∫ tanh(x)______∘ a + b cosh2(x) dx [78]

Optimal. Leaf size=26 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cosh ^2(x)}}{\sqrt {a}}\right )}{\sqrt {a}} \]

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Rubi [A]
time = 0.04, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3273, 65, 214} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cosh ^2(x)}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub

st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &

& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,

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

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = Free

Factors[Sin[e + f*x]^2, x]}, Dist[ff^((m + 1)/2)/(2*f), Subst[Int[x^((m - 1)/2)*((a + b*ff*x)^p/(1 - ff*x)^((m

+ 1)/2)), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

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

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Mathematica [A]
time = 0.02, size = 26, normalized size = 1.00 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cosh ^2(x)}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}

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method	result	size

derivativedivides	\(-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\cosh ^{2}\left (x \right )\right )}}{\cosh \left (x \right )}\right )}{\sqrt {a}}\)	\(31\)
default	\(-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\cosh ^{2}\left (x \right )\right )}}{\cosh \left (x \right )}\right )}{\sqrt {a}}\)	\(31\)

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (20) = 40\).
time = 0.44, size = 248, normalized size = 9.54 \begin {gather*} \left [\frac {\log \left (\frac {b \cosh \left (x\right )^{4} + 4 \, b \cosh \left (x\right ) \sinh \left (x\right )^{3} + b \sinh \left (x\right )^{4} + 2 \, {\left (4 \, a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b \cosh \left (x\right )^{2} + 4 \, a + b\right )} \sinh \left (x\right )^{2} - 4 \, \sqrt {2} \sqrt {a} \sqrt {\frac {b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a + b}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 4 \, {\left (b \cosh \left (x\right )^{3} + {\left (4 \, a + b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + b}{\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 )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {2} \sqrt {-a} \sqrt {\frac {b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a + b}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{2 \, {\left (a \cosh \left (x\right ) + a \sinh \left (x\right )\right )}}\right )}{a}\right ] \end {gather*}

[1/2*log((b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 + 2*(4*a + b)*cosh(x)^2 + 2*(3*b*cosh(x)^2 + 4*a +

b)*sinh(x)^2 - 4*sqrt(2)*sqrt(a)*sqrt((b*cosh(x)^2 + b*sinh(x)^2 + 2*a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) +

sinh(x)^2))*(cosh(x) + sinh(x)) + 4*(b*cosh(x)^3 + (4*a + b)*cosh(x))*sinh(x) + b)/(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))/sqrt(a

), sqrt(-a)*arctan(1/2*sqrt(2)*sqrt(-a)*sqrt((b*cosh(x)^2 + b*sinh(x)^2 + 2*a + b)/(cosh(x)^2 - 2*cosh(x)*sinh

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh {\left (x \right )}}{\sqrt {a + b \cosh ^{2}{\left (x \right )}}}\, dx \end {gather*}

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\mathrm {tanh}\left (x\right )}{\sqrt {b\,{\mathrm {cosh}\left (x\right )}^2+a}} \,d x \end {gather*}

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3.1.78 \(\int \frac {\tanh (x)}{\sqrt {a+b \cosh ^2(x)}} \, dx\) [78]