Integrand size = 11, antiderivative size = 95 \[ \int \frac {1}{(a \text {sech}(x)+b \tanh (x))^5} \, dx=\frac {\log (a+b \sinh (x))}{b^5}-\frac {\left (a^2+b^2\right )^2}{4 b^5 (a+b \sinh (x))^4}+\frac {4 a \left (a^2+b^2\right )}{3 b^5 (a+b \sinh (x))^3}-\frac {3 a^2+b^2}{b^5 (a+b \sinh (x))^2}+\frac {4 a}{b^5 (a+b \sinh (x))} \]
ln(a+b*sinh(x))/b^5-1/4*(a^2+b^2)^2/b^5/(a+b*sinh(x))^4+4/3*a*(a^2+b^2)/b^ 5/(a+b*sinh(x))^3+(-3*a^2-b^2)/b^5/(a+b*sinh(x))^2+4*a/b^5/(a+b*sinh(x))
Time = 0.17 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(a \text {sech}(x)+b \tanh (x))^5} \, dx=\frac {\log (a+b \sinh (x))-\frac {\left (a^2+b^2\right )^2}{4 (a+b \sinh (x))^4}+\frac {4 a \left (a^2+b^2\right )}{3 (a+b \sinh (x))^3}-\frac {3 a^2+b^2}{(a+b \sinh (x))^2}+\frac {4 a}{a+b \sinh (x)}}{b^5} \]
(Log[a + b*Sinh[x]] - (a^2 + b^2)^2/(4*(a + b*Sinh[x])^4) + (4*a*(a^2 + b^ 2))/(3*(a + b*Sinh[x])^3) - (3*a^2 + b^2)/(a + b*Sinh[x])^2 + (4*a)/(a + b *Sinh[x]))/b^5
Time = 0.35 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.87, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {3042, 4891, 3042, 3147, 476, 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.
\(\displaystyle \int \frac {1}{(a \text {sech}(x)+b \tanh (x))^5} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(a \sec (i x)-i b \tan (i x))^5}dx\) |
\(\Big \downarrow \) 4891 |
\(\displaystyle \int \frac {\cosh ^5(x)}{(a+b \sinh (x))^5}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (i x)^5}{(a-i b \sin (i x))^5}dx\) |
\(\Big \downarrow \) 3147 |
\(\displaystyle \frac {\int \frac {\left (\sinh ^2(x) b^2+b^2\right )^2}{(a+b \sinh (x))^5}d(b \sinh (x))}{b^5}\) |
\(\Big \downarrow \) 476 |
\(\displaystyle \frac {\int \left (\frac {\left (a^2+b^2\right )^2}{(a+b \sinh (x))^5}-\frac {4 a \left (a^2+b^2\right )}{(a+b \sinh (x))^4}+\frac {1}{a+b \sinh (x)}-\frac {4 a}{(a+b \sinh (x))^2}+\frac {2 \left (3 a^2+b^2\right )}{(a+b \sinh (x))^3}\right )d(b \sinh (x))}{b^5}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {\left (a^2+b^2\right )^2}{4 (a+b \sinh (x))^4}+\frac {4 a \left (a^2+b^2\right )}{3 (a+b \sinh (x))^3}-\frac {3 a^2+b^2}{(a+b \sinh (x))^2}+\frac {4 a}{a+b \sinh (x)}+\log (a+b \sinh (x))}{b^5}\) |
(Log[a + b*Sinh[x]] - (a^2 + b^2)^2/(4*(a + b*Sinh[x])^4) + (4*a*(a^2 + b^ 2))/(3*(a + b*Sinh[x])^3) - (3*a^2 + b^2)/(a + b*Sinh[x])^2 + (4*a)/(a + b *Sinh[x]))/b^5
3.7.23.3.1 Defintions of rubi rules used
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Int[(u_.)*((b_.)*sec[(c_.) + (d_.)*(x_)]^(n_.) + (a_.)*tan[(c_.) + (d_.)*(x _)]^(n_.))^(p_), x_Symbol] :> Int[ActivateTrig[u]*Sec[c + d*x]^(n*p)*(b + a *Sin[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]
Time = 289.96 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.85
method | result | size |
risch | \(-\frac {x}{b^{5}}+\frac {4 \left (6 a \,b^{3} {\mathrm e}^{6 x}+27 a^{2} b^{2} {\mathrm e}^{5 x}-3 \,{\mathrm e}^{5 x} b^{4}+44 a^{3} b \,{\mathrm e}^{4 x}-22 a \,b^{3} {\mathrm e}^{4 x}+25 a^{4} {\mathrm e}^{3 x}-56 a^{2} b^{2} {\mathrm e}^{3 x}+3 \,{\mathrm e}^{3 x} b^{4}-44 \,{\mathrm e}^{2 x} a^{3} b +22 \,{\mathrm e}^{2 x} a \,b^{3}+27 a^{2} b^{2} {\mathrm e}^{x}-3 \,{\mathrm e}^{x} b^{4}-6 a \,b^{3}\right ) {\mathrm e}^{x}}{3 b^{5} \left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}-b \right )^{4}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right )}{b^{5}}\) | \(176\) |
default | \(-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b^{5}}+\frac {\frac {2 \left (\frac {\left (a^{4}-b^{4}\right ) b \tanh \left (\frac {x}{2}\right )^{7}}{a}-\frac {b^{2} \left (7 a^{4}-3 b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{6}}{a^{2}}-\frac {b \left (9 a^{6}-52 a^{4} b^{2}-a^{2} b^{4}+12 b^{6}\right ) \tanh \left (\frac {x}{2}\right )^{5}}{3 a^{3}}+\frac {2 b^{2} \left (21 a^{6}-25 a^{4} b^{2}-7 a^{2} b^{4}+3 b^{6}\right ) \tanh \left (\frac {x}{2}\right )^{4}}{3 a^{4}}+\frac {b \left (9 a^{6}-52 a^{4} b^{2}-a^{2} b^{4}+12 b^{6}\right ) \tanh \left (\frac {x}{2}\right )^{3}}{3 a^{3}}-\frac {b^{2} \left (7 a^{4}-3 b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{2}}{a^{2}}-\frac {\left (a^{4}-b^{4}\right ) b \tanh \left (\frac {x}{2}\right )}{a}\right )}{\left (\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a \right )^{4}}+\ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a \right )}{b^{5}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b^{5}}\) | \(285\) |
-1/b^5*x+4/3*(6*a*b^3*exp(6*x)+27*a^2*b^2*exp(5*x)-3*exp(5*x)*b^4+44*a^3*b *exp(4*x)-22*a*b^3*exp(4*x)+25*a^4*exp(3*x)-56*a^2*b^2*exp(3*x)+3*exp(3*x) *b^4-44*exp(2*x)*a^3*b+22*exp(2*x)*a*b^3+27*a^2*b^2*exp(x)-3*exp(x)*b^4-6* a*b^3)/b^5*exp(x)/(b*exp(2*x)+2*a*exp(x)-b)^4+1/b^5*ln(exp(2*x)+2*a/b*exp( x)-1)
Leaf count of result is larger than twice the leaf count of optimal. 2640 vs. \(2 (91) = 182\).
Time = 0.28 (sec) , antiderivative size = 2640, normalized size of antiderivative = 27.79 \[ \int \frac {1}{(a \text {sech}(x)+b \tanh (x))^5} \, dx=\text {Too large to display} \]
-1/3*(3*b^4*x*cosh(x)^8 + 3*b^4*x*sinh(x)^8 + 24*(a*b^3*x - a*b^3)*cosh(x) ^7 + 24*(b^4*x*cosh(x) + a*b^3*x - a*b^3)*sinh(x)^7 - 12*(9*a^2*b^2 - b^4 - (6*a^2*b^2 - b^4)*x)*cosh(x)^6 + 12*(7*b^4*x*cosh(x)^2 - 9*a^2*b^2 + b^4 + (6*a^2*b^2 - b^4)*x + 14*(a*b^3*x - a*b^3)*cosh(x))*sinh(x)^6 - 8*(22*a ^3*b - 11*a*b^3 - 3*(4*a^3*b - 3*a*b^3)*x)*cosh(x)^5 + 8*(21*b^4*x*cosh(x) ^3 - 22*a^3*b + 11*a*b^3 + 63*(a*b^3*x - a*b^3)*cosh(x)^2 + 3*(4*a^3*b - 3 *a*b^3)*x - 9*(9*a^2*b^2 - b^4 - (6*a^2*b^2 - b^4)*x)*cosh(x))*sinh(x)^5 + 3*b^4*x - 2*(50*a^4 - 112*a^2*b^2 + 6*b^4 - 3*(8*a^4 - 24*a^2*b^2 + 3*b^4 )*x)*cosh(x)^4 + 2*(105*b^4*x*cosh(x)^4 - 50*a^4 + 112*a^2*b^2 - 6*b^4 + 4 20*(a*b^3*x - a*b^3)*cosh(x)^3 - 90*(9*a^2*b^2 - b^4 - (6*a^2*b^2 - b^4)*x )*cosh(x)^2 + 3*(8*a^4 - 24*a^2*b^2 + 3*b^4)*x - 20*(22*a^3*b - 11*a*b^3 - 3*(4*a^3*b - 3*a*b^3)*x)*cosh(x))*sinh(x)^4 + 8*(22*a^3*b - 11*a*b^3 - 3* (4*a^3*b - 3*a*b^3)*x)*cosh(x)^3 + 8*(21*b^4*x*cosh(x)^5 + 105*(a*b^3*x - a*b^3)*cosh(x)^4 + 22*a^3*b - 11*a*b^3 - 30*(9*a^2*b^2 - b^4 - (6*a^2*b^2 - b^4)*x)*cosh(x)^3 - 10*(22*a^3*b - 11*a*b^3 - 3*(4*a^3*b - 3*a*b^3)*x)*c osh(x)^2 - 3*(4*a^3*b - 3*a*b^3)*x - (50*a^4 - 112*a^2*b^2 + 6*b^4 - 3*(8* a^4 - 24*a^2*b^2 + 3*b^4)*x)*cosh(x))*sinh(x)^3 - 12*(9*a^2*b^2 - b^4 - (6 *a^2*b^2 - b^4)*x)*cosh(x)^2 + 4*(21*b^4*x*cosh(x)^6 + 126*(a*b^3*x - a*b^ 3)*cosh(x)^5 - 45*(9*a^2*b^2 - b^4 - (6*a^2*b^2 - b^4)*x)*cosh(x)^4 - 27*a ^2*b^2 + 3*b^4 - 20*(22*a^3*b - 11*a*b^3 - 3*(4*a^3*b - 3*a*b^3)*x)*cos...
Leaf count of result is larger than twice the leaf count of optimal. 2162 vs. \(2 (92) = 184\).
Time = 6.20 (sec) , antiderivative size = 2162, normalized size of antiderivative = 22.76 \[ \int \frac {1}{(a \text {sech}(x)+b \tanh (x))^5} \, dx=\text {Too large to display} \]
Piecewise((36*a**4*x*sech(x)**4/(36*a**4*b**5*sech(x)**4 + 144*a**3*b**6*t anh(x)*sech(x)**3 + 216*a**2*b**7*tanh(x)**2*sech(x)**2 + 144*a*b**8*tanh( x)**3*sech(x) + 36*b**9*tanh(x)**4) + 36*a**4*log(a*sech(x)/b + tanh(x))*s ech(x)**4/(36*a**4*b**5*sech(x)**4 + 144*a**3*b**6*tanh(x)*sech(x)**3 + 21 6*a**2*b**7*tanh(x)**2*sech(x)**2 + 144*a*b**8*tanh(x)**3*sech(x) + 36*b** 9*tanh(x)**4) - 36*a**4*log(tanh(x) + 1)*sech(x)**4/(36*a**4*b**5*sech(x)* *4 + 144*a**3*b**6*tanh(x)*sech(x)**3 + 216*a**2*b**7*tanh(x)**2*sech(x)** 2 + 144*a*b**8*tanh(x)**3*sech(x) + 36*b**9*tanh(x)**4) + 20*a**4*sech(x)* *4/(36*a**4*b**5*sech(x)**4 + 144*a**3*b**6*tanh(x)*sech(x)**3 + 216*a**2* b**7*tanh(x)**2*sech(x)**2 + 144*a*b**8*tanh(x)**3*sech(x) + 36*b**9*tanh( x)**4) + 144*a**3*b*x*tanh(x)*sech(x)**3/(36*a**4*b**5*sech(x)**4 + 144*a* *3*b**6*tanh(x)*sech(x)**3 + 216*a**2*b**7*tanh(x)**2*sech(x)**2 + 144*a*b **8*tanh(x)**3*sech(x) + 36*b**9*tanh(x)**4) + 144*a**3*b*log(a*sech(x)/b + tanh(x))*tanh(x)*sech(x)**3/(36*a**4*b**5*sech(x)**4 + 144*a**3*b**6*tan h(x)*sech(x)**3 + 216*a**2*b**7*tanh(x)**2*sech(x)**2 + 144*a*b**8*tanh(x) **3*sech(x) + 36*b**9*tanh(x)**4) - 144*a**3*b*log(tanh(x) + 1)*tanh(x)*se ch(x)**3/(36*a**4*b**5*sech(x)**4 + 144*a**3*b**6*tanh(x)*sech(x)**3 + 216 *a**2*b**7*tanh(x)**2*sech(x)**2 + 144*a*b**8*tanh(x)**3*sech(x) + 36*b**9 *tanh(x)**4) + 44*a**3*b*tanh(x)*sech(x)**3/(36*a**4*b**5*sech(x)**4 + 144 *a**3*b**6*tanh(x)*sech(x)**3 + 216*a**2*b**7*tanh(x)**2*sech(x)**2 + 1...
Leaf count of result is larger than twice the leaf count of optimal. 297 vs. \(2 (91) = 182\).
Time = 0.25 (sec) , antiderivative size = 297, normalized size of antiderivative = 3.13 \[ \int \frac {1}{(a \text {sech}(x)+b \tanh (x))^5} \, dx=\frac {4 \, {\left (6 \, a b^{3} e^{\left (-x\right )} - 6 \, a b^{3} e^{\left (-7 \, x\right )} + 3 \, {\left (9 \, a^{2} b^{2} - b^{4}\right )} e^{\left (-2 \, x\right )} + 22 \, {\left (2 \, a^{3} b - a b^{3}\right )} e^{\left (-3 \, x\right )} + {\left (25 \, a^{4} - 56 \, a^{2} b^{2} + 3 \, b^{4}\right )} e^{\left (-4 \, x\right )} - 22 \, {\left (2 \, a^{3} b - a b^{3}\right )} e^{\left (-5 \, x\right )} + 3 \, {\left (9 \, a^{2} b^{2} - b^{4}\right )} e^{\left (-6 \, x\right )}\right )}}{3 \, {\left (8 \, a b^{8} e^{\left (-x\right )} - 8 \, a b^{8} e^{\left (-7 \, x\right )} + b^{9} e^{\left (-8 \, x\right )} + b^{9} + 4 \, {\left (6 \, a^{2} b^{7} - b^{9}\right )} e^{\left (-2 \, x\right )} + 8 \, {\left (4 \, a^{3} b^{6} - 3 \, a b^{8}\right )} e^{\left (-3 \, x\right )} + 2 \, {\left (8 \, a^{4} b^{5} - 24 \, a^{2} b^{7} + 3 \, b^{9}\right )} e^{\left (-4 \, x\right )} - 8 \, {\left (4 \, a^{3} b^{6} - 3 \, a b^{8}\right )} e^{\left (-5 \, x\right )} + 4 \, {\left (6 \, a^{2} b^{7} - b^{9}\right )} e^{\left (-6 \, x\right )}\right )}} + \frac {x}{b^{5}} + \frac {\log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{b^{5}} \]
4/3*(6*a*b^3*e^(-x) - 6*a*b^3*e^(-7*x) + 3*(9*a^2*b^2 - b^4)*e^(-2*x) + 22 *(2*a^3*b - a*b^3)*e^(-3*x) + (25*a^4 - 56*a^2*b^2 + 3*b^4)*e^(-4*x) - 22* (2*a^3*b - a*b^3)*e^(-5*x) + 3*(9*a^2*b^2 - b^4)*e^(-6*x))/(8*a*b^8*e^(-x) - 8*a*b^8*e^(-7*x) + b^9*e^(-8*x) + b^9 + 4*(6*a^2*b^7 - b^9)*e^(-2*x) + 8*(4*a^3*b^6 - 3*a*b^8)*e^(-3*x) + 2*(8*a^4*b^5 - 24*a^2*b^7 + 3*b^9)*e^(- 4*x) - 8*(4*a^3*b^6 - 3*a*b^8)*e^(-5*x) + 4*(6*a^2*b^7 - b^9)*e^(-6*x)) + x/b^5 + log(-2*a*e^(-x) + b*e^(-2*x) - b)/b^5
Time = 0.28 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.60 \[ \int \frac {1}{(a \text {sech}(x)+b \tanh (x))^5} \, dx=\frac {\log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{b^{5}} - \frac {25 \, b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{4} - 104 \, a b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 168 \, a^{2} b {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 48 \, b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} - 96 \, a^{3} {\left (e^{\left (-x\right )} - e^{x}\right )} - 64 \, a b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )} + 32 \, a^{2} b + 48 \, b^{3}}{12 \, {\left (b {\left (e^{\left (-x\right )} - e^{x}\right )} - 2 \, a\right )}^{4} b^{4}} \]
log(abs(-b*(e^(-x) - e^x) + 2*a))/b^5 - 1/12*(25*b^3*(e^(-x) - e^x)^4 - 10 4*a*b^2*(e^(-x) - e^x)^3 + 168*a^2*b*(e^(-x) - e^x)^2 + 48*b^3*(e^(-x) - e ^x)^2 - 96*a^3*(e^(-x) - e^x) - 64*a*b^2*(e^(-x) - e^x) + 32*a^2*b + 48*b^ 3)/((b*(e^(-x) - e^x) - 2*a)^4*b^4)
Timed out. \[ \int \frac {1}{(a \text {sech}(x)+b \tanh (x))^5} \, dx=\int \frac {1}{{\left (b\,\mathrm {tanh}\left (x\right )+\frac {a}{\mathrm {cosh}\left (x\right )}\right )}^5} \,d x \]