Integrand size = 17, antiderivative size = 69 \[ \int \text {csch}^5(a+b x) \text {sech}^5(a+b x) \, dx=\frac {2 \coth ^2(a+b x)}{b}-\frac {\coth ^4(a+b x)}{4 b}+\frac {6 \log (\tanh (a+b x))}{b}-\frac {2 \tanh ^2(a+b x)}{b}+\frac {\tanh ^4(a+b x)}{4 b} \]
2*coth(b*x+a)^2/b-1/4*coth(b*x+a)^4/b+6*ln(tanh(b*x+a))/b-2*tanh(b*x+a)^2/ b+1/4*tanh(b*x+a)^4/b
Time = 0.06 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.32 \[ \int \text {csch}^5(a+b x) \text {sech}^5(a+b x) \, dx=32 \left (\frac {3 \text {csch}^2(a+b x)}{64 b}-\frac {\text {csch}^4(a+b x)}{128 b}-\frac {3 \log (\cosh (a+b x))}{16 b}+\frac {3 \log (\sinh (a+b x))}{16 b}+\frac {3 \text {sech}^2(a+b x)}{64 b}+\frac {\text {sech}^4(a+b x)}{128 b}\right ) \]
32*((3*Csch[a + b*x]^2)/(64*b) - Csch[a + b*x]^4/(128*b) - (3*Log[Cosh[a + b*x]])/(16*b) + (3*Log[Sinh[a + b*x]])/(16*b) + (3*Sech[a + b*x]^2)/(64*b ) + Sech[a + b*x]^4/(128*b))
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3042, 26, 3100, 243, 49, 2009}
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.
\(\displaystyle \int \text {csch}^5(a+b x) \text {sech}^5(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int i \csc (i a+i b x)^5 \sec (i a+i b x)^5dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \csc (i a+i b x)^5 \sec (i a+i b x)^5dx\) |
\(\Big \downarrow \) 3100 |
\(\displaystyle \frac {\int -i \coth ^5(a+b x) \left (1-\tanh ^2(a+b x)\right )^4d(i \tanh (a+b x))}{b}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\int i \coth ^3(a+b x) \left (1-\tanh ^2(a+b x)\right )^4d\left (-\tanh ^2(a+b x)\right )}{2 b}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\int \left (i \coth ^3(a+b x)-4 \coth ^2(a+b x)-6 i \coth (a+b x)-\tanh ^2(a+b x)+4\right )d\left (-\tanh ^2(a+b x)\right )}{2 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {1}{2} \tanh ^2(a+b x)+4 i \tanh (a+b x)+\frac {1}{2} \coth ^2(a+b x)+4 i \coth (a+b x)+6 \log \left (-\tanh ^2(a+b x)\right )}{2 b}\) |
((4*I)*Coth[a + b*x] + Coth[a + b*x]^2/2 + 6*Log[-Tanh[a + b*x]^2] + (4*I) *Tanh[a + b*x] - Tanh[a + b*x]^2/2)/(2*b)
3.1.48.3.1 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_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Simp[1/f Subst[Int[(1 + x^2)^((m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]] , x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]
Time = 242.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.01
method | result | size |
derivativedivides | \(\frac {-\frac {1}{4 \sinh \left (b x +a \right )^{4} \cosh \left (b x +a \right )^{4}}+\frac {1}{\sinh \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{4}}+\frac {3}{2 \cosh \left (b x +a \right )^{4}}+\frac {3}{\cosh \left (b x +a \right )^{2}}+6 \ln \left (\tanh \left (b x +a \right )\right )}{b}\) | \(70\) |
default | \(\frac {-\frac {1}{4 \sinh \left (b x +a \right )^{4} \cosh \left (b x +a \right )^{4}}+\frac {1}{\sinh \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{4}}+\frac {3}{2 \cosh \left (b x +a \right )^{4}}+\frac {3}{\cosh \left (b x +a \right )^{2}}+6 \ln \left (\tanh \left (b x +a \right )\right )}{b}\) | \(70\) |
risch | \(\frac {4 \,{\mathrm e}^{2 b x +2 a} \left (3 \,{\mathrm e}^{12 b x +12 a}-11 \,{\mathrm e}^{8 b x +8 a}-11 \,{\mathrm e}^{4 b x +4 a}+3\right )}{b \left (1+{\mathrm e}^{2 b x +2 a}\right )^{4} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{4}}+\frac {6 \ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b}-\frac {6 \ln \left (1+{\mathrm e}^{2 b x +2 a}\right )}{b}\) | \(111\) |
1/b*(-1/4/sinh(b*x+a)^4/cosh(b*x+a)^4+1/sinh(b*x+a)^2/cosh(b*x+a)^4+3/2/co sh(b*x+a)^4+3/cosh(b*x+a)^2+6*ln(tanh(b*x+a)))
Leaf count of result is larger than twice the leaf count of optimal. 2231 vs. \(2 (65) = 130\).
Time = 0.25 (sec) , antiderivative size = 2231, normalized size of antiderivative = 32.33 \[ \int \text {csch}^5(a+b x) \text {sech}^5(a+b x) \, dx=\text {Too large to display} \]
2*(6*cosh(b*x + a)^14 + 2184*cosh(b*x + a)^3*sinh(b*x + a)^11 + 546*cosh(b *x + a)^2*sinh(b*x + a)^12 + 84*cosh(b*x + a)*sinh(b*x + a)^13 + 6*sinh(b* x + a)^14 + 22*(273*cosh(b*x + a)^4 - 1)*sinh(b*x + a)^10 - 22*cosh(b*x + a)^10 + 44*(273*cosh(b*x + a)^5 - 5*cosh(b*x + a))*sinh(b*x + a)^9 + 198*( 91*cosh(b*x + a)^6 - 5*cosh(b*x + a)^2)*sinh(b*x + a)^8 + 528*(39*cosh(b*x + a)^7 - 5*cosh(b*x + a)^3)*sinh(b*x + a)^7 + 22*(819*cosh(b*x + a)^8 - 2 10*cosh(b*x + a)^4 - 1)*sinh(b*x + a)^6 - 22*cosh(b*x + a)^6 + 132*(91*cos h(b*x + a)^9 - 42*cosh(b*x + a)^5 - cosh(b*x + a))*sinh(b*x + a)^5 + 66*(9 1*cosh(b*x + a)^10 - 70*cosh(b*x + a)^6 - 5*cosh(b*x + a)^2)*sinh(b*x + a) ^4 + 8*(273*cosh(b*x + a)^11 - 330*cosh(b*x + a)^7 - 55*cosh(b*x + a)^3)*s inh(b*x + a)^3 + 6*(91*cosh(b*x + a)^12 - 165*cosh(b*x + a)^8 - 55*cosh(b* x + a)^4 + 1)*sinh(b*x + a)^2 + 6*cosh(b*x + a)^2 - 3*(cosh(b*x + a)^16 + 560*cosh(b*x + a)^3*sinh(b*x + a)^13 + 120*cosh(b*x + a)^2*sinh(b*x + a)^1 4 + 16*cosh(b*x + a)*sinh(b*x + a)^15 + sinh(b*x + a)^16 + 4*(455*cosh(b*x + a)^4 - 1)*sinh(b*x + a)^12 - 4*cosh(b*x + a)^12 + 48*(91*cosh(b*x + a)^ 5 - cosh(b*x + a))*sinh(b*x + a)^11 + 88*(91*cosh(b*x + a)^6 - 3*cosh(b*x + a)^2)*sinh(b*x + a)^10 + 880*(13*cosh(b*x + a)^7 - cosh(b*x + a)^3)*sinh (b*x + a)^9 + 6*(2145*cosh(b*x + a)^8 - 330*cosh(b*x + a)^4 + 1)*sinh(b*x + a)^8 + 6*cosh(b*x + a)^8 + 16*(715*cosh(b*x + a)^9 - 198*cosh(b*x + a)^5 + 3*cosh(b*x + a))*sinh(b*x + a)^7 + 56*(143*cosh(b*x + a)^10 - 66*cos...
\[ \int \text {csch}^5(a+b x) \text {sech}^5(a+b x) \, dx=\int \operatorname {csch}^{5}{\left (a + b x \right )} \operatorname {sech}^{5}{\left (a + b x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (65) = 130\).
Time = 0.27 (sec) , antiderivative size = 150, normalized size of antiderivative = 2.17 \[ \int \text {csch}^5(a+b x) \text {sech}^5(a+b x) \, dx=\frac {6 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {6 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{b} - \frac {6 \, \log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}{b} - \frac {4 \, {\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} - 11 \, e^{\left (-6 \, b x - 6 \, a\right )} - 11 \, e^{\left (-10 \, b x - 10 \, a\right )} + 3 \, e^{\left (-14 \, b x - 14 \, a\right )}\right )}}{b {\left (4 \, e^{\left (-4 \, b x - 4 \, a\right )} - 6 \, e^{\left (-8 \, b x - 8 \, a\right )} + 4 \, e^{\left (-12 \, b x - 12 \, a\right )} - e^{\left (-16 \, b x - 16 \, a\right )} - 1\right )}} \]
6*log(e^(-b*x - a) + 1)/b + 6*log(e^(-b*x - a) - 1)/b - 6*log(e^(-2*b*x - 2*a) + 1)/b - 4*(3*e^(-2*b*x - 2*a) - 11*e^(-6*b*x - 6*a) - 11*e^(-10*b*x - 10*a) + 3*e^(-14*b*x - 14*a))/(b*(4*e^(-4*b*x - 4*a) - 6*e^(-8*b*x - 8*a ) + 4*e^(-12*b*x - 12*a) - e^(-16*b*x - 16*a) - 1))
Time = 0.28 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.80 \[ \int \text {csch}^5(a+b x) \text {sech}^5(a+b x) \, dx=\frac {\frac {4 \, {\left (3 \, {\left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )}\right )}^{3} - 20 \, e^{\left (2 \, b x + 2 \, a\right )} - 20 \, e^{\left (-2 \, b x - 2 \, a\right )}\right )}}{{\left ({\left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )}\right )}^{2} - 4\right )}^{2}} - 3 \, \log \left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )} + 2\right ) + 3 \, \log \left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )} - 2\right )}{b} \]
(4*(3*(e^(2*b*x + 2*a) + e^(-2*b*x - 2*a))^3 - 20*e^(2*b*x + 2*a) - 20*e^( -2*b*x - 2*a))/((e^(2*b*x + 2*a) + e^(-2*b*x - 2*a))^2 - 4)^2 - 3*log(e^(2 *b*x + 2*a) + e^(-2*b*x - 2*a) + 2) + 3*log(e^(2*b*x + 2*a) + e^(-2*b*x - 2*a) - 2))/b
Time = 2.20 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.97 \[ \int \text {csch}^5(a+b x) \text {sech}^5(a+b x) \, dx=\frac {12\,{\mathrm {e}}^{2\,a+2\,b\,x}}{b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-1\right )}-\frac {12\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}-\frac {8\,{\mathrm {e}}^{2\,a+2\,b\,x}}{b\,\left ({\mathrm {e}}^{8\,a+8\,b\,x}-2\,{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )}-\frac {32\,{\mathrm {e}}^{2\,a+2\,b\,x}}{b\,\left (3\,{\mathrm {e}}^{4\,a+4\,b\,x}-3\,{\mathrm {e}}^{8\,a+8\,b\,x}+{\mathrm {e}}^{12\,a+12\,b\,x}-1\right )}-\frac {64\,{\mathrm {e}}^{6\,a+6\,b\,x}}{b\,\left (6\,{\mathrm {e}}^{8\,a+8\,b\,x}-4\,{\mathrm {e}}^{4\,a+4\,b\,x}-4\,{\mathrm {e}}^{12\,a+12\,b\,x}+{\mathrm {e}}^{16\,a+16\,b\,x}+1\right )} \]
(12*exp(2*a + 2*b*x))/(b*(exp(4*a + 4*b*x) - 1)) - (12*atan((exp(2*a)*exp( 2*b*x)*(-b^2)^(1/2))/b))/(-b^2)^(1/2) - (8*exp(2*a + 2*b*x))/(b*(exp(8*a + 8*b*x) - 2*exp(4*a + 4*b*x) + 1)) - (32*exp(2*a + 2*b*x))/(b*(3*exp(4*a + 4*b*x) - 3*exp(8*a + 8*b*x) + exp(12*a + 12*b*x) - 1)) - (64*exp(6*a + 6* b*x))/(b*(6*exp(8*a + 8*b*x) - 4*exp(4*a + 4*b*x) - 4*exp(12*a + 12*b*x) + exp(16*a + 16*b*x) + 1))