Integrand size = 21, antiderivative size = 49 \[ \int \frac {\tanh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {b}{2 a^2 d \left (b+a \cosh ^2(c+d x)\right )}+\frac {\log \left (b+a \cosh ^2(c+d x)\right )}{2 a^2 d} \] Output:
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
Time = 0.37 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.61 \[ \int \frac {\tanh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {2 b+(a+2 b) \log (a+2 b+a \cosh (2 (c+d x)))+a \cosh (2 (c+d x)) \log (a+2 b+a \cosh (2 (c+d x)))}{2 a^2 d (a+2 b+a \cosh (2 (c+d x)))} \] Input:
Integrate[Tanh[c + d*x]/(a + b*Sech[c + d*x]^2)^2,x]
Output:
(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) ]))
Time = 0.29 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 26, 4626, 243, 49, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\tanh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \tan (i c+i d x)}{\left (a+b \sec (i c+i d x)^2\right )^2}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\tan (i c+i d x)}{\left (b \sec (i c+i d x)^2+a\right )^2}dx\) |
\(\Big \downarrow \) 4626 |
\(\displaystyle \frac {\int \frac {\cosh ^3(c+d x)}{\left (a \cosh ^2(c+d x)+b\right )^2}d\cosh (c+d x)}{d}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\int \frac {\cosh ^2(c+d x)}{\left (a \cosh ^2(c+d x)+b\right )^2}d\cosh ^2(c+d x)}{2 d}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\int \left (\frac {1}{a \left (a \cosh ^2(c+d x)+b\right )}-\frac {b}{a \left (a \cosh ^2(c+d x)+b\right )^2}\right )d\cosh ^2(c+d x)}{2 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {b}{a^2 \left (a \cosh ^2(c+d x)+b\right )}+\frac {\log \left (a \cosh ^2(c+d x)+b\right )}{a^2}}{2 d}\) |
Input:
Int[Tanh[c + d*x]/(a + b*Sech[c + d*x]^2)^2,x]
Output:
(b/(a^2*(b + a*Cosh[c + d*x]^2)) + Log[b + a*Cosh[c + d*x]^2]/a^2)/(2*d)
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_ )]^(m_.), x_Symbol] :> Module[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-(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]
Time = 6.50 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.22
method | result | size |
derivativedivides | \(-\frac {\ln \left (\operatorname {sech}\left (d x +c \right )\right )}{d \,a^{2}}+\frac {\ln \left (a +b \operatorname {sech}\left (d x +c \right )^{2}\right )}{2 d \,a^{2}}-\frac {1}{2 d a \left (a +b \operatorname {sech}\left (d x +c \right )^{2}\right )}\) | \(60\) |
default | \(-\frac {\ln \left (\operatorname {sech}\left (d x +c \right )\right )}{d \,a^{2}}+\frac {\ln \left (a +b \operatorname {sech}\left (d x +c \right )^{2}\right )}{2 d \,a^{2}}-\frac {1}{2 d a \left (a +b \operatorname {sech}\left (d x +c \right )^{2}\right )}\) | \(60\) |
risch | \(-\frac {x}{a^{2}}-\frac {2 c}{a^{2} d}+\frac {2 b \,{\mathrm e}^{2 d x +2 c}}{a^{2} d \left ({\mathrm e}^{4 d x +4 c} a +2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )}+\frac {\ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a +2 b \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 a^{2} d}\) | \(113\) |
Input:
int(tanh(d*x+c)/(a+b*sech(d*x+c)^2)^2,x,method=_RETURNVERBOSE)
Output:
-1/d/a^2*ln(sech(d*x+c))+1/2/d/a^2*ln(a+b*sech(d*x+c)^2)-1/2/d/a/(a+b*sech (d*x+c)^2)
Leaf count of result is larger than twice the leaf count of optimal. 476 vs. \(2 (45) = 90\).
Time = 0.23 (sec) , antiderivative size = 476, normalized size of antiderivative = 9.71 \[ \int \frac {\tanh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=-\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 )}} \] Input:
integrate(tanh(d*x+c)/(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")
Output:
-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*cosh( 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*cosh(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*c osh(d*x + c)*sinh(d*x + c) + sinh(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))
Timed out. \[ \int \frac {\tanh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Timed out} \] Input:
integrate(tanh(d*x+c)/(a+b*sech(d*x+c)**2)**2,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (45) = 90\).
Time = 0.05 (sec) , antiderivative size = 106, normalized size of antiderivative = 2.16 \[ \int \frac {\tanh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {2 \, b e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (a^{3} e^{\left (-4 \, d x - 4 \, c\right )} + a^{3} + 2 \, {\left (a^{3} + 2 \, a^{2} b\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} d} + \frac {d x + c}{a^{2} d} + \frac {\log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{2 \, a^{2} d} \] Input:
integrate(tanh(d*x+c)/(a+b*sech(d*x+c)^2)^2,x, algorithm="maxima")
Output:
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)
Exception generated. \[ \int \frac {\tanh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(tanh(d*x+c)/(a+b*sech(d*x+c)^2)^2,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Time = 2.53 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.08 \[ \int \frac {\tanh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {\ln \left ({\mathrm {cosh}\left (c+d\,x\right )}^2\,\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )\right )}{2\,a^2\,d}-\frac {1}{2\,a\,d\,\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )} \] Input:
int(tanh(c + d*x)/(a + b/cosh(c + d*x)^2)^2,x)
Output:
log(cosh(c + d*x)^2*(a + b/cosh(c + d*x)^2))/(2*a^2*d) - 1/(2*a*d*(a + b/c osh(c + d*x)^2))
Time = 0.25 (sec) , antiderivative size = 1033, normalized size of antiderivative = 21.08 \[ \int \frac {\tanh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:
int(tanh(d*x+c)/(a+b*sech(d*x+c)^2)^2,x)
Output:
(e**(4*c + 4*d*x)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d *x)*sqrt(a))*a**2 + 2*e**(4*c + 4*d*x)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a*b + e**(4*c + 4*d*x)*log(sqrt(2*sqrt(b )*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**2 + 2*e**(4*c + 4*d*x) *log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a*b + e **(4*c + 4*d*x)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)* a**2 + 2*e**(4*c + 4*d*x)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a*b - 2*e**(4*c + 4*d*x)*a**2*d*x - 4*e**(4*c + 4*d*x)*a*b*d*x - 2*e**(4*c + 4*d*x)*a*b + 2*e**(2*c + 2*d*x)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**2 + 8*e**(2*c + 2*d*x)*log( - s qrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a*b + 8*e**(2 *c + 2*d*x)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sq rt(a))*b**2 + 2*e**(2*c + 2*d*x)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**2 + 8*e**(2*c + 2*d*x)*log(sqrt(2*sqrt(b)*sqrt (a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a*b + 8*e**(2*c + 2*d*x)*log(sq rt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*b**2 + 2*e**(2 *c + 2*d*x)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a**2 + 8*e**(2*c + 2*d*x)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a*b + 8*e**(2*c + 2*d*x)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x )*a + a + 2*b)*b**2 - 4*e**(2*c + 2*d*x)*a**2*d*x - 16*e**(2*c + 2*d*x)...