Integrand size = 15, antiderivative size = 55 \[ \int \coth ^4(a+b x) \text {csch}(a+b x) \, dx=-\frac {3 \text {arctanh}(\cosh (a+b x))}{8 b}-\frac {3 \coth (a+b x) \text {csch}(a+b x)}{8 b}-\frac {\coth ^3(a+b x) \text {csch}(a+b x)}{4 b} \]
Leaf count is larger than twice the leaf count of optimal. \(113\) vs. \(2(55)=110\).
Time = 0.11 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.05 \[ \int \coth ^4(a+b x) \text {csch}(a+b x) \, dx=-\frac {5 \text {csch}^2\left (\frac {1}{2} (a+b x)\right )}{32 b}-\frac {\text {csch}^4\left (\frac {1}{2} (a+b x)\right )}{64 b}-\frac {3 \log \left (\cosh \left (\frac {1}{2} (a+b x)\right )\right )}{8 b}+\frac {3 \log \left (\sinh \left (\frac {1}{2} (a+b x)\right )\right )}{8 b}-\frac {5 \text {sech}^2\left (\frac {1}{2} (a+b x)\right )}{32 b}+\frac {\text {sech}^4\left (\frac {1}{2} (a+b x)\right )}{64 b} \]
(-5*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) - (5*Sech[(a + b* x)/2]^2)/(32*b) + Sech[(a + b*x)/2]^4/(64*b)
Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.27, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {3042, 26, 3091, 26, 3042, 26, 3091, 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.
\(\displaystyle \int \coth ^4(a+b x) \text {csch}(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int i \tan \left (i a+i b x-\frac {\pi }{2}\right )^4 \sec \left (i a+i b x-\frac {\pi }{2}\right )dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \sec \left (\frac {1}{2} (2 i a-\pi )+i b x\right ) \tan \left (\frac {1}{2} (2 i a-\pi )+i b x\right )^4dx\) |
\(\Big \downarrow \) 3091 |
\(\displaystyle i \left (\frac {i \coth ^3(a+b x) \text {csch}(a+b x)}{4 b}-\frac {3}{4} \int i \coth ^2(a+b x) \text {csch}(a+b x)dx\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (\frac {i \coth ^3(a+b x) \text {csch}(a+b x)}{4 b}-\frac {3}{4} i \int \coth ^2(a+b x) \text {csch}(a+b x)dx\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle i \left (\frac {i \coth ^3(a+b x) \text {csch}(a+b x)}{4 b}-\frac {3}{4} i \int -i \sec \left (i a+i b x-\frac {\pi }{2}\right ) \tan \left (i a+i b x-\frac {\pi }{2}\right )^2dx\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (\frac {i \coth ^3(a+b x) \text {csch}(a+b x)}{4 b}-\frac {3}{4} \int \sec \left (\frac {1}{2} (2 i a-\pi )+i b x\right ) \tan \left (\frac {1}{2} (2 i a-\pi )+i b x\right )^2dx\right )\) |
\(\Big \downarrow \) 3091 |
\(\displaystyle i \left (\frac {i \coth ^3(a+b x) \text {csch}(a+b x)}{4 b}-\frac {3}{4} \left (-\frac {1}{2} \int -i \text {csch}(a+b x)dx-\frac {i \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (\frac {i \coth ^3(a+b x) \text {csch}(a+b x)}{4 b}-\frac {3}{4} \left (\frac {1}{2} i \int \text {csch}(a+b x)dx-\frac {i \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle i \left (\frac {i \coth ^3(a+b x) \text {csch}(a+b x)}{4 b}-\frac {3}{4} \left (\frac {1}{2} i \int i \csc (i a+i b x)dx-\frac {i \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (\frac {i \coth ^3(a+b x) \text {csch}(a+b x)}{4 b}-\frac {3}{4} \left (-\frac {1}{2} \int \csc (i a+i b x)dx-\frac {i \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\right )\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle i \left (\frac {i \coth ^3(a+b x) \text {csch}(a+b x)}{4 b}-\frac {3}{4} \left (-\frac {i \text {arctanh}(\cosh (a+b x))}{2 b}-\frac {i \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\right )\) |
I*(((I/4)*Coth[a + b*x]^3*Csch[a + b*x])/b - (3*(((-1/2*I)*ArcTanh[Cosh[a + b*x]])/b - ((I/2)*Coth[a + b*x]*Csch[a + b*x])/b))/4)
3.2.23.3.1 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[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1)) Int[(a*Sec[e + f*x])^m*( b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & & NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.97 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.35
method | result | size |
derivativedivides | \(\frac {-\frac {\cosh \left (b x +a \right )^{3}}{\sinh \left (b x +a \right )^{4}}+\frac {\cosh \left (b x +a \right )}{\sinh \left (b x +a \right )^{4}}+\left (-\frac {\operatorname {csch}\left (b x +a \right )^{3}}{4}+\frac {3 \,\operatorname {csch}\left (b x +a \right )}{8}\right ) \coth \left (b x +a \right )-\frac {3 \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{4}}{b}\) | \(74\) |
default | \(\frac {-\frac {\cosh \left (b x +a \right )^{3}}{\sinh \left (b x +a \right )^{4}}+\frac {\cosh \left (b x +a \right )}{\sinh \left (b x +a \right )^{4}}+\left (-\frac {\operatorname {csch}\left (b x +a \right )^{3}}{4}+\frac {3 \,\operatorname {csch}\left (b x +a \right )}{8}\right ) \coth \left (b x +a \right )-\frac {3 \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{4}}{b}\) | \(74\) |
risch | \(-\frac {{\mathrm e}^{b x +a} \left (5 \,{\mathrm e}^{6 b x +6 a}+3 \,{\mathrm e}^{4 b x +4 a}+3 \,{\mathrm e}^{2 b x +2 a}+5\right )}{4 b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{4}}+\frac {3 \ln \left ({\mathrm e}^{b x +a}-1\right )}{8 b}-\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right )}{8 b}\) | \(89\) |
1/b*(-cosh(b*x+a)^3/sinh(b*x+a)^4+1/sinh(b*x+a)^4*cosh(b*x+a)+(-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 = 0.26 (sec) , antiderivative size = 1114, normalized size of antiderivative = 20.25 \[ \int \coth ^4(a+b x) \text {csch}(a+b x) \, dx=\text {Too large to display} \]
-1/8*(10*cosh(b*x + a)^7 + 70*cosh(b*x + a)*sinh(b*x + a)^6 + 10*sinh(b*x + a)^7 + 6*(35*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^5 + 6*cosh(b*x + a)^5 + 10*(35*cosh(b*x + a)^3 + 3*cosh(b*x + a))*sinh(b*x + a)^4 + 2*(175*cosh(b* x + a)^4 + 30*cosh(b*x + a)^2 + 3)*sinh(b*x + a)^3 + 6*cosh(b*x + a)^3 + 6 *(35*cosh(b*x + a)^5 + 10*cosh(b*x + a)^3 + 3*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*c osh(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) + sinh(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*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)*sin h(b*x + a)^2 - 4*cosh(b*x + a)^2 + 8*(cosh(b*x + a)^7 - 3*cosh(b*x + a)...
\[ \int \coth ^4(a+b x) \text {csch}(a+b x) \, dx=\int \coth ^{4}{\left (a + b x \right )} \operatorname {csch}{\left (a + b x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (49) = 98\).
Time = 0.19 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.42 \[ \int \coth ^4(a+b x) \text {csch}(a+b x) \, dx=-\frac {3 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{8 \, b} + \frac {3 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{8 \, b} + \frac {5 \, e^{\left (-b x - a\right )} + 3 \, e^{\left (-3 \, b x - 3 \, a\right )} + 3 \, e^{\left (-5 \, b x - 5 \, a\right )} + 5 \, e^{\left (-7 \, b x - 7 \, a\right )}}{4 \, b {\left (4 \, e^{\left (-2 \, b x - 2 \, a\right )} - 6 \, e^{\left (-4 \, b x - 4 \, a\right )} + 4 \, e^{\left (-6 \, b x - 6 \, a\right )} - e^{\left (-8 \, b x - 8 \, a\right )} - 1\right )}} \]
-3/8*log(e^(-b*x - a) + 1)/b + 3/8*log(e^(-b*x - a) - 1)/b + 1/4*(5*e^(-b* x - a) + 3*e^(-3*b*x - 3*a) + 3*e^(-5*b*x - 5*a) + 5*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 = 0.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 2.00 \[ \int \coth ^4(a+b x) \text {csch}(a+b x) \, dx=-\frac {\frac {4 \, {\left (5 \, {\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{3} - 12 \, e^{\left (b x + a\right )} - 12 \, e^{\left (-b x - a\right )}\right )}}{{\left ({\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{2} - 4\right )}^{2}} + 3 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} + 2\right ) - 3 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} - 2\right )}{16 \, b} \]
-1/16*(4*(5*(e^(b*x + a) + e^(-b*x - a))^3 - 12*e^(b*x + a) - 12*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 = 2.05 (sec) , antiderivative size = 190, normalized size of antiderivative = 3.45 \[ \int \coth ^4(a+b x) \text {csch}(a+b x) \, dx=-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{4\,\sqrt {-b^2}}-\frac {9\,{\mathrm {e}}^{a+b\,x}}{2\,b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {6\,{\mathrm {e}}^{a+b\,x}}{b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}-3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}-1\right )}-\frac {4\,{\mathrm {e}}^{a+b\,x}}{b\,\left (6\,{\mathrm {e}}^{4\,a+4\,b\,x}-4\,{\mathrm {e}}^{2\,a+2\,b\,x}-4\,{\mathrm {e}}^{6\,a+6\,b\,x}+{\mathrm {e}}^{8\,a+8\,b\,x}+1\right )}-\frac {5\,{\mathrm {e}}^{a+b\,x}}{4\,b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \]
- (3*atan((exp(b*x)*exp(a)*(-b^2)^(1/2))/b))/(4*(-b^2)^(1/2)) - (9*exp(a + b*x))/(2*b*(exp(4*a + 4*b*x) - 2*exp(2*a + 2*b*x) + 1)) - (6*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(a + 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)) - (5*exp(a + b*x))/(4*b*(exp(2*a + 2*b*x) - 1))