Integrand size = 44, antiderivative size = 40 \[ \int \left (\left (1-b^2 n^2\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )+2 b^2 n^2 \text {sech}^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \text {sech}\left (a+b \log \left (c x^n\right )\right )+b n x \text {sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right ) \] Output:
x*sech(a+b*ln(c*x^n))+b*n*x*sech(a+b*ln(c*x^n))*tanh(a+b*ln(c*x^n))
Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.72 \[ \int \left (\left (1-b^2 n^2\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )+2 b^2 n^2 \text {sech}^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \text {sech}\left (a+b \log \left (c x^n\right )\right ) \left (1+b n \tanh \left (a+b \log \left (c x^n\right )\right )\right ) \] Input:
Integrate[(1 - b^2*n^2)*Sech[a + b*Log[c*x^n]] + 2*b^2*n^2*Sech[a + b*Log[ c*x^n]]^3,x]
Output:
x*Sech[a + b*Log[c*x^n]]*(1 + b*n*Tanh[a + b*Log[c*x^n]])
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.62 (sec) , antiderivative size = 139, normalized size of antiderivative = 3.48, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {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.
\(\displaystyle \int \left (2 b^2 n^2 \text {sech}^3\left (a+b \log \left (c x^n\right )\right )+\left (1-b^2 n^2\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {16 e^{3 a} b^2 n^2 x \left (c x^n\right )^{3 b} \operatorname {Hypergeometric2F1}\left (3,\frac {3 b+\frac {1}{n}}{2 b},\frac {1}{2} \left (5+\frac {1}{b n}\right ),-e^{2 a} \left (c x^n\right )^{2 b}\right )}{3 b n+1}+2 e^a x (1-b n) \left (c x^n\right )^b \operatorname {Hypergeometric2F1}\left (1,\frac {b+\frac {1}{n}}{2 b},\frac {1}{2} \left (3+\frac {1}{b n}\right ),-e^{2 a} \left (c x^n\right )^{2 b}\right )\) |
Input:
Int[(1 - b^2*n^2)*Sech[a + b*Log[c*x^n]] + 2*b^2*n^2*Sech[a + b*Log[c*x^n] ]^3,x]
Output:
2*E^a*(1 - b*n)*x*(c*x^n)^b*Hypergeometric2F1[1, (b + n^(-1))/(2*b), (3 + 1/(b*n))/2, -(E^(2*a)*(c*x^n)^(2*b))] + (16*b^2*E^(3*a)*n^2*x*(c*x^n)^(3*b )*Hypergeometric2F1[3, (3*b + n^(-1))/(2*b), (5 + 1/(b*n))/2, -(E^(2*a)*(c *x^n)^(2*b))])/(1 + 3*b*n)
Time = 28.71 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.75
method | result | size |
parallelrisch | \(\operatorname {sech}\left (a +b \ln \left (c \,x^{n}\right )\right ) \left (1+\tanh \left (a +b \ln \left (c \,x^{n}\right )\right ) b n \right ) x\) | \(30\) |
risch | \(\frac {2 c^{b} \left (x^{n}\right )^{b} x \left (n b \left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{3 a} {\mathrm e}^{-\frac {3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{\frac {3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{-\frac {3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}}-{\mathrm e}^{a} {\mathrm e}^{-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}} b n +\left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{3 a} {\mathrm e}^{-\frac {3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{\frac {3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{-\frac {3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}}+{\mathrm e}^{a} {\mathrm e}^{-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}}\right )}{{\left (\left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{2 a} {\mathrm e}^{i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}+1\right )}^{2}}\) | \(509\) |
Input:
int((-b^2*n^2+1)*sech(a+b*ln(c*x^n))+2*b^2*n^2*sech(a+b*ln(c*x^n))^3,x,met hod=_RETURNVERBOSE)
Output:
sech(a+b*ln(c*x^n))*(1+tanh(a+b*ln(c*x^n))*b*n)*x
Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (40) = 80\).
Time = 0.11 (sec) , antiderivative size = 189, normalized size of antiderivative = 4.72 \[ \int \left (\left (1-b^2 n^2\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )+2 b^2 n^2 \text {sech}^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {2 \, {\left ({\left (b n + 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, {\left (b n + 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + {\left (b n + 1\right )} x \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - {\left (b n - 1\right )} x\right )}}{\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + {\left (3 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 1\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 3 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \] Input:
integrate((-b^2*n^2+1)*sech(a+b*log(c*x^n))+2*b^2*n^2*sech(a+b*log(c*x^n)) ^3,x, algorithm="fricas")
Output:
2*((b*n + 1)*x*cosh(b*n*log(x) + b*log(c) + a)^2 + 2*(b*n + 1)*x*cosh(b*n* log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) + (b*n + 1)*x*sinh( b*n*log(x) + b*log(c) + a)^2 - (b*n - 1)*x)/(cosh(b*n*log(x) + b*log(c) + a)^3 + 3*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^2 + sinh(b*n*log(x) + b*log(c) + a)^3 + (3*cosh(b*n*log(x) + b*log(c) + a)^ 2 + 1)*sinh(b*n*log(x) + b*log(c) + a) + 3*cosh(b*n*log(x) + b*log(c) + a) )
\[ \int \left (\left (1-b^2 n^2\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )+2 b^2 n^2 \text {sech}^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \left (2 b^{2} n^{2} \operatorname {sech}^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} - b^{2} n^{2} + 1\right ) \operatorname {sech}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \] Input:
integrate((-b**2*n**2+1)*sech(a+b*ln(c*x**n))+2*b**2*n**2*sech(a+b*ln(c*x* *n))**3,x)
Output:
Integral((2*b**2*n**2*sech(a + b*log(c*x**n))**2 - b**2*n**2 + 1)*sech(a + b*log(c*x**n)), x)
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (40) = 80\).
Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.40 \[ \int \left (\left (1-b^2 n^2\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )+2 b^2 n^2 \text {sech}^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {2 \, {\left ({\left (b c^{3 \, b} n + c^{3 \, b}\right )} x e^{\left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right )} - {\left (b c^{b} n - c^{b}\right )} x e^{\left (b \log \left (x^{n}\right ) + a\right )}\right )}}{c^{4 \, b} e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 2 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 1} \] Input:
integrate((-b^2*n^2+1)*sech(a+b*log(c*x^n))+2*b^2*n^2*sech(a+b*log(c*x^n)) ^3,x, algorithm="maxima")
Output:
2*((b*c^(3*b)*n + c^(3*b))*x*e^(3*b*log(x^n) + 3*a) - (b*c^b*n - c^b)*x*e^ (b*log(x^n) + a))/(c^(4*b)*e^(4*b*log(x^n) + 4*a) + 2*c^(2*b)*e^(2*b*log(x ^n) + 2*a) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (40) = 80\).
Time = 0.27 (sec) , antiderivative size = 215, normalized size of antiderivative = 5.38 \[ \int \left (\left (1-b^2 n^2\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )+2 b^2 n^2 \text {sech}^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {2 \, b c^{3 \, b} n x x^{3 \, b n} e^{\left (3 \, a\right )}}{c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} + 2 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1} - \frac {2 \, b c^{b} n x x^{b n} e^{a}}{c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} + 2 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1} + \frac {2 \, c^{3 \, b} x x^{3 \, b n} e^{\left (3 \, a\right )}}{c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} + 2 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1} + \frac {2 \, c^{b} x x^{b n} e^{a}}{c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} + 2 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1} \] Input:
integrate((-b^2*n^2+1)*sech(a+b*log(c*x^n))+2*b^2*n^2*sech(a+b*log(c*x^n)) ^3,x, algorithm="giac")
Output:
2*b*c^(3*b)*n*x*x^(3*b*n)*e^(3*a)/(c^(4*b)*x^(4*b*n)*e^(4*a) + 2*c^(2*b)*x ^(2*b*n)*e^(2*a) + 1) - 2*b*c^b*n*x*x^(b*n)*e^a/(c^(4*b)*x^(4*b*n)*e^(4*a) + 2*c^(2*b)*x^(2*b*n)*e^(2*a) + 1) + 2*c^(3*b)*x*x^(3*b*n)*e^(3*a)/(c^(4* b)*x^(4*b*n)*e^(4*a) + 2*c^(2*b)*x^(2*b*n)*e^(2*a) + 1) + 2*c^b*x*x^(b*n)* e^a/(c^(4*b)*x^(4*b*n)*e^(4*a) + 2*c^(2*b)*x^(2*b*n)*e^(2*a) + 1)
Time = 2.31 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.65 \[ \int \left (\left (1-b^2 n^2\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )+2 b^2 n^2 \text {sech}^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {2\,x\,{\mathrm {e}}^a\,{\left (c\,x^n\right )}^b\,\left ({\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}-b\,n+b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+1\right )}{{\left ({\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+1\right )}^2} \] Input:
int((2*b^2*n^2)/cosh(a + b*log(c*x^n))^3 - (b^2*n^2 - 1)/cosh(a + b*log(c* x^n)),x)
Output:
(2*x*exp(a)*(c*x^n)^b*(exp(2*a)*(c*x^n)^(2*b) - b*n + b*n*exp(2*a)*(c*x^n) ^(2*b) + 1))/(exp(2*a)*(c*x^n)^(2*b) + 1)^2
Time = 0.25 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.38 \[ \int \left (\left (1-b^2 n^2\right ) \text {sech}\left (a+b \log \left (c x^n\right )\right )+2 b^2 n^2 \text {sech}^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {2 x^{b n} e^{a} c^{b} x \left (x^{2 b n} e^{2 a} c^{2 b} b n +x^{2 b n} e^{2 a} c^{2 b}-b n +1\right )}{x^{4 b n} e^{4 a} c^{4 b}+2 x^{2 b n} e^{2 a} c^{2 b}+1} \] Input:
int((-b^2*n^2+1)*sech(a+b*log(c*x^n))+2*b^2*n^2*sech(a+b*log(c*x^n))^3,x)
Output:
(2*x**(b*n)*e**a*c**b*x*(x**(2*b*n)*e**(2*a)*c**(2*b)*b*n + x**(2*b*n)*e** (2*a)*c**(2*b) - b*n + 1))/(x**(4*b*n)*e**(4*a)*c**(4*b) + 2*x**(2*b*n)*e* *(2*a)*c**(2*b) + 1)