Integrand size = 13, antiderivative size = 102 \[ \int \frac {\cosh ^5(x)}{a+b \text {csch}(x)} \, dx=-\frac {b \left (a^2+b^2\right )^2 \log (b+a \sinh (x))}{a^6}+\frac {\left (a^2+b^2\right )^2 \sinh (x)}{a^5}-\frac {b \left (2 a^2+b^2\right ) \sinh ^2(x)}{2 a^4}+\frac {\left (2 a^2+b^2\right ) \sinh ^3(x)}{3 a^3}-\frac {b \sinh ^4(x)}{4 a^2}+\frac {\sinh ^5(x)}{5 a} \]
-b*(a^2+b^2)^2*ln(b+a*sinh(x))/a^6+(a^2+b^2)^2*sinh(x)/a^5-1/2*b*(2*a^2+b^ 2)*sinh(x)^2/a^4+1/3*(2*a^2+b^2)*sinh(x)^3/a^3-1/4*b*sinh(x)^4/a^2+1/5*sin h(x)^5/a
Time = 0.20 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.95 \[ \int \frac {\cosh ^5(x)}{a+b \text {csch}(x)} \, dx=\frac {-60 b \left (a^2+b^2\right )^2 \log (b+a \sinh (x))+60 a \left (a^2+b^2\right )^2 \sinh (x)-30 a^2 b \left (2 a^2+b^2\right ) \sinh ^2(x)+20 a^3 \left (2 a^2+b^2\right ) \sinh ^3(x)-15 a^4 b \sinh ^4(x)+12 a^5 \sinh ^5(x)}{60 a^6} \]
(-60*b*(a^2 + b^2)^2*Log[b + a*Sinh[x]] + 60*a*(a^2 + b^2)^2*Sinh[x] - 30* a^2*b*(2*a^2 + b^2)*Sinh[x]^2 + 20*a^3*(2*a^2 + b^2)*Sinh[x]^3 - 15*a^4*b* Sinh[x]^4 + 12*a^5*Sinh[x]^5)/(60*a^6)
Time = 0.42 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {3042, 4360, 26, 26, 3042, 26, 3316, 26, 27, 522, 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.
\(\displaystyle \int \frac {\cosh ^5(x)}{a+b \text {csch}(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (i x)^5}{a+i b \csc (i x)}dx\) |
\(\Big \downarrow \) 4360 |
\(\displaystyle \int \frac {i \sinh (x) \cosh ^5(x)}{i a \sinh (x)+i b}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int -\frac {i \cosh ^5(x) \sinh (x)}{b+a \sinh (x)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \int \frac {\sinh (x) \cosh ^5(x)}{a \sinh (x)+b}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \sin (i x) \cos (i x)^5}{b-i a \sin (i x)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\cos (i x)^5 \sin (i x)}{b-i a \sin (i x)}dx\) |
\(\Big \downarrow \) 3316 |
\(\displaystyle -\frac {i \int \frac {i \sinh (x) \left (\sinh ^2(x) a^2+a^2\right )^2}{b+a \sinh (x)}d(a \sinh (x))}{a^5}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\int \frac {\sinh (x) \left (\sinh ^2(x) a^2+a^2\right )^2}{b+a \sinh (x)}d(a \sinh (x))}{a^5}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {a \sinh (x) \left (\sinh ^2(x) a^2+a^2\right )^2}{b+a \sinh (x)}d(a \sinh (x))}{a^6}\) |
\(\Big \downarrow \) 522 |
\(\displaystyle \frac {\int \left (a^4 \sinh ^4(x)-a^3 b \sinh ^3(x)+a^2 \left (2 a^2+b^2\right ) \sinh ^2(x)-a b \left (2 a^2+b^2\right ) \sinh (x)+\left (a^2+b^2\right )^2-\frac {b \left (a^2+b^2\right )^2}{b+a \sinh (x)}\right )d(a \sinh (x))}{a^6}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{5} a^5 \sinh ^5(x)-\frac {1}{4} a^4 b \sinh ^4(x)-\frac {1}{2} a^2 b \left (2 a^2+b^2\right ) \sinh ^2(x)+a \left (a^2+b^2\right )^2 \sinh (x)-b \left (a^2+b^2\right )^2 \log (a \sinh (x)+b)+\frac {1}{3} a^3 \left (2 a^2+b^2\right ) \sinh ^3(x)}{a^6}\) |
(-(b*(a^2 + b^2)^2*Log[b + a*Sinh[x]]) + a*(a^2 + b^2)^2*Sinh[x] - (a^2*b* (2*a^2 + b^2)*Sinh[x]^2)/2 + (a^3*(2*a^2 + b^2)*Sinh[x]^3)/3 - (a^4*b*Sinh [x]^4)/4 + (a^5*Sinh[x]^5)/5)/a^6
3.1.92.3.1 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_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^m*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x, b* Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1) /2] && NeQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Leaf count of result is larger than twice the leaf count of optimal. \(377\) vs. \(2(94)=188\).
Time = 0.13 (sec) , antiderivative size = 378, normalized size of antiderivative = 3.71
\[-\frac {1}{5 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{5}}-\frac {2 a +b}{4 a^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}-\frac {11 a^{2}+6 a b +4 b^{2}}{12 a^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {7 a^{3}+9 a^{2} b +4 a \,b^{2}+4 b^{3}}{8 a^{4} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {8 a^{4}+7 a^{3} b +16 a^{2} b^{2}+4 a \,b^{3}+8 b^{4}}{8 a^{5} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {b \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a^{6}}-\frac {2 b \left (\frac {1}{2} a^{4}+a^{2} b^{2}+\frac {1}{2} b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2} b -2 a \tanh \left (\frac {x}{2}\right )-b \right )}{a^{6}}-\frac {1}{5 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {-2 a +b}{4 a^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {11 a^{2}-6 a b +4 b^{2}}{12 a^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {-7 a^{3}+9 a^{2} b -4 a \,b^{2}+4 b^{3}}{8 a^{4} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {8 a^{4}-7 a^{3} b +16 a^{2} b^{2}-4 a \,b^{3}+8 b^{4}}{8 a^{5} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {b \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a^{6}}\]
-1/5/a/(tanh(1/2*x)-1)^5-1/4*(2*a+b)/a^2/(tanh(1/2*x)-1)^4-1/12*(11*a^2+6* a*b+4*b^2)/a^3/(tanh(1/2*x)-1)^3-1/8*(7*a^3+9*a^2*b+4*a*b^2+4*b^3)/a^4/(ta nh(1/2*x)-1)^2-1/8*(8*a^4+7*a^3*b+16*a^2*b^2+4*a*b^3+8*b^4)/a^5/(tanh(1/2* x)-1)+b*(a^4+2*a^2*b^2+b^4)/a^6*ln(tanh(1/2*x)-1)-2*b/a^6*(1/2*a^4+a^2*b^2 +1/2*b^4)*ln(tanh(1/2*x)^2*b-2*a*tanh(1/2*x)-b)-1/5/a/(tanh(1/2*x)+1)^5-1/ 4*(-2*a+b)/a^2/(tanh(1/2*x)+1)^4-1/12*(11*a^2-6*a*b+4*b^2)/a^3/(tanh(1/2*x )+1)^3-1/8*(-7*a^3+9*a^2*b-4*a*b^2+4*b^3)/a^4/(tanh(1/2*x)+1)^2-1/8*(8*a^4 -7*a^3*b+16*a^2*b^2-4*a*b^3+8*b^4)/a^5/(tanh(1/2*x)+1)+b*(a^4+2*a^2*b^2+b^ 4)/a^6*ln(tanh(1/2*x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 1398 vs. \(2 (94) = 188\).
Time = 0.26 (sec) , antiderivative size = 1398, normalized size of antiderivative = 13.71 \[ \int \frac {\cosh ^5(x)}{a+b \text {csch}(x)} \, dx=\text {Too large to display} \]
1/960*(6*a^5*cosh(x)^10 + 6*a^5*sinh(x)^10 - 15*a^4*b*cosh(x)^9 + 15*(4*a^ 5*cosh(x) - a^4*b)*sinh(x)^9 + 10*(5*a^5 + 4*a^3*b^2)*cosh(x)^8 + 5*(54*a^ 5*cosh(x)^2 - 27*a^4*b*cosh(x) + 10*a^5 + 8*a^3*b^2)*sinh(x)^8 - 60*(3*a^4 *b + 2*a^2*b^3)*cosh(x)^7 + 20*(36*a^5*cosh(x)^3 - 27*a^4*b*cosh(x)^2 - 9* a^4*b - 6*a^2*b^3 + 4*(5*a^5 + 4*a^3*b^2)*cosh(x))*sinh(x)^7 + 960*(a^4*b + 2*a^2*b^3 + b^5)*x*cosh(x)^5 + 60*(5*a^5 + 14*a^3*b^2 + 8*a*b^4)*cosh(x) ^6 + 20*(63*a^5*cosh(x)^4 - 63*a^4*b*cosh(x)^3 + 15*a^5 + 42*a^3*b^2 + 24* a*b^4 + 14*(5*a^5 + 4*a^3*b^2)*cosh(x)^2 - 21*(3*a^4*b + 2*a^2*b^3)*cosh(x ))*sinh(x)^6 - 15*a^4*b*cosh(x) + 2*(756*a^5*cosh(x)^5 - 945*a^4*b*cosh(x) ^4 + 280*(5*a^5 + 4*a^3*b^2)*cosh(x)^3 - 630*(3*a^4*b + 2*a^2*b^3)*cosh(x) ^2 + 480*(a^4*b + 2*a^2*b^3 + b^5)*x + 180*(5*a^5 + 14*a^3*b^2 + 8*a*b^4)* cosh(x))*sinh(x)^5 - 6*a^5 - 60*(5*a^5 + 14*a^3*b^2 + 8*a*b^4)*cosh(x)^4 + 10*(126*a^5*cosh(x)^6 - 189*a^4*b*cosh(x)^5 - 30*a^5 - 84*a^3*b^2 - 48*a* b^4 + 70*(5*a^5 + 4*a^3*b^2)*cosh(x)^4 - 210*(3*a^4*b + 2*a^2*b^3)*cosh(x) ^3 + 480*(a^4*b + 2*a^2*b^3 + b^5)*x*cosh(x) + 90*(5*a^5 + 14*a^3*b^2 + 8* a*b^4)*cosh(x)^2)*sinh(x)^4 - 60*(3*a^4*b + 2*a^2*b^3)*cosh(x)^3 + 20*(36* a^5*cosh(x)^7 - 63*a^4*b*cosh(x)^6 + 28*(5*a^5 + 4*a^3*b^2)*cosh(x)^5 - 9* a^4*b - 6*a^2*b^3 - 105*(3*a^4*b + 2*a^2*b^3)*cosh(x)^4 + 480*(a^4*b + 2*a ^2*b^3 + b^5)*x*cosh(x)^2 + 60*(5*a^5 + 14*a^3*b^2 + 8*a*b^4)*cosh(x)^3 - 12*(5*a^5 + 14*a^3*b^2 + 8*a*b^4)*cosh(x))*sinh(x)^3 - 10*(5*a^5 + 4*a^...
\[ \int \frac {\cosh ^5(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\cosh ^{5}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (94) = 188\).
Time = 0.18 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.37 \[ \int \frac {\cosh ^5(x)}{a+b \text {csch}(x)} \, dx=-\frac {{\left (15 \, a^{3} b e^{\left (-x\right )} - 6 \, a^{4} - 10 \, {\left (5 \, a^{4} + 4 \, a^{2} b^{2}\right )} e^{\left (-2 \, x\right )} + 60 \, {\left (3 \, a^{3} b + 2 \, a b^{3}\right )} e^{\left (-3 \, x\right )} - 60 \, {\left (5 \, a^{4} + 14 \, a^{2} b^{2} + 8 \, b^{4}\right )} e^{\left (-4 \, x\right )}\right )} e^{\left (5 \, x\right )}}{960 \, a^{5}} - \frac {15 \, a^{3} b e^{\left (-4 \, x\right )} + 6 \, a^{4} e^{\left (-5 \, x\right )} + 60 \, {\left (5 \, a^{4} + 14 \, a^{2} b^{2} + 8 \, b^{4}\right )} e^{\left (-x\right )} + 60 \, {\left (3 \, a^{3} b + 2 \, a b^{3}\right )} e^{\left (-2 \, x\right )} + 10 \, {\left (5 \, a^{4} + 4 \, a^{2} b^{2}\right )} e^{\left (-3 \, x\right )}}{960 \, a^{5}} - \frac {{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} x}{a^{6}} - \frac {{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \log \left (-2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} - a\right )}{a^{6}} \]
-1/960*(15*a^3*b*e^(-x) - 6*a^4 - 10*(5*a^4 + 4*a^2*b^2)*e^(-2*x) + 60*(3* a^3*b + 2*a*b^3)*e^(-3*x) - 60*(5*a^4 + 14*a^2*b^2 + 8*b^4)*e^(-4*x))*e^(5 *x)/a^5 - 1/960*(15*a^3*b*e^(-4*x) + 6*a^4*e^(-5*x) + 60*(5*a^4 + 14*a^2*b ^2 + 8*b^4)*e^(-x) + 60*(3*a^3*b + 2*a*b^3)*e^(-2*x) + 10*(5*a^4 + 4*a^2*b ^2)*e^(-3*x))/a^5 - (a^4*b + 2*a^2*b^3 + b^5)*x/a^6 - (a^4*b + 2*a^2*b^3 + b^5)*log(-2*b*e^(-x) + a*e^(-2*x) - a)/a^6
Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (94) = 188\).
Time = 0.29 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.90 \[ \int \frac {\cosh ^5(x)}{a+b \text {csch}(x)} \, dx=-\frac {6 \, a^{4} {\left (e^{\left (-x\right )} - e^{x}\right )}^{5} + 15 \, a^{3} b {\left (e^{\left (-x\right )} - e^{x}\right )}^{4} + 80 \, a^{4} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 40 \, a^{2} b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 240 \, a^{3} b {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 120 \, a b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 480 \, a^{4} {\left (e^{\left (-x\right )} - e^{x}\right )} + 960 \, a^{2} b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )} + 480 \, b^{4} {\left (e^{\left (-x\right )} - e^{x}\right )}}{960 \, a^{5}} - \frac {{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \log \left ({\left | -a {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b \right |}\right )}{a^{6}} \]
-1/960*(6*a^4*(e^(-x) - e^x)^5 + 15*a^3*b*(e^(-x) - e^x)^4 + 80*a^4*(e^(-x ) - e^x)^3 + 40*a^2*b^2*(e^(-x) - e^x)^3 + 240*a^3*b*(e^(-x) - e^x)^2 + 12 0*a*b^3*(e^(-x) - e^x)^2 + 480*a^4*(e^(-x) - e^x) + 960*a^2*b^2*(e^(-x) - e^x) + 480*b^4*(e^(-x) - e^x))/a^5 - (a^4*b + 2*a^2*b^3 + b^5)*log(abs(-a* (e^(-x) - e^x) + 2*b))/a^6
Time = 2.97 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.24 \[ \int \frac {\cosh ^5(x)}{a+b \text {csch}(x)} \, dx=\frac {{\mathrm {e}}^{5\,x}}{160\,a}-\frac {{\mathrm {e}}^{-5\,x}}{160\,a}-\frac {{\mathrm {e}}^{-2\,x}\,\left (3\,a^2\,b+2\,b^3\right )}{16\,a^4}-\frac {{\mathrm {e}}^{2\,x}\,\left (3\,a^2\,b+2\,b^3\right )}{16\,a^4}+\frac {{\mathrm {e}}^x\,\left (5\,a^4+14\,a^2\,b^2+8\,b^4\right )}{16\,a^5}-\frac {b\,{\mathrm {e}}^{-4\,x}}{64\,a^2}-\frac {b\,{\mathrm {e}}^{4\,x}}{64\,a^2}-\frac {\ln \left (2\,b\,{\mathrm {e}}^x-a+a\,{\mathrm {e}}^{2\,x}\right )\,\left (a^4\,b+2\,a^2\,b^3+b^5\right )}{a^6}-\frac {{\mathrm {e}}^{-x}\,\left (5\,a^4+14\,a^2\,b^2+8\,b^4\right )}{16\,a^5}-\frac {{\mathrm {e}}^{-3\,x}\,\left (5\,a^2+4\,b^2\right )}{96\,a^3}+\frac {{\mathrm {e}}^{3\,x}\,\left (5\,a^2+4\,b^2\right )}{96\,a^3}+\frac {b\,x\,{\left (a^2+b^2\right )}^2}{a^6} \]
exp(5*x)/(160*a) - exp(-5*x)/(160*a) - (exp(-2*x)*(3*a^2*b + 2*b^3))/(16*a ^4) - (exp(2*x)*(3*a^2*b + 2*b^3))/(16*a^4) + (exp(x)*(5*a^4 + 8*b^4 + 14* a^2*b^2))/(16*a^5) - (b*exp(-4*x))/(64*a^2) - (b*exp(4*x))/(64*a^2) - (log (2*b*exp(x) - a + a*exp(2*x))*(a^4*b + b^5 + 2*a^2*b^3))/a^6 - (exp(-x)*(5 *a^4 + 8*b^4 + 14*a^2*b^2))/(16*a^5) - (exp(-3*x)*(5*a^2 + 4*b^2))/(96*a^3 ) + (exp(3*x)*(5*a^2 + 4*b^2))/(96*a^3) + (b*x*(a^2 + b^2)^2)/a^6