Integrand size = 17, antiderivative size = 43 \[ \int \sinh ^2(a+b x) \tanh ^3(a+b x) \, dx=\frac {\cosh ^2(a+b x)}{2 b}-\frac {2 \log (\cosh (a+b x))}{b}-\frac {\text {sech}^2(a+b x)}{2 b} \]
Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \sinh ^2(a+b x) \tanh ^3(a+b x) \, dx=-\frac {4 \log (\cosh (a+b x))+\text {sech}^2(a+b x)-\sinh ^2(a+b x)}{2 b} \]
Time = 0.23 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3042, 26, 3070, 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.
\(\displaystyle \int \sinh ^2(a+b x) \tanh ^3(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -i \sin (i a+i b x)^2 \tan (i a+i b x)^3dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \sin (i a+i b x)^2 \tan (i a+i b x)^3dx\) |
\(\Big \downarrow \) 3070 |
\(\displaystyle \frac {\int \left (1-\cosh ^2(a+b x)\right )^2 \text {sech}^3(a+b x)d\cosh (a+b x)}{b}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\int \left (1-\cosh ^2(a+b x)\right )^2 \text {sech}^2(a+b x)d\cosh ^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\int \left (\text {sech}^2(a+b x)-2 \text {sech}(a+b x)+1\right )d\cosh ^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\cosh ^2(a+b x)-\text {sech}(a+b x)-2 \log \left (\cosh ^2(a+b x)\right )}{2 b}\) |
3.1.76.3.1 Defintions of rubi rules used
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[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Simp[-f^(-1) Subst[Int[(1 - x^2)^((m + n - 1)/2)/x^n, x], x, Cos[e + f *x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]
Time = 0.43 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {\frac {\sinh \left (b x +a \right )^{4}}{2 \cosh \left (b x +a \right )^{2}}-2 \ln \left (\cosh \left (b x +a \right )\right )+\tanh \left (b x +a \right )^{2}}{b}\) | \(41\) |
default | \(\frac {\frac {\sinh \left (b x +a \right )^{4}}{2 \cosh \left (b x +a \right )^{2}}-2 \ln \left (\cosh \left (b x +a \right )\right )+\tanh \left (b x +a \right )^{2}}{b}\) | \(41\) |
risch | \(2 x +\frac {{\mathrm e}^{2 b x +2 a}}{8 b}+\frac {{\mathrm e}^{-2 b x -2 a}}{8 b}+\frac {4 a}{b}-\frac {2 \,{\mathrm e}^{2 b x +2 a}}{b \left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}-\frac {2 \ln \left (1+{\mathrm e}^{2 b x +2 a}\right )}{b}\) | \(83\) |
Leaf count of result is larger than twice the leaf count of optimal. 742 vs. \(2 (39) = 78\).
Time = 0.26 (sec) , antiderivative size = 742, normalized size of antiderivative = 17.26 \[ \int \sinh ^2(a+b x) \tanh ^3(a+b x) \, dx=\frac {\cosh \left (b x + a\right )^{8} + 8 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + \sinh \left (b x + a\right )^{8} + 2 \, {\left (8 \, b x + 1\right )} \cosh \left (b x + a\right )^{6} + 2 \, {\left (8 \, b x + 14 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{6} + 4 \, {\left (14 \, \cosh \left (b x + a\right )^{3} + 3 \, {\left (8 \, b x + 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 2 \, {\left (16 \, b x - 7\right )} \cosh \left (b x + a\right )^{4} + 2 \, {\left (35 \, \cosh \left (b x + a\right )^{4} + 15 \, {\left (8 \, b x + 1\right )} \cosh \left (b x + a\right )^{2} + 16 \, b x - 7\right )} \sinh \left (b x + a\right )^{4} + 8 \, {\left (7 \, \cosh \left (b x + a\right )^{5} + 5 \, {\left (8 \, b x + 1\right )} \cosh \left (b x + a\right )^{3} + {\left (16 \, b x - 7\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 2 \, {\left (8 \, b x + 1\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (14 \, \cosh \left (b x + a\right )^{6} + 15 \, {\left (8 \, b x + 1\right )} \cosh \left (b x + a\right )^{4} + 6 \, {\left (16 \, b x - 7\right )} \cosh \left (b x + a\right )^{2} + 8 \, b x + 1\right )} \sinh \left (b x + a\right )^{2} - 16 \, {\left (\cosh \left (b x + a\right )^{6} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + \sinh \left (b x + a\right )^{6} + {\left (15 \, \cosh \left (b x + a\right )^{2} + 2\right )} \sinh \left (b x + a\right )^{4} + 2 \, \cosh \left (b x + a\right )^{4} + 4 \, {\left (5 \, \cosh \left (b x + a\right )^{3} + 2 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + {\left (15 \, \cosh \left (b x + a\right )^{4} + 12 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{5} + 4 \, \cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} \log \left (\frac {2 \, \cosh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) + 4 \, {\left (2 \, \cosh \left (b x + a\right )^{7} + 3 \, {\left (8 \, b x + 1\right )} \cosh \left (b x + a\right )^{5} + 2 \, {\left (16 \, b x - 7\right )} \cosh \left (b x + a\right )^{3} + {\left (8 \, b x + 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1}{8 \, {\left (b \cosh \left (b x + a\right )^{6} + 6 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + b \sinh \left (b x + a\right )^{6} + 2 \, b \cosh \left (b x + a\right )^{4} + {\left (15 \, b \cosh \left (b x + a\right )^{2} + 2 \, b\right )} \sinh \left (b x + a\right )^{4} + 4 \, {\left (5 \, b \cosh \left (b x + a\right )^{3} + 2 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )^{2} + {\left (15 \, b \cosh \left (b x + a\right )^{4} + 12 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{5} + 4 \, b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )}} \]
1/8*(cosh(b*x + a)^8 + 8*cosh(b*x + a)*sinh(b*x + a)^7 + sinh(b*x + a)^8 + 2*(8*b*x + 1)*cosh(b*x + a)^6 + 2*(8*b*x + 14*cosh(b*x + a)^2 + 1)*sinh(b *x + a)^6 + 4*(14*cosh(b*x + a)^3 + 3*(8*b*x + 1)*cosh(b*x + a))*sinh(b*x + a)^5 + 2*(16*b*x - 7)*cosh(b*x + a)^4 + 2*(35*cosh(b*x + a)^4 + 15*(8*b* x + 1)*cosh(b*x + a)^2 + 16*b*x - 7)*sinh(b*x + a)^4 + 8*(7*cosh(b*x + a)^ 5 + 5*(8*b*x + 1)*cosh(b*x + a)^3 + (16*b*x - 7)*cosh(b*x + a))*sinh(b*x + a)^3 + 2*(8*b*x + 1)*cosh(b*x + a)^2 + 2*(14*cosh(b*x + a)^6 + 15*(8*b*x + 1)*cosh(b*x + a)^4 + 6*(16*b*x - 7)*cosh(b*x + a)^2 + 8*b*x + 1)*sinh(b* x + a)^2 - 16*(cosh(b*x + a)^6 + 6*cosh(b*x + a)*sinh(b*x + a)^5 + sinh(b* x + a)^6 + (15*cosh(b*x + a)^2 + 2)*sinh(b*x + a)^4 + 2*cosh(b*x + a)^4 + 4*(5*cosh(b*x + a)^3 + 2*cosh(b*x + a))*sinh(b*x + a)^3 + (15*cosh(b*x + a )^4 + 12*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^2 + cosh(b*x + a)^2 + 2*(3*cos h(b*x + a)^5 + 4*cosh(b*x + a)^3 + cosh(b*x + a))*sinh(b*x + a))*log(2*cos h(b*x + a)/(cosh(b*x + a) - sinh(b*x + a))) + 4*(2*cosh(b*x + a)^7 + 3*(8* b*x + 1)*cosh(b*x + a)^5 + 2*(16*b*x - 7)*cosh(b*x + a)^3 + (8*b*x + 1)*co sh(b*x + a))*sinh(b*x + a) + 1)/(b*cosh(b*x + a)^6 + 6*b*cosh(b*x + a)*sin h(b*x + a)^5 + b*sinh(b*x + a)^6 + 2*b*cosh(b*x + a)^4 + (15*b*cosh(b*x + a)^2 + 2*b)*sinh(b*x + a)^4 + 4*(5*b*cosh(b*x + a)^3 + 2*b*cosh(b*x + a))* sinh(b*x + a)^3 + b*cosh(b*x + a)^2 + (15*b*cosh(b*x + a)^4 + 12*b*cosh(b* x + a)^2 + b)*sinh(b*x + a)^2 + 2*(3*b*cosh(b*x + a)^5 + 4*b*cosh(b*x +...
\[ \int \sinh ^2(a+b x) \tanh ^3(a+b x) \, dx=\int \sinh ^{2}{\left (a + b x \right )} \tanh ^{3}{\left (a + b x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (39) = 78\).
Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.40 \[ \int \sinh ^2(a+b x) \tanh ^3(a+b x) \, dx=-\frac {2 \, {\left (b x + a\right )}}{b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} - \frac {2 \, \log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}{b} + \frac {2 \, e^{\left (-2 \, b x - 2 \, a\right )} - 15 \, e^{\left (-4 \, b x - 4 \, a\right )} + 1}{8 \, b {\left (e^{\left (-2 \, b x - 2 \, a\right )} + 2 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )}\right )}} \]
-2*(b*x + a)/b + 1/8*e^(-2*b*x - 2*a)/b - 2*log(e^(-2*b*x - 2*a) + 1)/b + 1/8*(2*e^(-2*b*x - 2*a) - 15*e^(-4*b*x - 4*a) + 1)/(b*(e^(-2*b*x - 2*a) + 2*e^(-4*b*x - 4*a) + e^(-6*b*x - 6*a)))
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (39) = 78\).
Time = 0.32 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.30 \[ \int \sinh ^2(a+b x) \tanh ^3(a+b x) \, dx=\frac {16 \, b x - {\left (8 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-2 \, b x - 2 \, a\right )} + 16 \, a + \frac {8 \, {\left (3 \, e^{\left (4 \, b x + 4 \, a\right )} + 4 \, e^{\left (2 \, b x + 2 \, a\right )} + 3\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{2}} + e^{\left (2 \, b x + 2 \, a\right )} - 16 \, \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}{8 \, b} \]
1/8*(16*b*x - (8*e^(2*b*x + 2*a) - 1)*e^(-2*b*x - 2*a) + 16*a + 8*(3*e^(4* b*x + 4*a) + 4*e^(2*b*x + 2*a) + 3)/(e^(2*b*x + 2*a) + 1)^2 + e^(2*b*x + 2 *a) - 16*log(e^(2*b*x + 2*a) + 1))/b
Time = 2.24 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.26 \[ \int \sinh ^2(a+b x) \tanh ^3(a+b x) \, dx=2\,x-\frac {2\,\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1\right )}{b}-\frac {2}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}+\frac {2}{b\,\left (2\,{\mathrm {e}}^{2\,a+2\,b\,x}+{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )}+\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}}{8\,b}+\frac {{\mathrm {e}}^{2\,a+2\,b\,x}}{8\,b} \]