Integrand size = 8, antiderivative size = 55 \[ \int \text {csch}^5(a+b x) \, dx=-\frac {3 \text {arctanh}(\cosh (a+b x))}{8 b}+\frac {3 \coth (a+b x) \text {csch}(a+b x)}{8 b}-\frac {\coth (a+b x) \text {csch}^3(a+b x)}{4 b} \] Output:
-3/8*arctanh(cosh(b*x+a))/b+3/8*coth(b*x+a)*csch(b*x+a)/b-1/4*coth(b*x+a)* csch(b*x+a)^3/b
Leaf count is larger than twice the leaf count of optimal. \(113\) vs. \(2(55)=110\).
Time = 0.02 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.05 \[ \int \text {csch}^5(a+b x) \, dx=\frac {3 \text {csch}^2\left (\frac {1}{2} (a+b x)\right )}{32 b}-\frac {\text {csch}^4\left (\frac {1}{2} (a+b x)\right )}{64 b}-\frac {3 \log \left (\cosh \left (\frac {1}{2} (a+b x)\right )\right )}{8 b}+\frac {3 \log \left (\sinh \left (\frac {1}{2} (a+b x)\right )\right )}{8 b}+\frac {3 \text {sech}^2\left (\frac {1}{2} (a+b x)\right )}{32 b}+\frac {\text {sech}^4\left (\frac {1}{2} (a+b x)\right )}{64 b} \] Input:
Integrate[Csch[a + b*x]^5,x]
Output:
(3*Csch[(a + b*x)/2]^2)/(32*b) - Csch[(a + b*x)/2]^4/(64*b) - (3*Log[Cosh[ (a + b*x)/2]])/(8*b) + (3*Log[Sinh[(a + b*x)/2]])/(8*b) + (3*Sech[(a + b*x )/2]^2)/(32*b) + Sech[(a + b*x)/2]^4/(64*b)
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.27, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.375, Rules used = {3042, 26, 4255, 26, 3042, 26, 4255, 26, 3042, 26, 4257}
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.
\(\displaystyle \int \text {csch}^5(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int i \csc (i a+i b x)^5dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \csc (i a+i b x)^5dx\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle i \left (\frac {3}{4} \int i \text {csch}^3(a+b x)dx+\frac {i \coth (a+b x) \text {csch}^3(a+b x)}{4 b}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (\frac {3}{4} i \int \text {csch}^3(a+b x)dx+\frac {i \coth (a+b x) \text {csch}^3(a+b x)}{4 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle i \left (\frac {3}{4} i \int -i \csc (i a+i b x)^3dx+\frac {i \coth (a+b x) \text {csch}^3(a+b x)}{4 b}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (\frac {3}{4} \int \csc (i a+i b x)^3dx+\frac {i \coth (a+b x) \text {csch}^3(a+b x)}{4 b}\right )\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle i \left (\frac {3}{4} \left (\frac {1}{2} \int -i \text {csch}(a+b x)dx-\frac {i \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )+\frac {i \coth (a+b x) \text {csch}^3(a+b x)}{4 b}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (\frac {3}{4} \left (-\frac {1}{2} i \int \text {csch}(a+b x)dx-\frac {i \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )+\frac {i \coth (a+b x) \text {csch}^3(a+b x)}{4 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle i \left (\frac {3}{4} \left (-\frac {1}{2} i \int i \csc (i a+i b x)dx-\frac {i \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )+\frac {i \coth (a+b x) \text {csch}^3(a+b x)}{4 b}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (\frac {3}{4} \left (\frac {1}{2} \int \csc (i a+i b x)dx-\frac {i \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )+\frac {i \coth (a+b x) \text {csch}^3(a+b x)}{4 b}\right )\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle i \left (\frac {3}{4} \left (\frac {i \text {arctanh}(\cosh (a+b x))}{2 b}-\frac {i \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )+\frac {i \coth (a+b x) \text {csch}^3(a+b x)}{4 b}\right )\) |
Input:
Int[Csch[a + b*x]^5,x]
Output:
I*(((I/4)*Coth[a + b*x]*Csch[a + b*x]^3)/b + (3*(((I/2)*ArcTanh[Cosh[a + b *x]])/b - ((I/2)*Coth[a + b*x]*Csch[a + b*x])/b))/4)
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[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.31 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {\left (-\frac {\operatorname {csch}\left (b x +a \right )^{3}}{4}+\frac {3 \,\operatorname {csch}\left (b x +a \right )}{8}\right ) \coth \left (b x +a \right )-\frac {3 \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{4}}{b}\) | \(41\) |
default | \(\frac {\left (-\frac {\operatorname {csch}\left (b x +a \right )^{3}}{4}+\frac {3 \,\operatorname {csch}\left (b x +a \right )}{8}\right ) \coth \left (b x +a \right )-\frac {3 \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{4}}{b}\) | \(41\) |
parallelrisch | \(\frac {\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}-\coth \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}-8 \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+8 \coth \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+24 \ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{64 b}\) | \(69\) |
risch | \(\frac {{\mathrm e}^{b x +a} \left (3 \,{\mathrm e}^{6 b x +6 a}-11 \,{\mathrm e}^{4 b x +4 a}-11 \,{\mathrm e}^{2 b x +2 a}+3\right )}{4 b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{4}}+\frac {3 \ln \left ({\mathrm e}^{b x +a}-1\right )}{8 b}-\frac {3 \ln \left (1+{\mathrm e}^{b x +a}\right )}{8 b}\) | \(89\) |
Input:
int(csch(b*x+a)^5,x,method=_RETURNVERBOSE)
Output:
1/b*((-1/4*csch(b*x+a)^3+3/8*csch(b*x+a))*coth(b*x+a)-3/4*arctanh(exp(b*x+ a)))
Leaf count of result is larger than twice the leaf count of optimal. 1114 vs. \(2 (49) = 98\).
Time = 0.08 (sec) , antiderivative size = 1114, normalized size of antiderivative = 20.25 \[ \int \text {csch}^5(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(csch(b*x+a)^5,x, algorithm="fricas")
Output:
1/8*(6*cosh(b*x + a)^7 + 42*cosh(b*x + a)*sinh(b*x + a)^6 + 6*sinh(b*x + a )^7 + 2*(63*cosh(b*x + a)^2 - 11)*sinh(b*x + a)^5 - 22*cosh(b*x + a)^5 + 1 0*(21*cosh(b*x + a)^3 - 11*cosh(b*x + a))*sinh(b*x + a)^4 + 2*(105*cosh(b* x + a)^4 - 110*cosh(b*x + a)^2 - 11)*sinh(b*x + a)^3 - 22*cosh(b*x + a)^3 + 2*(63*cosh(b*x + a)^5 - 110*cosh(b*x + a)^3 - 33*cosh(b*x + a))*sinh(b*x + a)^2 - 3*(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(cosh(b*x + a) + si nh(b*x + a) + 1) + 3*(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*co sh(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...
\[ \int \text {csch}^5(a+b x) \, dx=\int \operatorname {csch}^{5}{\left (a + b x \right )}\, dx \] Input:
integrate(csch(b*x+a)**5,x)
Output:
Integral(csch(a + b*x)**5, x)
Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (49) = 98\).
Time = 0.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.42 \[ \int \text {csch}^5(a+b x) \, dx=-\frac {3 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{8 \, b} + \frac {3 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{8 \, b} - \frac {3 \, e^{\left (-b x - a\right )} - 11 \, e^{\left (-3 \, b x - 3 \, a\right )} - 11 \, e^{\left (-5 \, b x - 5 \, a\right )} + 3 \, e^{\left (-7 \, b x - 7 \, a\right )}}{4 \, b {\left (4 \, e^{\left (-2 \, b x - 2 \, a\right )} - 6 \, e^{\left (-4 \, b x - 4 \, a\right )} + 4 \, e^{\left (-6 \, b x - 6 \, a\right )} - e^{\left (-8 \, b x - 8 \, a\right )} - 1\right )}} \] Input:
integrate(csch(b*x+a)^5,x, algorithm="maxima")
Output:
-3/8*log(e^(-b*x - a) + 1)/b + 3/8*log(e^(-b*x - a) - 1)/b - 1/4*(3*e^(-b* x - a) - 11*e^(-3*b*x - 3*a) - 11*e^(-5*b*x - 5*a) + 3*e^(-7*b*x - 7*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))
Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (49) = 98\).
Time = 0.11 (sec) , antiderivative size = 110, normalized size of antiderivative = 2.00 \[ \int \text {csch}^5(a+b x) \, dx=\frac {\frac {4 \, {\left (3 \, {\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{3} - 20 \, e^{\left (b x + a\right )} - 20 \, e^{\left (-b x - a\right )}\right )}}{{\left ({\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{2} - 4\right )}^{2}} - 3 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} + 2\right ) + 3 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} - 2\right )}{16 \, b} \] Input:
integrate(csch(b*x+a)^5,x, algorithm="giac")
Output:
1/16*(4*(3*(e^(b*x + a) + e^(-b*x - a))^3 - 20*e^(b*x + a) - 20*e^(-b*x - a))/((e^(b*x + a) + e^(-b*x - a))^2 - 4)^2 - 3*log(e^(b*x + a) + e^(-b*x - a) + 2) + 3*log(e^(b*x + a) + e^(-b*x - a) - 2))/b
Time = 0.06 (sec) , antiderivative size = 193, normalized size of antiderivative = 3.51 \[ \int \text {csch}^5(a+b x) \, dx=\frac {3\,{\mathrm {e}}^{a+b\,x}}{4\,b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}-\frac {{\mathrm {e}}^{a+b\,x}}{2\,b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}-3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}-1\right )}-\frac {4\,{\mathrm {e}}^{3\,a+3\,b\,x}}{b\,\left (6\,{\mathrm {e}}^{4\,a+4\,b\,x}-4\,{\mathrm {e}}^{2\,a+2\,b\,x}-4\,{\mathrm {e}}^{6\,a+6\,b\,x}+{\mathrm {e}}^{8\,a+8\,b\,x}+1\right )}-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{4\,\sqrt {-b^2}} \] Input:
int(1/sinh(a + b*x)^5,x)
Output:
(3*exp(a + b*x))/(4*b*(exp(2*a + 2*b*x) - 1)) - exp(a + b*x)/(2*b*(exp(4*a + 4*b*x) - 2*exp(2*a + 2*b*x) + 1)) - (2*exp(a + b*x))/(b*(3*exp(2*a + 2* b*x) - 3*exp(4*a + 4*b*x) + exp(6*a + 6*b*x) - 1)) - (4*exp(3*a + 3*b*x))/ (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)) - (3*atan((exp(b*x)*exp(a)*(-b^2)^(1/2))/b))/(4*(-b^2)^(1/ 2))
Time = 0.23 (sec) , antiderivative size = 301, normalized size of antiderivative = 5.47 \[ \int \text {csch}^5(a+b x) \, dx=\frac {3 e^{8 b x +8 a} \mathrm {log}\left (e^{b x +a}-1\right )-3 e^{8 b x +8 a} \mathrm {log}\left (e^{b x +a}+1\right )+6 e^{7 b x +7 a}-12 e^{6 b x +6 a} \mathrm {log}\left (e^{b x +a}-1\right )+12 e^{6 b x +6 a} \mathrm {log}\left (e^{b x +a}+1\right )-22 e^{5 b x +5 a}+18 e^{4 b x +4 a} \mathrm {log}\left (e^{b x +a}-1\right )-18 e^{4 b x +4 a} \mathrm {log}\left (e^{b x +a}+1\right )-22 e^{3 b x +3 a}-12 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}-1\right )+12 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}+1\right )+6 e^{b x +a}+3 \,\mathrm {log}\left (e^{b x +a}-1\right )-3 \,\mathrm {log}\left (e^{b x +a}+1\right )}{8 b \left (e^{8 b x +8 a}-4 e^{6 b x +6 a}+6 e^{4 b x +4 a}-4 e^{2 b x +2 a}+1\right )} \] Input:
int(csch(b*x+a)^5,x)
Output:
(3*e**(8*a + 8*b*x)*log(e**(a + b*x) - 1) - 3*e**(8*a + 8*b*x)*log(e**(a + b*x) + 1) + 6*e**(7*a + 7*b*x) - 12*e**(6*a + 6*b*x)*log(e**(a + b*x) - 1 ) + 12*e**(6*a + 6*b*x)*log(e**(a + b*x) + 1) - 22*e**(5*a + 5*b*x) + 18*e **(4*a + 4*b*x)*log(e**(a + b*x) - 1) - 18*e**(4*a + 4*b*x)*log(e**(a + b* x) + 1) - 22*e**(3*a + 3*b*x) - 12*e**(2*a + 2*b*x)*log(e**(a + b*x) - 1) + 12*e**(2*a + 2*b*x)*log(e**(a + b*x) + 1) + 6*e**(a + b*x) + 3*log(e**(a + b*x) - 1) - 3*log(e**(a + b*x) + 1))/(8*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))