Integrand size = 9, antiderivative size = 34 \[ \int (-\cosh (x)+\text {sech}(x))^3 \, dx=-\frac {5}{2} \arctan (\sinh (x))+\frac {5 \sinh (x)}{2}-\frac {5 \sinh ^3(x)}{6}+\frac {1}{2} \sinh ^3(x) \tanh ^2(x) \]
Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.09 \[ \int (-\cosh (x)+\text {sech}(x))^3 \, dx=-\frac {1}{48} \text {sech}^2(x) (60 \arctan (\sinh (x))+60 \arctan (\sinh (x)) \cosh (2 x)-50 \sinh (x)-25 \sinh (3 x)+\sinh (5 x)) \]
-1/48*(Sech[x]^2*(60*ArcTan[Sinh[x]] + 60*ArcTan[Sinh[x]]*Cosh[2*x] - 50*S inh[x] - 25*Sinh[3*x] + Sinh[5*x]))
Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.111, Rules used = {3042, 4897, 25, 3042, 25, 3072, 25, 252, 254, 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.
\(\displaystyle \int (\text {sech}(x)-\cosh (x))^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (\sec (i x)-\cos (i x))^3dx\) |
\(\Big \downarrow \) 4897 |
\(\displaystyle \int -\sinh ^3(x) \tanh ^3(x)dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \sinh ^3(x) \tanh ^3(x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int -\sin (i x)^3 \tan (i x)^3dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \sin (i x)^3 \tan (i x)^3dx\) |
\(\Big \downarrow \) 3072 |
\(\displaystyle \int -\frac {\sinh ^6(x)}{\left (\sinh ^2(x)+1\right )^2}d\sinh (x)\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\sinh ^6(x)}{\left (\sinh ^2(x)+1\right )^2}d\sinh (x)\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {\sinh ^5(x)}{2 \left (\sinh ^2(x)+1\right )}-\frac {5}{2} \int \frac {\sinh ^4(x)}{\sinh ^2(x)+1}d\sinh (x)\) |
\(\Big \downarrow \) 254 |
\(\displaystyle \frac {\sinh ^5(x)}{2 \left (\sinh ^2(x)+1\right )}-\frac {5}{2} \int \left (\sinh ^2(x)+\frac {1}{\sinh ^2(x)+1}-1\right )d\sinh (x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sinh ^5(x)}{2 \left (\sinh ^2(x)+1\right )}-\frac {5}{2} \left (\arctan (\sinh (x))+\frac {\sinh ^3(x)}{3}-\sinh (x)\right )\) |
3.7.82.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[ (ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, a*(Sin[e + f*x]/ff)], x ]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]
Time = 3.72 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85
method | result | size |
default | \(-\left (\frac {2}{3}+\frac {\cosh \left (x \right )^{2}}{3}\right ) \sinh \left (x \right )+3 \sinh \left (x \right )-5 \arctan \left ({\mathrm e}^{x}\right )+\frac {\operatorname {sech}\left (x \right ) \tanh \left (x \right )}{2}\) | \(29\) |
parts | \(-\left (\frac {2}{3}+\frac {\cosh \left (x \right )^{2}}{3}\right ) \sinh \left (x \right )+3 \sinh \left (x \right )-5 \arctan \left ({\mathrm e}^{x}\right )+\frac {\operatorname {sech}\left (x \right ) \tanh \left (x \right )}{2}\) | \(29\) |
parallelrisch | \(\frac {\operatorname {sech}\left (x \right ) \tanh \left (x \right )}{2}+\frac {5 i \ln \left (-i+\coth \left (x \right )-\operatorname {csch}\left (x \right )\right )}{2}-\frac {5 i \ln \left (i+\coth \left (x \right )-\operatorname {csch}\left (x \right )\right )}{2}+\frac {9 \sinh \left (x \right )}{4}-\frac {\sinh \left (3 x \right )}{12}\) | \(44\) |
risch | \(-\frac {{\mathrm e}^{3 x}}{24}+\frac {9 \,{\mathrm e}^{x}}{8}-\frac {9 \,{\mathrm e}^{-x}}{8}+\frac {{\mathrm e}^{-3 x}}{24}+\frac {\left ({\mathrm e}^{2 x}-1\right ) {\mathrm e}^{x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {5 i \ln \left ({\mathrm e}^{x}-i\right )}{2}-\frac {5 i \ln \left ({\mathrm e}^{x}+i\right )}{2}\) | \(59\) |
Leaf count of result is larger than twice the leaf count of optimal. 486 vs. \(2 (26) = 52\).
Time = 0.26 (sec) , antiderivative size = 486, normalized size of antiderivative = 14.29 \[ \int (-\cosh (x)+\text {sech}(x))^3 \, dx=-\frac {\cosh \left (x\right )^{10} + 10 \, \cosh \left (x\right ) \sinh \left (x\right )^{9} + \sinh \left (x\right )^{10} + 5 \, {\left (9 \, \cosh \left (x\right )^{2} - 5\right )} \sinh \left (x\right )^{8} - 25 \, \cosh \left (x\right )^{8} + 40 \, {\left (3 \, \cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{7} + 10 \, {\left (21 \, \cosh \left (x\right )^{4} - 70 \, \cosh \left (x\right )^{2} - 5\right )} \sinh \left (x\right )^{6} - 50 \, \cosh \left (x\right )^{6} + 4 \, {\left (63 \, \cosh \left (x\right )^{5} - 350 \, \cosh \left (x\right )^{3} - 75 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 10 \, {\left (21 \, \cosh \left (x\right )^{6} - 175 \, \cosh \left (x\right )^{4} - 75 \, \cosh \left (x\right )^{2} + 5\right )} \sinh \left (x\right )^{4} + 50 \, \cosh \left (x\right )^{4} + 40 \, {\left (3 \, \cosh \left (x\right )^{7} - 35 \, \cosh \left (x\right )^{5} - 25 \, \cosh \left (x\right )^{3} + 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 5 \, {\left (9 \, \cosh \left (x\right )^{8} - 140 \, \cosh \left (x\right )^{6} - 150 \, \cosh \left (x\right )^{4} + 60 \, \cosh \left (x\right )^{2} + 5\right )} \sinh \left (x\right )^{2} + 120 \, {\left (\cosh \left (x\right )^{7} + 7 \, \cosh \left (x\right ) \sinh \left (x\right )^{6} + \sinh \left (x\right )^{7} + {\left (21 \, \cosh \left (x\right )^{2} + 2\right )} \sinh \left (x\right )^{5} + 2 \, \cosh \left (x\right )^{5} + 5 \, {\left (7 \, \cosh \left (x\right )^{3} + 2 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{4} + {\left (35 \, \cosh \left (x\right )^{4} + 20 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{3} + \cosh \left (x\right )^{3} + {\left (21 \, \cosh \left (x\right )^{5} + 20 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + {\left (7 \, \cosh \left (x\right )^{6} + 10 \, \cosh \left (x\right )^{4} + 3 \, \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + 25 \, \cosh \left (x\right )^{2} + 10 \, {\left (\cosh \left (x\right )^{9} - 20 \, \cosh \left (x\right )^{7} - 30 \, \cosh \left (x\right )^{5} + 20 \, \cosh \left (x\right )^{3} + 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1}{24 \, {\left (\cosh \left (x\right )^{7} + 7 \, \cosh \left (x\right ) \sinh \left (x\right )^{6} + \sinh \left (x\right )^{7} + {\left (21 \, \cosh \left (x\right )^{2} + 2\right )} \sinh \left (x\right )^{5} + 2 \, \cosh \left (x\right )^{5} + 5 \, {\left (7 \, \cosh \left (x\right )^{3} + 2 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{4} + {\left (35 \, \cosh \left (x\right )^{4} + 20 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{3} + \cosh \left (x\right )^{3} + {\left (21 \, \cosh \left (x\right )^{5} + 20 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + {\left (7 \, \cosh \left (x\right )^{6} + 10 \, \cosh \left (x\right )^{4} + 3 \, \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )\right )}} \]
-1/24*(cosh(x)^10 + 10*cosh(x)*sinh(x)^9 + sinh(x)^10 + 5*(9*cosh(x)^2 - 5 )*sinh(x)^8 - 25*cosh(x)^8 + 40*(3*cosh(x)^3 - 5*cosh(x))*sinh(x)^7 + 10*( 21*cosh(x)^4 - 70*cosh(x)^2 - 5)*sinh(x)^6 - 50*cosh(x)^6 + 4*(63*cosh(x)^ 5 - 350*cosh(x)^3 - 75*cosh(x))*sinh(x)^5 + 10*(21*cosh(x)^6 - 175*cosh(x) ^4 - 75*cosh(x)^2 + 5)*sinh(x)^4 + 50*cosh(x)^4 + 40*(3*cosh(x)^7 - 35*cos h(x)^5 - 25*cosh(x)^3 + 5*cosh(x))*sinh(x)^3 + 5*(9*cosh(x)^8 - 140*cosh(x )^6 - 150*cosh(x)^4 + 60*cosh(x)^2 + 5)*sinh(x)^2 + 120*(cosh(x)^7 + 7*cos h(x)*sinh(x)^6 + sinh(x)^7 + (21*cosh(x)^2 + 2)*sinh(x)^5 + 2*cosh(x)^5 + 5*(7*cosh(x)^3 + 2*cosh(x))*sinh(x)^4 + (35*cosh(x)^4 + 20*cosh(x)^2 + 1)* sinh(x)^3 + cosh(x)^3 + (21*cosh(x)^5 + 20*cosh(x)^3 + 3*cosh(x))*sinh(x)^ 2 + (7*cosh(x)^6 + 10*cosh(x)^4 + 3*cosh(x)^2)*sinh(x))*arctan(cosh(x) + s inh(x)) + 25*cosh(x)^2 + 10*(cosh(x)^9 - 20*cosh(x)^7 - 30*cosh(x)^5 + 20* cosh(x)^3 + 5*cosh(x))*sinh(x) - 1)/(cosh(x)^7 + 7*cosh(x)*sinh(x)^6 + sin h(x)^7 + (21*cosh(x)^2 + 2)*sinh(x)^5 + 2*cosh(x)^5 + 5*(7*cosh(x)^3 + 2*c osh(x))*sinh(x)^4 + (35*cosh(x)^4 + 20*cosh(x)^2 + 1)*sinh(x)^3 + cosh(x)^ 3 + (21*cosh(x)^5 + 20*cosh(x)^3 + 3*cosh(x))*sinh(x)^2 + (7*cosh(x)^6 + 1 0*cosh(x)^4 + 3*cosh(x)^2)*sinh(x))
\[ \int (-\cosh (x)+\text {sech}(x))^3 \, dx=- \int 3 \cosh {\left (x \right )} \operatorname {sech}^{2}{\left (x \right )}\, dx - \int \left (- 3 \cosh ^{2}{\left (x \right )} \operatorname {sech}{\left (x \right )}\right )\, dx - \int \cosh ^{3}{\left (x \right )}\, dx - \int \left (- \operatorname {sech}^{3}{\left (x \right )}\right )\, dx \]
-Integral(3*cosh(x)*sech(x)**2, x) - Integral(-3*cosh(x)**2*sech(x), x) - Integral(cosh(x)**3, x) - Integral(-sech(x)**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (26) = 52\).
Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.65 \[ \int (-\cosh (x)+\text {sech}(x))^3 \, dx=\frac {e^{\left (-x\right )} - e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} + 5 \, \arctan \left (e^{\left (-x\right )}\right ) - \frac {1}{24} \, e^{\left (3 \, x\right )} - \frac {9}{8} \, e^{\left (-x\right )} + \frac {1}{24} \, e^{\left (-3 \, x\right )} + \frac {9}{8} \, e^{x} \]
(e^(-x) - e^(-3*x))/(2*e^(-2*x) + e^(-4*x) + 1) + 5*arctan(e^(-x)) - 1/24* e^(3*x) - 9/8*e^(-x) + 1/24*e^(-3*x) + 9/8*e^x
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (26) = 52\).
Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.94 \[ \int (-\cosh (x)+\text {sech}(x))^3 \, dx=-\frac {5}{4} \, \pi + \frac {1}{24} \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} - \frac {e^{\left (-x\right )} - e^{x}}{{\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4} - \frac {5}{2} \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) - e^{\left (-x\right )} + e^{x} \]
-5/4*pi + 1/24*(e^(-x) - e^x)^3 - (e^(-x) - e^x)/((e^(-x) - e^x)^2 + 4) - 5/2*arctan(1/2*(e^(2*x) - 1)*e^(-x)) - e^(-x) + e^x
Time = 0.06 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.68 \[ \int (-\cosh (x)+\text {sech}(x))^3 \, dx=\frac {{\mathrm {e}}^{-3\,x}}{24}-\frac {9\,{\mathrm {e}}^{-x}}{8}-\frac {{\mathrm {e}}^{3\,x}}{24}-5\,\mathrm {atan}\left ({\mathrm {e}}^x\right )+\frac {9\,{\mathrm {e}}^x}{8}+\frac {{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}+1}-\frac {2\,{\mathrm {e}}^x}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1} \]