Integrand size = 17, antiderivative size = 43 \[ \int \frac {\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\log \left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
Time = 0.16 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\log \left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
Result contains complex when optimal does not.
Time = 0.30 (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 |
| \(\displaystyle \frac {i \left (\frac {i \tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b}-\frac {i \log \left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b}\right )}{n}\) |
3.2.86.3.1 Defintions of rubi rules used
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 = 0.17 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.09
| method | result | size |
| parallelrisch | \(-\frac {2 \ln \left (x \right ) b n +{\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}+2 \ln \left (1-\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{2 b n}\) | \(47\) |
| 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}{b n}-\frac {2 \ln \left (c \right )}{n}-\frac {2 \ln \left (x^{n}\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 {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 {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}\) | \(449\) |
Leaf count of result is larger than twice the leaf count of optimal. 566 vs. \(2 (41) = 82\).
Time = 0.26 (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=-\frac {b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} \log \left (x\right ) + 4 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \log \left (x\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + b n \log \left (x\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 2 \, {\left (b n \log \left (x\right ) - 1\right )} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + b n \log \left (x\right ) + 2 \, {\left (3 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} \log \left (x\right ) + b n \log \left (x\right ) - 1\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 4 \, \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 )^{3} + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 2 \, {\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 )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 4 \, {\left (\cosh \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 )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 1\right )} \log \left (\frac {2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\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 )}\right ) + 4 \, {\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} \log \left (x\right ) + {\left (b n \log \left (x\right ) - 1\right )} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 4 \, b n \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 )^{3} + b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 2 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, {\left (3 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + b n\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + b n + 4 \, {\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \]
-(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 = 0.75 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.58 \[ \int \frac {\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\begin {cases} \log {\left (x \right )} \tanh ^{3}{\left (a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\left (x \right )} \tanh ^{3}{\left (a + b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\frac {\log {\left (c x^{n} \right )}}{n} - \frac {\log {\left (\tanh {\left (a + b \log {\left (c x^{n} \right )} \right )} + 1 \right )}}{b n} - \frac {\tanh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{2 b n} & \text {otherwise} \end {cases} \]
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 = 0.27 (sec) , antiderivative size = 304, normalized size of antiderivative = 7.07 \[ \int \frac {\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {4 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 3}{4 \, {\left (b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} - \frac {2 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 3}{4 \, {\left (b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} + \frac {3 \, {\left (2 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 1\right )}}{4 \, {\left (b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} - \frac {3}{4 \, {\left (b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} + \frac {\log \left (\frac {{\left (c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 1\right )} e^{\left (-2 \, a\right )}}{c^{2 \, b}}\right )}{b n} - \log \left (x\right ) \]
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. 127 vs. \(2 (41) = 82\).
Time = 0.35 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.95 \[ \int \frac {\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\log \left (\sqrt {2 \, x^{2 \, b n} {\left | c \right |}^{2 \, b} \cos \left (\pi b \mathrm {sgn}\left (c\right ) - \pi b\right ) e^{\left (2 \, a\right )} + x^{4 \, b n} {\left | c \right |}^{4 \, b} e^{\left (4 \, a\right )} + 1}\right )}{b n} - \frac {3 \, c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} + 2 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 3}{2 \, {\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1\right )}^{2} b n} - \log \left (x\right ) \]
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) - log(x)
Time = 1.83 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.19 \[ \int \frac {\tanh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2}{b\,n+b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}}-\ln \left (x\right )-\frac {2}{b\,n+2\,b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+b\,n\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}}+\frac {\ln \left ({\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+1\right )}{b\,n} \]