Integrand size = 8, antiderivative size = 42 \[ \int \coth ^5(a+b x) \, dx=-\frac {\coth ^2(a+b x)}{2 b}-\frac {\coth ^4(a+b x)}{4 b}+\frac {\log (\sinh (a+b x))}{b} \] Output:
-1/2*coth(b*x+a)^2/b-1/4*coth(b*x+a)^4/b+ln(sinh(b*x+a))/b
Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.83 \[ \int \coth ^5(a+b x) \, dx=-\frac {4 \text {csch}^2(a+b x)+\text {csch}^4(a+b x)-4 \log (\sinh (a+b x))}{4 b} \] Input:
Integrate[Coth[a + b*x]^5,x]
Output:
-1/4*(4*Csch[a + b*x]^2 + Csch[a + b*x]^4 - 4*Log[Sinh[a + b*x]])/b
Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.36, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.375, Rules used = {3042, 26, 3954, 26, 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.
\(\displaystyle \int \coth ^5(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -i \tan \left (i a+i b x+\frac {\pi }{2}\right )^5dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )^5dx\) |
\(\Big \downarrow \) 3954 |
\(\displaystyle -i \left (-\int -i \coth ^3(a+b x)dx-\frac {i \coth ^4(a+b x)}{4 b}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \left (i \int \coth ^3(a+b x)dx-\frac {i \coth ^4(a+b x)}{4 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \left (i \int i \tan \left (i a+i b x+\frac {\pi }{2}\right )^3dx-\frac {i \coth ^4(a+b x)}{4 b}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \left (-\int \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )^3dx-\frac {i \coth ^4(a+b x)}{4 b}\right )\) |
\(\Big \downarrow \) 3954 |
\(\displaystyle -i \left (\int i \coth (a+b x)dx-\frac {i \coth ^4(a+b x)}{4 b}-\frac {i \coth ^2(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \left (i \int \coth (a+b x)dx-\frac {i \coth ^4(a+b x)}{4 b}-\frac {i \coth ^2(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \left (i \int -i \tan \left (i a+i b x+\frac {\pi }{2}\right )dx-\frac {i \coth ^4(a+b x)}{4 b}-\frac {i \coth ^2(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \left (\int \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )dx-\frac {i \coth ^4(a+b x)}{4 b}-\frac {i \coth ^2(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle -i \left (-\frac {i \coth ^4(a+b x)}{4 b}-\frac {i \coth ^2(a+b x)}{2 b}+\frac {i \log (-i \sinh (a+b x))}{b}\right )\) |
Input:
Int[Coth[a + b*x]^5,x]
Output:
(-I)*(((-1/2*I)*Coth[a + b*x]^2)/b - ((I/4)*Coth[a + b*x]^4)/b + (I*Log[(- I)*Sinh[a + b*x]])/b)
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[((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 = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.14
method | result | size |
derivativedivides | \(\frac {-\frac {\coth \left (b x +a \right )^{4}}{4}-\frac {\coth \left (b x +a \right )^{2}}{2}-\frac {\ln \left (\coth \left (b x +a \right )-1\right )}{2}-\frac {\ln \left (\coth \left (b x +a \right )+1\right )}{2}}{b}\) | \(48\) |
default | \(\frac {-\frac {\coth \left (b x +a \right )^{4}}{4}-\frac {\coth \left (b x +a \right )^{2}}{2}-\frac {\ln \left (\coth \left (b x +a \right )-1\right )}{2}-\frac {\ln \left (\coth \left (b x +a \right )+1\right )}{2}}{b}\) | \(48\) |
parallelrisch | \(\frac {-\coth \left (b x +a \right )^{4}-2 \coth \left (b x +a \right )^{2}-4 b x +4 \ln \left (\tanh \left (b x +a \right )\right )-4 \ln \left (1-\tanh \left (b x +a \right )\right )}{4 b}\) | \(53\) |
risch | \(-x -\frac {2 a}{b}-\frac {4 \,{\mathrm e}^{2 b x +2 a} \left ({\mathrm e}^{4 b x +4 a}-{\mathrm e}^{2 b x +2 a}+1\right )}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b}\) | \(76\) |
Input:
int(coth(b*x+a)^5,x,method=_RETURNVERBOSE)
Output:
1/b*(-1/4*coth(b*x+a)^4-1/2*coth(b*x+a)^2-1/2*ln(coth(b*x+a)-1)-1/2*ln(cot h(b*x+a)+1))
Leaf count of result is larger than twice the leaf count of optimal. 978 vs. \(2 (38) = 76\).
Time = 0.10 (sec) , antiderivative size = 978, normalized size of antiderivative = 23.29 \[ \int \coth ^5(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(coth(b*x+a)^5,x, algorithm="fricas")
Output:
-(b*x*cosh(b*x + a)^8 + 8*b*x*cosh(b*x + a)*sinh(b*x + a)^7 + b*x*sinh(b*x + a)^8 - 4*(b*x - 1)*cosh(b*x + a)^6 + 4*(7*b*x*cosh(b*x + a)^2 - b*x + 1 )*sinh(b*x + a)^6 + 8*(7*b*x*cosh(b*x + a)^3 - 3*(b*x - 1)*cosh(b*x + a))* sinh(b*x + a)^5 + 2*(3*b*x - 2)*cosh(b*x + a)^4 + 2*(35*b*x*cosh(b*x + a)^ 4 - 30*(b*x - 1)*cosh(b*x + a)^2 + 3*b*x - 2)*sinh(b*x + a)^4 + 8*(7*b*x*c osh(b*x + a)^5 - 10*(b*x - 1)*cosh(b*x + a)^3 + (3*b*x - 2)*cosh(b*x + a)) *sinh(b*x + a)^3 - 4*(b*x - 1)*cosh(b*x + a)^2 + 4*(7*b*x*cosh(b*x + a)^6 - 15*(b*x - 1)*cosh(b*x + a)^4 + 3*(3*b*x - 2)*cosh(b*x + a)^2 - b*x + 1)* sinh(b*x + a)^2 + b*x - (cosh(b*x + a)^8 + 8*cosh(b*x + a)*sinh(b*x + a)^7 + sinh(b*x + a)^8 + 4*(7*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^6 - 4*cosh(b* x + a)^6 + 8*(7*cosh(b*x + a)^3 - 3*cosh(b*x + a))*sinh(b*x + a)^5 + 2*(35 *cosh(b*x + a)^4 - 30*cosh(b*x + a)^2 + 3)*sinh(b*x + a)^4 + 6*cosh(b*x + a)^4 + 8*(7*cosh(b*x + a)^5 - 10*cosh(b*x + a)^3 + 3*cosh(b*x + a))*sinh(b *x + a)^3 + 4*(7*cosh(b*x + a)^6 - 15*cosh(b*x + a)^4 + 9*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^2 - 4*cosh(b*x + a)^2 + 8*(cosh(b*x + a)^7 - 3*cosh(b*x + a)^5 + 3*cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a) + 1)*log(2*sinh (b*x + a)/(cosh(b*x + a) - sinh(b*x + a))) + 8*(b*x*cosh(b*x + a)^7 - 3*(b *x - 1)*cosh(b*x + a)^5 + (3*b*x - 2)*cosh(b*x + a)^3 - (b*x - 1)*cosh(b*x + a))*sinh(b*x + a))/(b*cosh(b*x + a)^8 + 8*b*cosh(b*x + a)*sinh(b*x + a) ^7 + b*sinh(b*x + a)^8 - 4*b*cosh(b*x + a)^6 + 4*(7*b*cosh(b*x + a)^2 -...
Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (32) = 64\).
Time = 1.61 (sec) , antiderivative size = 104, normalized size of antiderivative = 2.48 \[ \int \coth ^5(a+b x) \, dx=\begin {cases} x \coth ^{5}{\left (a \right )} & \text {for}\: b = 0 \\x \coth ^{5}{\left (b x + \log {\left (- e^{- b x} \right )} \right )} & \text {for}\: a = \log {\left (- e^{- b x} \right )} \\x \coth ^{5}{\left (b x + \log {\left (e^{- b x} \right )} \right )} & \text {for}\: a = \log {\left (e^{- b x} \right )} \\x - \frac {\log {\left (\tanh {\left (a + b x \right )} + 1 \right )}}{b} + \frac {\log {\left (\tanh {\left (a + b x \right )} \right )}}{b} - \frac {1}{2 b \tanh ^{2}{\left (a + b x \right )}} - \frac {1}{4 b \tanh ^{4}{\left (a + b x \right )}} & \text {otherwise} \end {cases} \] Input:
integrate(coth(b*x+a)**5,x)
Output:
Piecewise((x*coth(a)**5, Eq(b, 0)), (x*coth(b*x + log(-exp(-b*x)))**5, Eq( a, log(-exp(-b*x)))), (x*coth(b*x + log(exp(-b*x)))**5, Eq(a, log(exp(-b*x )))), (x - log(tanh(a + b*x) + 1)/b + log(tanh(a + b*x))/b - 1/(2*b*tanh(a + b*x)**2) - 1/(4*b*tanh(a + b*x)**4), True))
Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (38) = 76\).
Time = 0.04 (sec) , antiderivative size = 122, normalized size of antiderivative = 2.90 \[ \int \coth ^5(a+b x) \, dx=x + \frac {a}{b} + \frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} + \frac {4 \, {\left (e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )}\right )}}{b {\left (4 \, e^{\left (-2 \, b x - 2 \, a\right )} - 6 \, e^{\left (-4 \, b x - 4 \, a\right )} + 4 \, e^{\left (-6 \, b x - 6 \, a\right )} - e^{\left (-8 \, b x - 8 \, a\right )} - 1\right )}} \] Input:
integrate(coth(b*x+a)^5,x, algorithm="maxima")
Output:
x + a/b + log(e^(-b*x - a) + 1)/b + log(e^(-b*x - a) - 1)/b + 4*(e^(-2*b*x - 2*a) - e^(-4*b*x - 4*a) + e^(-6*b*x - 6*a))/(b*(4*e^(-2*b*x - 2*a) - 6* e^(-4*b*x - 4*a) + 4*e^(-6*b*x - 6*a) - e^(-8*b*x - 8*a) - 1))
Time = 0.13 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.67 \[ \int \coth ^5(a+b x) \, dx=-\frac {b x + a + \frac {4 \, {\left (e^{\left (6 \, b x + 6 \, a\right )} - e^{\left (4 \, b x + 4 \, a\right )} + e^{\left (2 \, b x + 2 \, a\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{4}} - \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{b} \] Input:
integrate(coth(b*x+a)^5,x, algorithm="giac")
Output:
-(b*x + a + 4*(e^(6*b*x + 6*a) - e^(4*b*x + 4*a) + e^(2*b*x + 2*a))/(e^(2* b*x + 2*a) - 1)^4 - log(abs(e^(2*b*x + 2*a) - 1)))/b
Time = 2.09 (sec) , antiderivative size = 159, normalized size of antiderivative = 3.79 \[ \int \coth ^5(a+b x) \, dx=\frac {\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1\right )}{b}-x-\frac {4}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}-\frac {8}{b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {8}{b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}-3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}-1\right )}-\frac {4}{b\,\left (6\,{\mathrm {e}}^{4\,a+4\,b\,x}-4\,{\mathrm {e}}^{2\,a+2\,b\,x}-4\,{\mathrm {e}}^{6\,a+6\,b\,x}+{\mathrm {e}}^{8\,a+8\,b\,x}+1\right )} \] Input:
int(coth(a + b*x)^5,x)
Output:
log(exp(2*a)*exp(2*b*x) - 1)/b - x - 4/(b*(exp(2*a + 2*b*x) - 1)) - 8/(b*( exp(4*a + 4*b*x) - 2*exp(2*a + 2*b*x) + 1)) - 8/(b*(3*exp(2*a + 2*b*x) - 3 *exp(4*a + 4*b*x) + exp(6*a + 6*b*x) - 1)) - 4/(b*(6*exp(4*a + 4*b*x) - 4* exp(2*a + 2*b*x) - 4*exp(6*a + 6*b*x) + exp(8*a + 8*b*x) + 1))
Time = 0.26 (sec) , antiderivative size = 334, normalized size of antiderivative = 7.95 \[ \int \coth ^5(a+b x) \, dx=\frac {e^{8 b x +8 a} \mathrm {log}\left (e^{b x +a}-1\right )+e^{8 b x +8 a} \mathrm {log}\left (e^{b x +a}+1\right )-e^{8 b x +8 a} b x -e^{8 b x +8 a}-4 e^{6 b x +6 a} \mathrm {log}\left (e^{b x +a}-1\right )-4 e^{6 b x +6 a} \mathrm {log}\left (e^{b x +a}+1\right )+4 e^{6 b x +6 a} b x +6 e^{4 b x +4 a} \mathrm {log}\left (e^{b x +a}-1\right )+6 e^{4 b x +4 a} \mathrm {log}\left (e^{b x +a}+1\right )-6 e^{4 b x +4 a} b x -2 e^{4 b x +4 a}-4 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}-1\right )-4 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}+1\right )+4 e^{2 b x +2 a} b x +\mathrm {log}\left (e^{b x +a}-1\right )+\mathrm {log}\left (e^{b x +a}+1\right )-b x -1}{b \left (e^{8 b x +8 a}-4 e^{6 b x +6 a}+6 e^{4 b x +4 a}-4 e^{2 b x +2 a}+1\right )} \] Input:
int(coth(b*x+a)^5,x)
Output:
(e**(8*a + 8*b*x)*log(e**(a + b*x) - 1) + e**(8*a + 8*b*x)*log(e**(a + b*x ) + 1) - e**(8*a + 8*b*x)*b*x - e**(8*a + 8*b*x) - 4*e**(6*a + 6*b*x)*log( e**(a + b*x) - 1) - 4*e**(6*a + 6*b*x)*log(e**(a + b*x) + 1) + 4*e**(6*a + 6*b*x)*b*x + 6*e**(4*a + 4*b*x)*log(e**(a + b*x) - 1) + 6*e**(4*a + 4*b*x )*log(e**(a + b*x) + 1) - 6*e**(4*a + 4*b*x)*b*x - 2*e**(4*a + 4*b*x) - 4* e**(2*a + 2*b*x)*log(e**(a + b*x) - 1) - 4*e**(2*a + 2*b*x)*log(e**(a + b* x) + 1) + 4*e**(2*a + 2*b*x)*b*x + log(e**(a + b*x) - 1) + log(e**(a + b*x ) + 1) - b*x - 1)/(b*(e**(8*a + 8*b*x) - 4*e**(6*a + 6*b*x) + 6*e**(4*a + 4*b*x) - 4*e**(2*a + 2*b*x) + 1))