(cosh(x)^4 + 4*cosh(
x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 + 1)*sinh(x)^2 + 2*cosh(x)^2 + 4*(cosh(x)^3 + cosh(x))*sinh(x) + 1)*
log((cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 2*sqrt((cosh(x)^2 + sinh(x)^2)/(cosh(x)^2 - 2*cosh(x)*sinh(x)
+ sinh(x)^2)) - 1)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)) + 5*(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x
)^4 + 2*(3*cosh(x)^2 + 1)*sinh(x)^2 + 2*cosh(x)^2 + 4*(cosh(x)^3 + cosh(x))*sinh(x) + 1)*log((cosh(x)^2 + 2*co
sh(x)*sinh(x) + sinh(x)^2 - 2*sqrt((cosh(x)^2 + sinh(x)^2)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) - 1)/(
cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)) - 4*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt((cosh(x)^
2 + sinh(x)^2)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*
(3*cosh(x)^2 + 1)*sinh(x)^2 + 2*cosh(x)^2 + 4*(cosh(x)^3 + cosh(x))*sinh(x) + 1)

Sympy [F]

\[ \int \left (1+\tanh ^2(x)\right )^{3/2} \, dx=\int \left (\tanh ^{2}{\left (x \right )} + 1\right )^{\frac {3}{2}}\, dx \]

[In]

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

[Out]

Integral((tanh(x)**2 + 1)**(3/2), x)

Maxima [F]

\[ \int \left (1+\tanh ^2(x)\right )^{3/2} \, dx=\int { {\left (\tanh \left (x\right )^{2} + 1\right )}^{\frac {3}{2}} \,d x } \]

[In]

integrate((1+tanh(x)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((tanh(x)^2 + 1)^(3/2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (38) = 76\).

Time = 0.28 (sec) , antiderivative size = 202, normalized size of antiderivative = 4.04 \[ \int \left (1+\tanh ^2(x)\right )^{3/2} \, dx=-\frac {1}{4} \, \sqrt {2} {\left (5 \, \sqrt {2} \log \left (\frac {\sqrt {2} - \sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1}{\sqrt {2} + \sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} - 1}\right ) - \frac {4 \, {\left (3 \, {\left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{3} - {\left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} - \sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} - 1\right )}}{{\left ({\left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} - 2 \, \sqrt {e^{\left (4 \, x\right )} + 1} + 2 \, e^{\left (2 \, x\right )} - 1\right )}^{2}} + 4 \, \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) + 4 \, \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) - 4 \, \log \left (-\sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right )\right )} \]

[In]

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

[Out]

-1/4*sqrt(2)*(5*sqrt(2)*log((sqrt(2) - sqrt(e^(4*x) + 1) + e^(2*x) + 1)/(sqrt(2) + sqrt(e^(4*x) + 1) - e^(2*x)
- 1)) - 4*(3*(sqrt(e^(4*x) + 1) - e^(2*x))^3 - (sqrt(e^(4*x) + 1) - e^(2*x))^2 - sqrt(e^(4*x) + 1) + e^(2*x)
- 1)/((sqrt(e^(4*x) + 1) - e^(2*x))^2 - 2*sqrt(e^(4*x) + 1) + 2*e^(2*x) - 1)^2 + 4*log(sqrt(e^(4*x) + 1) - e^(
2*x) + 1) + 4*log(sqrt(e^(4*x) + 1) - e^(2*x)) - 4*log(-sqrt(e^(4*x) + 1) + e^(2*x) + 1))

Mupad [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.56 \[ \int \left (1+\tanh ^2(x)\right )^{3/2} \, dx=\sqrt {2}\,\left (\ln \left (\mathrm {tanh}\left (x\right )+1\right )-\ln \left (\sqrt {2}\,\sqrt {{\mathrm {tanh}\left (x\right )}^2+1}-\mathrm {tanh}\left (x\right )+1\right )\right )-\frac {5\,\mathrm {asinh}\left (\mathrm {tanh}\left (x\right )\right )}{2}-\frac {\mathrm {tanh}\left (x\right )\,\sqrt {{\mathrm {tanh}\left (x\right )}^2+1}}{2}+\sqrt {2}\,\left (\ln \left (\mathrm {tanh}\left (x\right )+\sqrt {2}\,\sqrt {{\mathrm {tanh}\left (x\right )}^2+1}+1\right )-\ln \left (\mathrm {tanh}\left (x\right )-1\right )\right ) \]

[In]

int((tanh(x)^2 + 1)^(3/2),x)

[Out]

2^(1/2)*(log(tanh(x) + 1) - log(2^(1/2)*(tanh(x)^2 + 1)^(1/2) - tanh(x) + 1)) - (5*asinh(tanh(x)))/2 - (tanh(x
)*(tanh(x)^2 + 1)^(1/2))/2 + 2^(1/2)*(log(tanh(x) + 2^(1/2)*(tanh(x)^2 + 1)^(1/2) + 1) - log(tanh(x) - 1))