Integrand size = 13, antiderivative size = 44 \[ \int \frac {\sinh ^4(x)}{a+a \text {sech}(x)} \, dx=-\frac {x}{8 a}-\frac {\cosh (x) \sinh (x)}{8 a}+\frac {\cosh ^3(x) \sinh (x)}{4 a}-\frac {\sinh ^3(x)}{3 a} \] Output:
-1/8*x/a-1/8*cosh(x)*sinh(x)/a+1/4*cosh(x)^3*sinh(x)/a-1/3*sinh(x)^3/a
Time = 0.17 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.64 \[ \int \frac {\sinh ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {24 \sinh (x)-8 \sinh (3 x)+3 (-4 x+\sinh (4 x))}{96 a} \] Input:
Integrate[Sinh[x]^4/(a + a*Sech[x]),x]
Output:
(24*Sinh[x] - 8*Sinh[3*x] + 3*(-4*x + Sinh[4*x]))/(96*a)
Time = 0.83 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.05, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.154, Rules used = {3042, 4360, 25, 25, 3042, 3318, 25, 3042, 25, 3044, 15, 3048, 3042, 3115, 24}
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.
\(\displaystyle \int \frac {\sinh ^4(x)}{a \text {sech}(x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos \left (-\frac {\pi }{2}+i x\right )^4}{a-a \csc \left (-\frac {\pi }{2}+i x\right )}dx\) |
\(\Big \downarrow \) 4360 |
\(\displaystyle \int -\frac {\sinh ^4(x) \cosh (x)}{a (-\cosh (x))-a}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int -\frac {\cosh (x) \sinh ^4(x)}{\cosh (x) a+a}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {\sinh ^4(x) \cosh (x)}{a \cosh (x)+a}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin \left (\frac {\pi }{2}+i x\right ) \cos \left (\frac {\pi }{2}+i x\right )^4}{a+a \sin \left (\frac {\pi }{2}+i x\right )}dx\) |
\(\Big \downarrow \) 3318 |
\(\displaystyle \frac {\int -\cosh (x) \sinh ^2(x)dx}{a}-\frac {\int -\cosh ^2(x) \sinh ^2(x)dx}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \cosh ^2(x) \sinh ^2(x)dx}{a}-\frac {\int \cosh (x) \sinh ^2(x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int -\cos (i x)^2 \sin (i x)^2dx}{a}-\frac {\int -\cos (i x) \sin (i x)^2dx}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \cos (i x) \sin (i x)^2dx}{a}-\frac {\int \cos (i x)^2 \sin (i x)^2dx}{a}\) |
\(\Big \downarrow \) 3044 |
\(\displaystyle -\frac {i \int -\sinh ^2(x)d(i \sinh (x))}{a}-\frac {\int \cos (i x)^2 \sin (i x)^2dx}{a}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {\sinh ^3(x)}{3 a}-\frac {\int \cos (i x)^2 \sin (i x)^2dx}{a}\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle -\frac {\frac {1}{4} \int \cosh ^2(x)dx-\frac {1}{4} \sinh (x) \cosh ^3(x)}{a}-\frac {\sinh ^3(x)}{3 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\sinh ^3(x)}{3 a}-\frac {-\frac {1}{4} \sinh (x) \cosh ^3(x)+\frac {1}{4} \int \sin \left (i x+\frac {\pi }{2}\right )^2dx}{a}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle -\frac {\frac {1}{4} \left (\frac {\int 1dx}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)}{a}-\frac {\sinh ^3(x)}{3 a}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -\frac {\sinh ^3(x)}{3 a}-\frac {\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)}{a}\) |
Input:
Int[Sinh[x]^4/(a + a*Sech[x]),x]
Output:
-1/3*Sinh[x]^3/a - (-1/4*(Cosh[x]^3*Sinh[x]) + (x/2 + (Cosh[x]*Sinh[x])/2) /4)/a
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[1/(a*f) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a *Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, n])
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g^2/a Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[g^2/(b*d) Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 15.18 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.36
method | result | size |
risch | \(-\frac {x}{8 a}+\frac {{\mathrm e}^{4 x}}{64 a}-\frac {{\mathrm e}^{3 x}}{24 a}+\frac {{\mathrm e}^{x}}{8 a}-\frac {{\mathrm e}^{-x}}{8 a}+\frac {{\mathrm e}^{-3 x}}{24 a}-\frac {{\mathrm e}^{-4 x}}{64 a}\) | \(60\) |
default | \(\frac {-\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {5}{6 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {7}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {16}{128 \tanh \left (\frac {x}{2}\right )+128}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8}+\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {5}{6 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {7}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {16}{128 \tanh \left (\frac {x}{2}\right )-128}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8}}{a}\) | \(105\) |
Input:
int(sinh(x)^4/(a+a*sech(x)),x,method=_RETURNVERBOSE)
Output:
-1/8*x/a+1/64/a*exp(4*x)-1/24/a*exp(3*x)+1/8/a*exp(x)-1/8/a*exp(-x)+1/24/a *exp(-3*x)-1/64/a*exp(-4*x)
Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.82 \[ \int \frac {\sinh ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {{\left (3 \, \cosh \left (x\right ) - 2\right )} \sinh \left (x\right )^{3} + 3 \, {\left (\cosh \left (x\right )^{3} - 2 \, \cosh \left (x\right )^{2} + 2\right )} \sinh \left (x\right ) - 3 \, x}{24 \, a} \] Input:
integrate(sinh(x)^4/(a+a*sech(x)),x, algorithm="fricas")
Output:
1/24*((3*cosh(x) - 2)*sinh(x)^3 + 3*(cosh(x)^3 - 2*cosh(x)^2 + 2)*sinh(x) - 3*x)/a
\[ \int \frac {\sinh ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {\int \frac {\sinh ^{4}{\left (x \right )}}{\operatorname {sech}{\left (x \right )} + 1}\, dx}{a} \] Input:
integrate(sinh(x)**4/(a+a*sech(x)),x)
Output:
Integral(sinh(x)**4/(sech(x) + 1), x)/a
Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.23 \[ \int \frac {\sinh ^4(x)}{a+a \text {sech}(x)} \, dx=-\frac {{\left (8 \, e^{\left (-x\right )} - 24 \, e^{\left (-3 \, x\right )} - 3\right )} e^{\left (4 \, x\right )}}{192 \, a} - \frac {x}{8 \, a} - \frac {24 \, e^{\left (-x\right )} - 8 \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )}}{192 \, a} \] Input:
integrate(sinh(x)^4/(a+a*sech(x)),x, algorithm="maxima")
Output:
-1/192*(8*e^(-x) - 24*e^(-3*x) - 3)*e^(4*x)/a - 1/8*x/a - 1/192*(24*e^(-x) - 8*e^(-3*x) + 3*e^(-4*x))/a
Time = 0.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.95 \[ \int \frac {\sinh ^4(x)}{a+a \text {sech}(x)} \, dx=-\frac {{\left (24 \, e^{\left (3 \, x\right )} - 8 \, e^{x} + 3\right )} e^{\left (-4 \, x\right )} + 24 \, x - 3 \, e^{\left (4 \, x\right )} + 8 \, e^{\left (3 \, x\right )} - 24 \, e^{x}}{192 \, a} \] Input:
integrate(sinh(x)^4/(a+a*sech(x)),x, algorithm="giac")
Output:
-1/192*((24*e^(3*x) - 8*e^x + 3)*e^(-4*x) + 24*x - 3*e^(4*x) + 8*e^(3*x) - 24*e^x)/a
Time = 2.34 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.34 \[ \int \frac {\sinh ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {{\mathrm {e}}^{-3\,x}}{24\,a}-\frac {{\mathrm {e}}^{-x}}{8\,a}-\frac {{\mathrm {e}}^{3\,x}}{24\,a}-\frac {{\mathrm {e}}^{-4\,x}}{64\,a}+\frac {{\mathrm {e}}^{4\,x}}{64\,a}-\frac {x}{8\,a}+\frac {{\mathrm {e}}^x}{8\,a} \] Input:
int(sinh(x)^4/(a + a/cosh(x)),x)
Output:
exp(-3*x)/(24*a) - exp(-x)/(8*a) - exp(3*x)/(24*a) - exp(-4*x)/(64*a) + ex p(4*x)/(64*a) - x/(8*a) + exp(x)/(8*a)
Time = 0.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.25 \[ \int \frac {\sinh ^4(x)}{a+a \text {sech}(x)} \, dx=\frac {3 e^{8 x}-8 e^{7 x}+24 e^{5 x}-24 e^{4 x} x -24 e^{3 x}+8 e^{x}-3}{192 e^{4 x} a} \] Input:
int(sinh(x)^4/(a+a*sech(x)),x)
Output:
(3*e**(8*x) - 8*e**(7*x) + 24*e**(5*x) - 24*e**(4*x)*x - 24*e**(3*x) + 8*e **x - 3)/(192*e**(4*x)*a)