3.2.52 \(\int \frac {\tanh (c+d x)}{(a+b \text {sech}^2(c+d x))^2} \, dx\) [152]

Optimal. Leaf size=49 \[ \frac {b}{2 a^2 d \left (b+a \cosh ^2(c+d x)\right )}+\frac {\log \left (b+a \cosh ^2(c+d x)\right )}{2 a^2 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4223, 272, 45} \begin {gather*} \frac {b}{2 a^2 d \left (a \cosh ^2(c+d x)+b\right )}+\frac {\log \left (a \cosh ^2(c+d x)+b\right )}{2 a^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 4223

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)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {x^3}{\left (b+a x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {x}{(b+a x)^2} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {b}{a (b+a x)^2}+\frac {1}{a (b+a x)}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {b}{2 a^2 d \left (b+a \cosh ^2(c+d x)\right )}+\frac {\log \left (b+a \cosh ^2(c+d x)\right )}{2 a^2 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.41, size = 79, normalized size = 1.61 \begin {gather*} \frac {2 b+(a+2 b) \log (a+2 b+a \cosh (2 (c+d x)))+a \cosh (2 (c+d x)) \log (a+2 b+a \cosh (2 (c+d x)))}{2 a^2 d (a+2 b+a \cosh (2 (c+d x)))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.95, size = 62, normalized size = 1.27

method	result	size
derivativedivides	\(-\frac {\frac {\ln \left (\mathrm {sech}\left (d x +c \right )\right )}{a^{2}}-\frac {b \left (-\frac {a}{b \left (a +b \mathrm {sech}\left (d x +c \right )^{2}\right )}+\frac {\ln \left (a +b \mathrm {sech}\left (d x +c \right )^{2}\right )}{b}\right )}{2 a^{2}}}{d}\)	\(62\)
default	\(-\frac {\frac {\ln \left (\mathrm {sech}\left (d x +c \right )\right )}{a^{2}}-\frac {b \left (-\frac {a}{b \left (a +b \mathrm {sech}\left (d x +c \right )^{2}\right )}+\frac {\ln \left (a +b \mathrm {sech}\left (d x +c \right )^{2}\right )}{b}\right )}{2 a^{2}}}{d}\)	\(62\)
risch	\(-\frac {x}{a^{2}}-\frac {2 c}{a^{2} d}+\frac {2 b \,{\mathrm e}^{2 d x +2 c}}{a^{2} d \left (a \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )}+\frac {\ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (2 b +a \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 a^{2} d}\)	\(113\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (45) = 90\).
time = 0.27, size = 106, normalized size = 2.16 \begin {gather*} \frac {2 \, b e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (a^{3} e^{\left (-4 \, d x - 4 \, c\right )} + a^{3} + 2 \, {\left (a^{3} + 2 \, a^{2} b\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} d} + \frac {d x + c}{a^{2} 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^{2} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b*e^(-2*d*x - 2*c)/((a^3*e^(-4*d*x - 4*c) + a^3 + 2*(a^3 + 2*a^2*b)*e^(-2*d*x - 2*c))*d) + (d*x + c)/(a^2*d)
+ 1/2*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/(a^2*d)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 476 vs. \(2 (45) = 90\).
time = 0.44, size = 476, normalized size = 9.71 \begin {gather*} -\frac {2 \, a d x \cosh \left (d x + c\right )^{4} + 8 \, a d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 2 \, a d x \sinh \left (d x + c\right )^{4} + 2 \, a d x + 4 \, {\left ({\left (a + 2 \, b\right )} d x - b\right )} \cosh \left (d x + c\right )^{2} + 4 \, {\left (3 \, a d x \cosh \left (d x + c\right )^{2} + {\left (a + 2 \, b\right )} d x - b\right )} \sinh \left (d x + c\right )^{2} - {\left (a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} + 2 \, {\left (a + 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a \cosh \left (d x + c\right )^{2} + a + 2 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a \cosh \left (d x + c\right )^{3} + {\left (a + 2 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a\right )} \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 ) + 8 \, {\left (a d x \cosh \left (d x + c\right )^{3} + {\left ({\left (a + 2 \, b\right )} d x - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left (a^{3} d \cosh \left (d x + c\right )^{4} + 4 \, a^{3} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{3} d \sinh \left (d x + c\right )^{4} + a^{3} d + 2 \, {\left (a^{3} + 2 \, a^{2} b\right )} d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{3} d \cosh \left (d x + c\right )^{2} + {\left (a^{3} + 2 \, a^{2} b\right )} d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a^{3} d \cosh \left (d x + c\right )^{3} + {\left (a^{3} + 2 \, a^{2} b\right )} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*a*d*x*cosh(d*x + c)^4 + 8*a*d*x*cosh(d*x + c)*sinh(d*x + c)^3 + 2*a*d*x*sinh(d*x + c)^4 + 2*a*d*x + 4*
((a + 2*b)*d*x - b)*cosh(d*x + c)^2 + 4*(3*a*d*x*cosh(d*x + c)^2 + (a + 2*b)*d*x - b)*sinh(d*x + c)^2 - (a*cos
h(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 + 2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*co
sh(d*x + c)^2 + a + 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 + (a + 2*b)*cosh(d*x + c))*sinh(d*x + c) + a)*
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) + sin
h(d*x + c)^2)) + 8*(a*d*x*cosh(d*x + c)^3 + ((a + 2*b)*d*x - b)*cosh(d*x + c))*sinh(d*x + c))/(a^3*d*cosh(d*x
+ c)^4 + 4*a^3*d*cosh(d*x + c)*sinh(d*x + c)^3 + a^3*d*sinh(d*x + c)^4 + a^3*d + 2*(a^3 + 2*a^2*b)*d*cosh(d*x
+ c)^2 + 2*(3*a^3*d*cosh(d*x + c)^2 + (a^3 + 2*a^2*b)*d)*sinh(d*x + c)^2 + 4*(a^3*d*cosh(d*x + c)^3 + (a^3 + 2
*a^2*b)*d*cosh(d*x + c))*sinh(d*x + c))

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

________________________________________________________________________________________

Mupad [B]
time = 1.73, size = 53, normalized size = 1.08 \begin {gather*} \frac {\ln \left ({\mathrm {cosh}\left (c+d\,x\right )}^2\,\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )\right )}{2\,a^2\,d}-\frac {1}{2\,a\,d\,\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________