Integrand size = 17, antiderivative size = 43 \[ \int \cosh ^2(a+b x) \coth ^3(a+b x) \, dx=-\frac {\text {csch}^2(a+b x)}{2 b}+\frac {2 \log (\sinh (a+b x))}{b}+\frac {\sinh ^2(a+b x)}{2 b} \]
Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \cosh ^2(a+b x) \coth ^3(a+b x) \, dx=-\frac {\text {csch}^2(a+b x)-4 \log (\sinh (a+b x))-\sinh ^2(a+b x)}{2 b} \]
Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, 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.
\(\displaystyle \int \cosh ^2(a+b x) \coth ^3(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int i \sin \left (i a+i b x+\frac {\pi }{2}\right )^2 \tan \left (i a+i b x+\frac {\pi }{2}\right )^3dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \sin \left (\frac {1}{2} (2 i a+\pi )+i b x\right )^2 \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )^3dx\) |
\(\Big \downarrow \) 3070 |
\(\displaystyle -\frac {\int -i \text {csch}^3(a+b x) \left (\sinh ^2(a+b x)+1\right )^2d(-i \sinh (a+b x))}{b}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -\frac {\int -\text {csch}^2(a+b x) (i \sinh (a+b x)+1)^2d\left (-\sinh ^2(a+b x)\right )}{2 b}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle -\frac {\int \left (-\text {csch}^2(a+b x)-2 i \text {csch}(a+b x)+1\right )d\left (-\sinh ^2(a+b x)\right )}{2 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-\sinh ^2(a+b x)-i \text {csch}(a+b x)-2 \log \left (-\sinh ^2(a+b x)\right )}{2 b}\) |
3.2.7.3.1 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 = 0.77 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {\frac {\cosh \left (b x +a \right )^{4}}{2 \sinh \left (b x +a \right )^{2}}+2 \ln \left (\sinh \left (b x +a \right )\right )-\coth \left (b x +a \right )^{2}}{b}\) | \(43\) |
default | \(\frac {\frac {\cosh \left (b x +a \right )^{4}}{2 \sinh \left (b x +a \right )^{2}}+2 \ln \left (\sinh \left (b x +a \right )\right )-\coth \left (b x +a \right )^{2}}{b}\) | \(43\) |
risch | \(-2 x +\frac {{\mathrm e}^{2 b x +2 a}}{8 b}+\frac {{\mathrm e}^{-2 b x -2 a}}{8 b}-\frac {4 a}{b}-\frac {2 \,{\mathrm e}^{2 b x +2 a}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}+\frac {2 \ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b}\) | \(83\) |
Leaf count of result is larger than twice the leaf count of optimal. 743 vs. \(2 (39) = 78\).
Time = 0.26 (sec) , antiderivative size = 743, normalized size of antiderivative = 17.28 \[ \int \cosh ^2(a+b x) \coth ^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 \, \sinh \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*sin 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 +...
\[ \int \cosh ^2(a+b x) \coth ^3(a+b x) \, dx=\int \cosh ^{2}{\left (a + b x \right )} \coth ^{3}{\left (a + b x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (39) = 78\).
Time = 0.20 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.79 \[ \int \cosh ^2(a+b x) \coth ^3(a+b x) \, dx=\frac {2 \, {\left (b x + a\right )}}{b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} + \frac {2 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {2 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{b} - \frac {2 \, e^{\left (-2 \, b x - 2 \, a\right )} + 15 \, e^{\left (-4 \, b x - 4 \, a\right )} - 1}{8 \, b {\left (e^{\left (-2 \, b x - 2 \, a\right )} - 2 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )}\right )}} \]
2*(b*x + a)/b + 1/8*e^(-2*b*x - 2*a)/b + 2*log(e^(-b*x - a) + 1)/b + 2*log (e^(-b*x - 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. 102 vs. \(2 (39) = 78\).
Time = 0.30 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.37 \[ \int \cosh ^2(a+b x) \coth ^3(a+b x) \, dx=-\frac {16 \, b x - {\left (8 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )} + 16 \, a + \frac {8 \, {\left (3 \, e^{\left (4 \, b x + 4 \, a\right )} - 4 \, e^{\left (2 \, b x + 2 \, a\right )} + 3\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{2}} - e^{\left (2 \, b x + 2 \, a\right )} - 16 \, \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{8 \, b} \]
-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(abs(e^(2*b*x + 2*a) - 1)))/b
Time = 2.10 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.26 \[ \int \cosh ^2(a+b x) \coth ^3(a+b x) \, dx=\frac {2\,\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1\right )}{b}-2\,x-\frac {2}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}-\frac {2}{b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}+\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}}{8\,b}+\frac {{\mathrm {e}}^{2\,a+2\,b\,x}}{8\,b} \]