Integrand size = 7, antiderivative size = 28 \[ \int (\text {csch}(x)+\sinh (x))^3 \, dx=-\frac {5}{2} \text {arctanh}(\cosh (x))+2 \cosh (x)+\frac {\cosh ^3(x)}{3}-\frac {1}{2} \coth (x) \text {csch}(x) \] Output:
-5/2*arctanh(cosh(x))+2*cosh(x)+1/3*cosh(x)^3-1/2*coth(x)*csch(x)
Leaf count is larger than twice the leaf count of optimal. \(67\) vs. \(2(28)=56\).
Time = 0.29 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.39 \[ \int (\text {csch}(x)+\sinh (x))^3 \, dx=\frac {1}{48} \text {csch}^2(x) \left (-50 \cosh (x)+25 \cosh (3 x)+\cosh (5 x)+60 \log \left (\cosh \left (\frac {x}{2}\right )\right )-60 \cosh (2 x) \log \left (\cosh \left (\frac {x}{2}\right )\right )-60 \log \left (\sinh \left (\frac {x}{2}\right )\right )+60 \cosh (2 x) \log \left (\sinh \left (\frac {x}{2}\right )\right )\right ) \] Input:
Integrate[(Csch[x] + Sinh[x])^3,x]
Output:
(Csch[x]^2*(-50*Cosh[x] + 25*Cosh[3*x] + Cosh[5*x] + 60*Log[Cosh[x/2]] - 6 0*Cosh[2*x]*Log[Cosh[x/2]] - 60*Log[Sinh[x/2]] + 60*Cosh[2*x]*Log[Sinh[x/2 ]]))/48
Time = 0.29 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.143, Rules used = {3042, 4897, 3042, 26, 3072, 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.
\(\displaystyle \int (\sinh (x)+\text {csch}(x))^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (i \csc (i x)-i \sin (i x))^3dx\) |
\(\Big \downarrow \) 4897 |
\(\displaystyle \int \cosh ^3(x) \coth ^3(x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int i \sin \left (\frac {\pi }{2}+i x\right )^3 \tan \left (\frac {\pi }{2}+i x\right )^3dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \sin \left (i x+\frac {\pi }{2}\right )^3 \tan \left (i x+\frac {\pi }{2}\right )^3dx\) |
\(\Big \downarrow \) 3072 |
\(\displaystyle \int \frac {\cosh ^6(x)}{\left (1-\cosh ^2(x)\right )^2}d\cosh (x)\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {\cosh ^5(x)}{2 \left (1-\cosh ^2(x)\right )}-\frac {5}{2} \int \frac {\cosh ^4(x)}{1-\cosh ^2(x)}d\cosh (x)\) |
\(\Big \downarrow \) 254 |
\(\displaystyle \frac {\cosh ^5(x)}{2 \left (1-\cosh ^2(x)\right )}-\frac {5}{2} \int \left (-\cosh ^2(x)+\frac {1}{1-\cosh ^2(x)}-1\right )d\cosh (x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\cosh ^5(x)}{2 \left (1-\cosh ^2(x)\right )}-\frac {5}{2} \left (\text {arctanh}(\cosh (x))-\frac {1}{3} \cosh ^3(x)-\cosh (x)\right )\) |
Input:
Int[(Csch[x] + Sinh[x])^3,x]
Output:
Cosh[x]^5/(2*(1 - Cosh[x]^2)) - (5*(ArcTanh[Cosh[x]] - Cosh[x] - Cosh[x]^3 /3))/2
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[((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 = 0.38 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {\coth \left (x \right ) \operatorname {csch}\left (x \right )}{2}-5 \,\operatorname {arctanh}\left ({\mathrm e}^{x}\right )+3 \cosh \left (x \right )+\left (-\frac {2}{3}+\frac {\sinh \left (x \right )^{2}}{3}\right ) \cosh \left (x \right )\) | \(28\) |
parts | \(-\frac {\coth \left (x \right ) \operatorname {csch}\left (x \right )}{2}-5 \,\operatorname {arctanh}\left ({\mathrm e}^{x}\right )+3 \cosh \left (x \right )+\left (-\frac {2}{3}+\frac {\sinh \left (x \right )^{2}}{3}\right ) \cosh \left (x \right )\) | \(28\) |
parallelrisch | \(\frac {\operatorname {sech}\left (\frac {x}{2}\right )^{2} \operatorname {csch}\left (\frac {x}{2}\right )^{2} \left (-10+\cosh \left (5 x \right )+25 \cosh \left (3 x \right )+10 \cosh \left (2 x \right )-50 \cosh \left (x \right )+60 \ln \left (\tanh \left (\frac {x}{2}\right )\right ) \cosh \left (2 x \right )-60 \ln \left (\tanh \left (\frac {x}{2}\right )\right )\right )}{192}\) | \(55\) |
risch | \(\frac {{\mathrm e}^{3 x}}{24}+\frac {9 \,{\mathrm e}^{x}}{8}+\frac {9 \,{\mathrm e}^{-x}}{8}+\frac {{\mathrm e}^{-3 x}}{24}-\frac {{\mathrm e}^{x} \left (1+{\mathrm e}^{2 x}\right )}{\left ({\mathrm e}^{2 x}-1\right )^{2}}+\frac {5 \ln \left ({\mathrm e}^{x}-1\right )}{2}-\frac {5 \ln \left (1+{\mathrm e}^{x}\right )}{2}\) | \(56\) |
Input:
int((csch(x)+sinh(x))^3,x,method=_RETURNVERBOSE)
Output:
-1/2*coth(x)*csch(x)-5*arctanh(exp(x))+3*cosh(x)+(-2/3+1/3*sinh(x)^2)*cosh (x)
Leaf count of result is larger than twice the leaf count of optimal. 616 vs. \(2 (22) = 44\).
Time = 0.09 (sec) , antiderivative size = 616, normalized size of antiderivative = 22.00 \[ \int (\text {csch}(x)+\sinh (x))^3 \, dx=\text {Too large to display} \] Input:
integrate((csch(x)+sinh(x))^3,x, algorithm="fricas")
Output:
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*(2 1*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*cosh (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 + 25*cosh(x)^2 - 60*(cosh (x)^7 + 7*cosh(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*co sh(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))*log(c osh(x) + sinh(x) + 1) + 60*(cosh(x)^7 + 7*cosh(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))*s inh(x)^4 + (35*cosh(x)^4 - 20*cosh(x)^2 + 1)*sinh(x)^3 + cosh(x)^3 + (21*c osh(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))*log(cosh(x) + sinh(x) - 1) + 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 + sinh(x)^7 + (21*cosh(x)^2 - 2)*sinh(x)^5 - 2*c osh(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*cos...
\[ \int (\text {csch}(x)+\sinh (x))^3 \, dx=\int \left (\sinh {\left (x \right )} + \operatorname {csch}{\left (x \right )}\right )^{3}\, dx \] Input:
integrate((csch(x)+sinh(x))**3,x)
Output:
Integral((sinh(x) + csch(x))**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (22) = 44\).
Time = 0.05 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.39 \[ \int (\text {csch}(x)+\sinh (x))^3 \, dx=\frac {e^{\left (-x\right )} + e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + \frac {1}{24} \, e^{\left (3 \, x\right )} + \frac {9}{8} \, e^{\left (-x\right )} + \frac {1}{24} \, e^{\left (-3 \, x\right )} + \frac {9}{8} \, e^{x} - \frac {5}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {5}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \] Input:
integrate((csch(x)+sinh(x))^3,x, algorithm="maxima")
Output:
(e^(-x) + e^(-3*x))/(2*e^(-2*x) - e^(-4*x) - 1) + 1/24*e^(3*x) + 9/8*e^(-x ) + 1/24*e^(-3*x) + 9/8*e^x - 5/2*log(e^(-x) + 1) + 5/2*log(e^(-x) - 1)
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (22) = 44\).
Time = 0.12 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.21 \[ \int (\text {csch}(x)+\sinh (x))^3 \, dx=\frac {1}{24} \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - \frac {e^{\left (-x\right )} + e^{x}}{{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4} + e^{\left (-x\right )} + e^{x} - \frac {5}{4} \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + \frac {5}{4} \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right ) \] Input:
integrate((csch(x)+sinh(x))^3,x, algorithm="giac")
Output:
1/24*(e^(-x) + e^x)^3 - (e^(-x) + e^x)/((e^(-x) + e^x)^2 - 4) + e^(-x) + e ^x - 5/4*log(e^(-x) + e^x + 2) + 5/4*log(e^(-x) + e^x - 2)
Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.54 \[ \int (\text {csch}(x)+\sinh (x))^3 \, dx=\frac {5\,\ln \left (5-5\,{\mathrm {e}}^x\right )}{2}-\frac {5\,\ln \left (-5\,{\mathrm {e}}^x-5\right )}{2}+\frac {9\,{\mathrm {e}}^{-x}}{8}+\frac {{\mathrm {e}}^{-3\,x}}{24}+\frac {{\mathrm {e}}^{3\,x}}{24}+\frac {9\,{\mathrm {e}}^x}{8}-\frac {{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}-1}-\frac {2\,{\mathrm {e}}^x}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1} \] Input:
int((sinh(x) + 1/sinh(x))^3,x)
Output:
(5*log(5 - 5*exp(x)))/2 - (5*log(- 5*exp(x) - 5))/2 + (9*exp(-x))/8 + exp( -3*x)/24 + exp(3*x)/24 + (9*exp(x))/8 - exp(x)/(exp(2*x) - 1) - (2*exp(x)) /(exp(4*x) - 2*exp(2*x) + 1)
Time = 0.22 (sec) , antiderivative size = 138, normalized size of antiderivative = 4.93 \[ \int (\text {csch}(x)+\sinh (x))^3 \, dx=\frac {e^{10 x}+25 e^{8 x}+60 e^{7 x} \mathrm {log}\left (e^{x}-1\right )-60 e^{7 x} \mathrm {log}\left (e^{x}+1\right )-50 e^{6 x}-120 e^{5 x} \mathrm {log}\left (e^{x}-1\right )+120 e^{5 x} \mathrm {log}\left (e^{x}+1\right )-50 e^{4 x}+60 e^{3 x} \mathrm {log}\left (e^{x}-1\right )-60 e^{3 x} \mathrm {log}\left (e^{x}+1\right )+25 e^{2 x}+1}{24 e^{3 x} \left (e^{4 x}-2 e^{2 x}+1\right )} \] Input:
int((csch(x)+sinh(x))^3,x)
Output:
(e**(10*x) + 25*e**(8*x) + 60*e**(7*x)*log(e**x - 1) - 60*e**(7*x)*log(e** x + 1) - 50*e**(6*x) - 120*e**(5*x)*log(e**x - 1) + 120*e**(5*x)*log(e**x + 1) - 50*e**(4*x) + 60*e**(3*x)*log(e**x - 1) - 60*e**(3*x)*log(e**x + 1) + 25*e**(2*x) + 1)/(24*e**(3*x)*(e**(4*x) - 2*e**(2*x) + 1))