Integrand size = 17, antiderivative size = 43 \[ \int \frac {\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\log \left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \] Output:
ln(cosh(a+b*ln(c*x^n)))/b/n-1/2*tanh(a+b*ln(c*x^n))^2/b/n
Time = 0.11 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.88 \[ \int \frac {\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2 \log \left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )+\text {sech}^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \] Input:
Integrate[Tanh[a + b*Log[c*x^n]]^3/x,x]
Output:
(2*Log[Cosh[a + b*Log[c*x^n]]] + Sech[a + b*Log[c*x^n]]^2)/(2*b*n)
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {3039, 3042, 26, 3954, 26, 3042, 26, 3956}
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.
\(\displaystyle \int \frac {\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \frac {\int \tanh ^3\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int i \tan \left (i a+i b \log \left (c x^n\right )\right )^3d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {i \int \tan \left (i a+i b \log \left (c x^n\right )\right )^3d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 3954 |
\(\displaystyle \frac {i \left (\frac {i \tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b}-\int i \tanh \left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )\right )}{n}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {i \left (\frac {i \tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b}-i \int \tanh \left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )\right )}{n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {i \left (\frac {i \tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b}-i \int -i \tan \left (i a+i b \log \left (c x^n\right )\right )d\log \left (c x^n\right )\right )}{n}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {i \left (\frac {i \tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b}-\int \tan \left (i a+i b \log \left (c x^n\right )\right )d\log \left (c x^n\right )\right )}{n}\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle \frac {i \left (\frac {i \tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b}-\frac {i \log \left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b}\right )}{n}\) |
Input:
Int[Tanh[a + b*Log[c*x^n]]^3/x,x]
Output:
(I*(((-I)*Log[Cosh[a + b*Log[c*x^n]]])/b + ((I/2)*Tanh[a + b*Log[c*x^n]]^2 )/b))/n
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[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst [[3]] Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /; !FalseQ[lst]] /; NonsumQ[u]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.54 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.09
method | result | size |
parallelrisch | \(-\frac {2 \ln \left (x \right ) b n +{\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}+2 \ln \left (1-\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{2 n b}\) | \(47\) |
derivativedivides | \(\frac {-\frac {{\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{2}-\frac {\ln \left (\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )-1\right )}{2}-\frac {\ln \left (\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )+1\right )}{2}}{n b}\) | \(56\) |
default | \(\frac {-\frac {{\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{2}-\frac {\ln \left (\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )-1\right )}{2}-\frac {\ln \left (\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )+1\right )}{2}}{n b}\) | \(56\) |
risch | \(\ln \left (x \right )-\frac {2 a}{n b}-\frac {2 \ln \left (c \right )}{n}-\frac {2 \ln \left (x^{n}\right )}{n}+\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{n}-\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{n}-\frac {i \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{n}+\frac {i \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{n}+\frac {2 \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 )}}{b n {\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}}+\frac {\ln \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 )}{b n}\) | \(449\) |
Input:
int(tanh(a+b*ln(c*x^n))^3/x,x,method=_RETURNVERBOSE)
Output:
-1/2*(2*ln(x)*b*n+tanh(a+b*ln(c*x^n))^2+2*ln(1-tanh(a+b*ln(c*x^n))))/n/b
Leaf count of result is larger than twice the leaf count of optimal. 566 vs. \(2 (41) = 82\).
Time = 0.10 (sec) , antiderivative size = 566, normalized size of antiderivative = 13.16 \[ \int \frac {\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx =\text {Too large to display} \] Input:
integrate(tanh(a+b*log(c*x^n))^3/x,x, algorithm="fricas")
Output:
-(b*n*cosh(b*n*log(x) + b*log(c) + a)^4*log(x) + 4*b*n*cosh(b*n*log(x) + b *log(c) + a)*log(x)*sinh(b*n*log(x) + b*log(c) + a)^3 + b*n*log(x)*sinh(b* n*log(x) + b*log(c) + a)^4 + 2*(b*n*log(x) - 1)*cosh(b*n*log(x) + b*log(c) + a)^2 + b*n*log(x) + 2*(3*b*n*cosh(b*n*log(x) + b*log(c) + a)^2*log(x) + b*n*log(x) - 1)*sinh(b*n*log(x) + b*log(c) + a)^2 - (cosh(b*n*log(x) + b* log(c) + a)^4 + 4*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log( c) + a)^3 + sinh(b*n*log(x) + b*log(c) + a)^4 + 2*(3*cosh(b*n*log(x) + b*l og(c) + a)^2 + 1)*sinh(b*n*log(x) + b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*log(c) + a)^2 + 4*(cosh(b*n*log(x) + b*log(c) + a)^3 + cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a) + 1)*log(2*cosh(b*n*log(x) + b*log(c) + a)/(cosh(b*n*log(x) + b*log(c) + a) - sinh(b*n*log(x) + b*lo g(c) + a))) + 4*(b*n*cosh(b*n*log(x) + b*log(c) + a)^3*log(x) + (b*n*log(x ) - 1)*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a))/( b*n*cosh(b*n*log(x) + b*log(c) + a)^4 + 4*b*n*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^3 + b*n*sinh(b*n*log(x) + b*log(c) + a )^4 + 2*b*n*cosh(b*n*log(x) + b*log(c) + a)^2 + 2*(3*b*n*cosh(b*n*log(x) + b*log(c) + a)^2 + b*n)*sinh(b*n*log(x) + b*log(c) + a)^2 + b*n + 4*(b*n*c osh(b*n*log(x) + b*log(c) + a)^3 + b*n*cosh(b*n*log(x) + b*log(c) + a))*si nh(b*n*log(x) + b*log(c) + a))
Time = 0.77 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.58 \[ \int \frac {\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\begin {cases} \log {\left (x \right )} \tanh ^{3}{\left (a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\left (x \right )} \tanh ^{3}{\left (a + b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\frac {\log {\left (c x^{n} \right )}}{n} - \frac {\log {\left (\tanh {\left (a + b \log {\left (c x^{n} \right )} \right )} + 1 \right )}}{b n} - \frac {\tanh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{2 b n} & \text {otherwise} \end {cases} \] Input:
integrate(tanh(a+b*ln(c*x**n))**3/x,x)
Output:
Piecewise((log(x)*tanh(a)**3, Eq(b, 0) & (Eq(b, 0) | Eq(n, 0))), (log(x)*t anh(a + b*log(c))**3, Eq(n, 0)), (log(c*x**n)/n - log(tanh(a + b*log(c*x** n)) + 1)/(b*n) - tanh(a + b*log(c*x**n))**2/(2*b*n), True))
Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (41) = 82\).
Time = 0.10 (sec) , antiderivative size = 304, normalized size of antiderivative = 7.07 \[ \int \frac {\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {4 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 3}{4 \, {\left (b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} - \frac {2 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 3}{4 \, {\left (b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} + \frac {3 \, {\left (2 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 1\right )}}{4 \, {\left (b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} - \frac {3}{4 \, {\left (b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} + \frac {\log \left (\frac {{\left (c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 1\right )} e^{\left (-2 \, a\right )}}{c^{2 \, b}}\right )}{b n} - \log \left (x\right ) \] Input:
integrate(tanh(a+b*log(c*x^n))^3/x,x, algorithm="maxima")
Output:
1/4*(4*c^(2*b)*e^(2*b*log(x^n) + 2*a) + 3)/(b*c^(4*b)*n*e^(4*b*log(x^n) + 4*a) + 2*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) + b*n) - 1/4*(2*c^(2*b)*e^(2*b *log(x^n) + 2*a) + 3)/(b*c^(4*b)*n*e^(4*b*log(x^n) + 4*a) + 2*b*c^(2*b)*n* e^(2*b*log(x^n) + 2*a) + b*n) + 3/4*(2*c^(2*b)*e^(2*b*log(x^n) + 2*a) + 1) /(b*c^(4*b)*n*e^(4*b*log(x^n) + 4*a) + 2*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a ) + b*n) - 3/4/(b*c^(4*b)*n*e^(4*b*log(x^n) + 4*a) + 2*b*c^(2*b)*n*e^(2*b* log(x^n) + 2*a) + b*n) + log((c^(2*b)*e^(2*b*log(x^n) + 2*a) + 1)*e^(-2*a) /c^(2*b))/(b*n) - log(x)
Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (41) = 82\).
Time = 0.20 (sec) , antiderivative size = 137, normalized size of antiderivative = 3.19 \[ \int \frac {\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\log \left (x^{b n}\right )}{b n} + \frac {\log \left (\sqrt {2 \, x^{2 \, b n} {\left | c \right |}^{2 \, b} \cos \left (\pi b \mathrm {sgn}\left (c\right ) - \pi b\right ) e^{\left (2 \, a\right )} + x^{4 \, b n} {\left | c \right |}^{4 \, b} e^{\left (4 \, a\right )} + 1}\right )}{b n} - \frac {3 \, c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} + 2 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 3}{2 \, {\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1\right )}^{2} b n} \] Input:
integrate(tanh(a+b*log(c*x^n))^3/x,x, algorithm="giac")
Output:
-log(x^(b*n))/(b*n) + log(sqrt(2*x^(2*b*n)*abs(c)^(2*b)*cos(pi*b*sgn(c) - pi*b)*e^(2*a) + x^(4*b*n)*abs(c)^(4*b)*e^(4*a) + 1))/(b*n) - 1/2*(3*c^(4*b )*x^(4*b*n)*e^(4*a) + 2*c^(2*b)*x^(2*b*n)*e^(2*a) + 3)/((c^(2*b)*x^(2*b*n) *e^(2*a) + 1)^2*b*n)
Time = 2.24 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.19 \[ \int \frac {\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2}{b\,n+b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}}-\ln \left (x\right )-\frac {2}{b\,n+2\,b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+b\,n\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}}+\frac {\ln \left ({\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+1\right )}{b\,n} \] Input:
int(tanh(a + b*log(c*x^n))^3/x,x)
Output:
2/(b*n + b*n*exp(2*a)*(c*x^n)^(2*b)) - log(x) - 2/(b*n + 2*b*n*exp(2*a)*(c *x^n)^(2*b) + b*n*exp(4*a)*(c*x^n)^(4*b)) + log(exp(2*a)*(c*x^n)^(2*b) + 1 )/(b*n)
Time = 0.25 (sec) , antiderivative size = 211, normalized size of antiderivative = 4.91 \[ \int \frac {\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {x^{4 b n} e^{4 a} c^{4 b} \mathrm {log}\left (x^{2 b n} e^{2 a} c^{2 b}+1\right )-x^{4 b n} e^{4 a} c^{4 b} \mathrm {log}\left (x \right ) b n -x^{4 b n} e^{4 a} c^{4 b}+2 x^{2 b n} e^{2 a} c^{2 b} \mathrm {log}\left (x^{2 b n} e^{2 a} c^{2 b}+1\right )-2 x^{2 b n} e^{2 a} c^{2 b} \mathrm {log}\left (x \right ) b n +\mathrm {log}\left (x^{2 b n} e^{2 a} c^{2 b}+1\right )-\mathrm {log}\left (x \right ) b n -1}{b n \left (x^{4 b n} e^{4 a} c^{4 b}+2 x^{2 b n} e^{2 a} c^{2 b}+1\right )} \] Input:
int(tanh(a+b*log(c*x^n))^3/x,x)
Output:
(x**(4*b*n)*e**(4*a)*c**(4*b)*log(x**(2*b*n)*e**(2*a)*c**(2*b) + 1) - x**( 4*b*n)*e**(4*a)*c**(4*b)*log(x)*b*n - x**(4*b*n)*e**(4*a)*c**(4*b) + 2*x** (2*b*n)*e**(2*a)*c**(2*b)*log(x**(2*b*n)*e**(2*a)*c**(2*b) + 1) - 2*x**(2* b*n)*e**(2*a)*c**(2*b)*log(x)*b*n + log(x**(2*b*n)*e**(2*a)*c**(2*b) + 1) - log(x)*b*n - 1)/(b*n*(x**(4*b*n)*e**(4*a)*c**(4*b) + 2*x**(2*b*n)*e**(2* a)*c**(2*b) + 1))