3.142 \(\int \frac {\tanh (c+d x)}{a+b \text {sech}^2(c+d x)} \, dx\)

Optimal. Leaf size=23 \[ \frac {\log \left (a \cosh ^2(c+d x)+b\right )}{2 a d} \]

[Out]

1/2*ln(b+a*cosh(d*x+c)^2)/a/d

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {4138, 260} \[ \frac {\log \left (a \cosh ^2(c+d x)+b\right )}{2 a d} \]

Antiderivative was successfully verified.

[In]

Int[Tanh[c + d*x]/(a + b*Sech[c + d*x]^2),x]

[Out]

Log[b + a*Cosh[c + d*x]^2]/(2*a*d)

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 4138

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> Module[{ff =
FreeFactors[Cos[e + f*x], x]}, -Dist[(f*ff^(m + n*p - 1))^(-1), Subst[Int[((1 - ff^2*x^2)^((m - 1)/2)*(b + a*
(ff*x)^n)^p)/x^(m + n*p), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[(m - 1)/2] &&
IntegerQ[n] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {\tanh (c+d x)}{a+b \text {sech}^2(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x}{b+a x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\log \left (b+a \cosh ^2(c+d x)\right )}{2 a d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.18, size = 26, normalized size = 1.13 \[ \frac {\log (a \cosh (2 (c+d x))+a+2 b)}{2 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[c + d*x]/(a + b*Sech[c + d*x]^2),x]

[Out]

Log[a + 2*b + a*Cosh[2*(c + d*x)]]/(2*a*d)

________________________________________________________________________________________

fricas [B]  time = 0.42, size = 76, normalized size = 3.30 \[ -\frac {2 \, d x - \log \left (\frac {2 \, {\left (a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + a + 2 \, b\right )}}{\cosh \left (d x + c\right )^{2} - 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2}}\right )}{2 \, a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)/(a+b*sech(d*x+c)^2),x, algorithm="fricas")

[Out]

-1/2*(2*d*x - log(2*(a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 + a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(
d*x + c) + sinh(d*x + c)^2)))/(a*d)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)/(a+b*sech(d*x+c)^2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Erro
r index.cc index_gcd Error: Bad Argument ValueError index.cc index_gcd Error: Bad Argument ValueDone

________________________________________________________________________________________

maple [A]  time = 0.15, size = 38, normalized size = 1.65 \[ \frac {\ln \left (a +b \mathrm {sech}\left (d x +c \right )^{2}\right )}{2 d a}-\frac {\ln \left (\mathrm {sech}\left (d x +c \right )\right )}{d a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(d*x+c)/(a+b*sech(d*x+c)^2),x)

[Out]

1/2/d/a*ln(a+b*sech(d*x+c)^2)-1/d/a*ln(sech(d*x+c))

________________________________________________________________________________________

maxima [B]  time = 0.38, size = 51, normalized size = 2.22 \[ \frac {d x + c}{a d} + \frac {\log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{2 \, a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)/(a+b*sech(d*x+c)^2),x, algorithm="maxima")

[Out]

(d*x + c)/(a*d) + 1/2*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/(a*d)

________________________________________________________________________________________

mupad [B]  time = 0.35, size = 51, normalized size = 2.22 \[ \frac {\ln \left (a+2\,a\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+a\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}+4\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )-2\,d\,x}{2\,a\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(c + d*x)/(a + b/cosh(c + d*x)^2),x)

[Out]

(log(a + 2*a*exp(2*c)*exp(2*d*x) + a*exp(4*c)*exp(4*d*x) + 4*b*exp(2*c)*exp(2*d*x)) - 2*d*x)/(2*a*d)

________________________________________________________________________________________

sympy [A]  time = 7.04, size = 124, normalized size = 5.39 \[ \begin {cases} \frac {\tilde {\infty } x \tanh {\relax (c )}}{\operatorname {sech}^{2}{\relax (c )}} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac {1}{2 b d \operatorname {sech}^{2}{\left (c + d x \right )}} & \text {for}\: a = 0 \\\frac {x \tanh {\relax (c )}}{a + b \operatorname {sech}^{2}{\relax (c )}} & \text {for}\: d = 0 \\\frac {x - \frac {\log {\left (\tanh {\left (c + d x \right )} + 1 \right )}}{d}}{a} & \text {for}\: b = 0 \\\frac {x}{a} + \frac {\log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \operatorname {sech}{\left (c + d x \right )} \right )}}{2 a d} + \frac {\log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \operatorname {sech}{\left (c + d x \right )} \right )}}{2 a d} - \frac {\log {\left (\tanh {\left (c + d x \right )} + 1 \right )}}{a d} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)/(a+b*sech(d*x+c)**2),x)

[Out]

Piecewise((zoo*x*tanh(c)/sech(c)**2, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), (1/(2*b*d*sech(c + d*x)**2), Eq(a, 0)),
(x*tanh(c)/(a + b*sech(c)**2), Eq(d, 0)), ((x - log(tanh(c + d*x) + 1)/d)/a, Eq(b, 0)), (x/a + log(-I*sqrt(a)*
sqrt(1/b) + sech(c + d*x))/(2*a*d) + log(I*sqrt(a)*sqrt(1/b) + sech(c + d*x))/(2*a*d) - log(tanh(c + d*x) + 1)
/(a*d), True))

________________________________________________________________________________________