Integrand size = 21, antiderivative size = 73 \[ \int \frac {\tanh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=-\frac {b^2}{4 a^3 d \left (b+a \cosh ^2(c+d x)\right )^2}+\frac {b}{a^3 d \left (b+a \cosh ^2(c+d x)\right )}+\frac {\log \left (b+a \cosh ^2(c+d x)\right )}{2 a^3 d} \] Output:
-1/4*b^2/a^3/d/(b+a*cosh(d*x+c)^2)^2+b/a^3/d/(b+a*cosh(d*x+c)^2)+1/2*ln(b+ a*cosh(d*x+c)^2)/a^3/d
Time = 1.15 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.77 \[ \int \frac {\tanh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {2 b (2 a+3 b)+(a+2 b)^2 \log (a+2 b+a \cosh (2 (c+d x)))+a^2 \cosh ^2(2 (c+d x)) \log (a+2 b+a \cosh (2 (c+d x)))+2 a \cosh (2 (c+d x)) (2 b+(a+2 b) \log (a+2 b+a \cosh (2 (c+d x))))}{2 a^3 d (a+2 b+a \cosh (2 (c+d x)))^2} \] Input:
Integrate[Tanh[c + d*x]/(a + b*Sech[c + d*x]^2)^3,x]
Output:
(2*b*(2*a + 3*b) + (a + 2*b)^2*Log[a + 2*b + a*Cosh[2*(c + d*x)]] + a^2*Co sh[2*(c + d*x)]^2*Log[a + 2*b + a*Cosh[2*(c + d*x)]] + 2*a*Cosh[2*(c + d*x )]*(2*b + (a + 2*b)*Log[a + 2*b + a*Cosh[2*(c + d*x)]]))/(2*a^3*d*(a + 2*b + a*Cosh[2*(c + d*x)])^2)
Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.95, 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.
\(\displaystyle \int \frac {\tanh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \tan (i c+i d x)}{\left (a+b \sec (i c+i d x)^2\right )^3}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\tan (i c+i d x)}{\left (b \sec (i c+i d x)^2+a\right )^3}dx\) |
\(\Big \downarrow \) 4626 |
\(\displaystyle \frac {\int \frac {\cosh ^5(c+d x)}{\left (a \cosh ^2(c+d x)+b\right )^3}d\cosh (c+d x)}{d}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\int \frac {\cosh ^4(c+d x)}{\left (a \cosh ^2(c+d x)+b\right )^3}d\cosh ^2(c+d x)}{2 d}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\int \left (\frac {b^2}{a^2 \left (a \cosh ^2(c+d x)+b\right )^3}-\frac {2 b}{a^2 \left (a \cosh ^2(c+d x)+b\right )^2}+\frac {1}{a^2 \left (a \cosh ^2(c+d x)+b\right )}\right )d\cosh ^2(c+d x)}{2 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {b^2}{2 a^3 \left (a \cosh ^2(c+d x)+b\right )^2}+\frac {2 b}{a^3 \left (a \cosh ^2(c+d x)+b\right )}+\frac {\log \left (a \cosh ^2(c+d x)+b\right )}{a^3}}{2 d}\) |
Input:
Int[Tanh[c + d*x]/(a + b*Sech[c + d*x]^2)^3,x]
Output:
(-1/2*b^2/(a^3*(b + a*Cosh[c + d*x]^2)^2) + (2*b)/(a^3*(b + a*Cosh[c + d*x ]^2)) + Log[b + a*Cosh[c + d*x]^2]/a^3)/(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 = 63.74 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.12
method | result | size |
derivativedivides | \(-\frac {\ln \left (\operatorname {sech}\left (d x +c \right )\right )}{d \,a^{3}}-\frac {1}{2 d \,a^{2} \left (a +b \operatorname {sech}\left (d x +c \right )^{2}\right )}+\frac {\ln \left (a +b \operatorname {sech}\left (d x +c \right )^{2}\right )}{2 d \,a^{3}}-\frac {1}{4 d a \left (a +b \operatorname {sech}\left (d x +c \right )^{2}\right )^{2}}\) | \(82\) |
default | \(-\frac {\ln \left (\operatorname {sech}\left (d x +c \right )\right )}{d \,a^{3}}-\frac {1}{2 d \,a^{2} \left (a +b \operatorname {sech}\left (d x +c \right )^{2}\right )}+\frac {\ln \left (a +b \operatorname {sech}\left (d x +c \right )^{2}\right )}{2 d \,a^{3}}-\frac {1}{4 d a \left (a +b \operatorname {sech}\left (d x +c \right )^{2}\right )^{2}}\) | \(82\) |
risch | \(-\frac {x}{a^{3}}-\frac {2 c}{a^{3} d}+\frac {4 \left ({\mathrm e}^{4 d x +4 c} a +2 a \,{\mathrm e}^{2 d x +2 c}+3 b \,{\mathrm e}^{2 d x +2 c}+a \right ) {\mathrm e}^{2 d x +2 c} b}{a^{3} \left ({\mathrm e}^{4 d x +4 c} a +2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{2} d}+\frac {\ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a +2 b \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 a^{3} d}\) | \(150\) |
Input:
int(tanh(d*x+c)/(a+b*sech(d*x+c)^2)^3,x,method=_RETURNVERBOSE)
Output:
-1/d/a^3*ln(sech(d*x+c))-1/2/d/a^2/(a+b*sech(d*x+c)^2)+1/2/d/a^3*ln(a+b*se ch(d*x+c)^2)-1/4/d/a/(a+b*sech(d*x+c)^2)^2
Leaf count of result is larger than twice the leaf count of optimal. 1666 vs. \(2 (69) = 138\).
Time = 0.24 (sec) , antiderivative size = 1666, normalized size of antiderivative = 22.82 \[ \int \frac {\tanh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(tanh(d*x+c)/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")
Output:
-1/2*(2*a^2*d*x*cosh(d*x + c)^8 + 16*a^2*d*x*cosh(d*x + c)*sinh(d*x + c)^7 + 2*a^2*d*x*sinh(d*x + c)^8 + 8*((a^2 + 2*a*b)*d*x - a*b)*cosh(d*x + c)^6 + 8*(7*a^2*d*x*cosh(d*x + c)^2 + (a^2 + 2*a*b)*d*x - a*b)*sinh(d*x + c)^6 + 16*(7*a^2*d*x*cosh(d*x + c)^3 + 3*((a^2 + 2*a*b)*d*x - a*b)*cosh(d*x + c))*sinh(d*x + c)^5 + 4*((3*a^2 + 8*a*b + 8*b^2)*d*x - 4*a*b - 6*b^2)*cosh (d*x + c)^4 + 4*(35*a^2*d*x*cosh(d*x + c)^4 + (3*a^2 + 8*a*b + 8*b^2)*d*x + 30*((a^2 + 2*a*b)*d*x - a*b)*cosh(d*x + c)^2 - 4*a*b - 6*b^2)*sinh(d*x + c)^4 + 2*a^2*d*x + 16*(7*a^2*d*x*cosh(d*x + c)^5 + 10*((a^2 + 2*a*b)*d*x - a*b)*cosh(d*x + c)^3 + ((3*a^2 + 8*a*b + 8*b^2)*d*x - 4*a*b - 6*b^2)*cos h(d*x + c))*sinh(d*x + c)^3 + 8*((a^2 + 2*a*b)*d*x - a*b)*cosh(d*x + c)^2 + 8*(7*a^2*d*x*cosh(d*x + c)^6 + 15*((a^2 + 2*a*b)*d*x - a*b)*cosh(d*x + c )^4 + (a^2 + 2*a*b)*d*x + 3*((3*a^2 + 8*a*b + 8*b^2)*d*x - 4*a*b - 6*b^2)* cosh(d*x + c)^2 - a*b)*sinh(d*x + c)^2 - (a^2*cosh(d*x + c)^8 + 8*a^2*cosh (d*x + c)*sinh(d*x + c)^7 + a^2*sinh(d*x + c)^8 + 4*(a^2 + 2*a*b)*cosh(d*x + c)^6 + 4*(7*a^2*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^6 + 8*(7*a ^2*cosh(d*x + c)^3 + 3*(a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3 *a^2 + 8*a*b + 8*b^2)*cosh(d*x + c)^4 + 2*(35*a^2*cosh(d*x + c)^4 + 30*(a^ 2 + 2*a*b)*cosh(d*x + c)^2 + 3*a^2 + 8*a*b + 8*b^2)*sinh(d*x + c)^4 + 8*(7 *a^2*cosh(d*x + c)^5 + 10*(a^2 + 2*a*b)*cosh(d*x + c)^3 + (3*a^2 + 8*a*b + 8*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(a^2 + 2*a*b)*cosh(d*x + c)^...
Timed out. \[ \int \frac {\tanh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Timed out} \] Input:
integrate(tanh(d*x+c)/(a+b*sech(d*x+c)**2)**3,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (69) = 138\).
Time = 0.05 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.64 \[ \int \frac {\tanh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {4 \, {\left (a b e^{\left (-2 \, d x - 2 \, c\right )} + a b e^{\left (-6 \, d x - 6 \, c\right )} + {\left (2 \, a b + 3 \, b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )}}{{\left (a^{5} e^{\left (-8 \, d x - 8 \, c\right )} + a^{5} + 4 \, {\left (a^{5} + 2 \, a^{4} b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (3 \, a^{5} + 8 \, a^{4} b + 8 \, a^{3} b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, {\left (a^{5} + 2 \, a^{4} b\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )} d} + \frac {d x + c}{a^{3} d} + \frac {\log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{2 \, a^{3} d} \] Input:
integrate(tanh(d*x+c)/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")
Output:
4*(a*b*e^(-2*d*x - 2*c) + a*b*e^(-6*d*x - 6*c) + (2*a*b + 3*b^2)*e^(-4*d*x - 4*c))/((a^5*e^(-8*d*x - 8*c) + a^5 + 4*(a^5 + 2*a^4*b)*e^(-2*d*x - 2*c) + 2*(3*a^5 + 8*a^4*b + 8*a^3*b^2)*e^(-4*d*x - 4*c) + 4*(a^5 + 2*a^4*b)*e^ (-6*d*x - 6*c))*d) + (d*x + c)/(a^3*d) + 1/2*log(2*(a + 2*b)*e^(-2*d*x - 2 *c) + a*e^(-4*d*x - 4*c) + a)/(a^3*d)
Exception generated. \[ \int \frac {\tanh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(tanh(d*x+c)/(a+b*sech(d*x+c)^2)^3,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 = 2.40 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.29 \[ \int \frac {\tanh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {\ln \left ({\mathrm {cosh}\left (c+d\,x\right )}^2\,\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )\right )}{2\,a^3\,d}-\frac {b^2}{4\,a^3\,d\,{\mathrm {cosh}\left (c+d\,x\right )}^4\,{\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )}^2}+\frac {b}{a^3\,d\,{\mathrm {cosh}\left (c+d\,x\right )}^2\,\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )} \] Input:
int(tanh(c + d*x)/(a + b/cosh(c + d*x)^2)^3,x)
Output:
log(cosh(c + d*x)^2*(a + b/cosh(c + d*x)^2))/(2*a^3*d) - b^2/(4*a^3*d*cosh (c + d*x)^4*(a + b/cosh(c + d*x)^2)^2) + b/(a^3*d*cosh(c + d*x)^2*(a + b/c osh(c + d*x)^2))
Time = 0.23 (sec) , antiderivative size = 2266, normalized size of antiderivative = 31.04 \[ \int \frac {\tanh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:
int(tanh(d*x+c)/(a+b*sech(d*x+c)^2)^3,x)
Output:
(e**(8*c + 8*d*x)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d *x)*sqrt(a))*a**3 + 2*e**(8*c + 8*d*x)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**2*b + e**(8*c + 8*d*x)*log(sqrt(2*sqr t(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**3 + 2*e**(8*c + 8*d *x)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**2 *b + e**(8*c + 8*d*x)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a**3 + 2*e**(8*c + 8*d*x)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d* x)*a + a + 2*b)*a**2*b - 2*e**(8*c + 8*d*x)*a**3*d*x - 4*e**(8*c + 8*d*x)* a**2*b*d*x - 2*e**(8*c + 8*d*x)*a**2*b + 4*e**(6*c + 6*d*x)*log( - sqrt(2* sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**3 + 16*e**(6*c + 6*d*x)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a ))*a**2*b + 16*e**(6*c + 6*d*x)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2* b) + e**(c + d*x)*sqrt(a))*a*b**2 + 4*e**(6*c + 6*d*x)*log(sqrt(2*sqrt(b)* sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**3 + 16*e**(6*c + 6*d*x)* log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**2*b + 16*e**(6*c + 6*d*x)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d *x)*sqrt(a))*a*b**2 + 4*e**(6*c + 6*d*x)*log(2*sqrt(b)*sqrt(a + b) + e**(2 *c + 2*d*x)*a + a + 2*b)*a**3 + 16*e**(6*c + 6*d*x)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a**2*b + 16*e**(6*c + 6*d*x)*log(2*sqr t(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a*b**2 - 8*e**(6*c + 6...