Integrand size = 15, antiderivative size = 89 \[ \int \frac {\text {csch}^6(x)}{a+b \cosh ^2(x)} \, dx=-\frac {b^3 \text {arctanh}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{\sqrt {a} (a+b)^{7/2}}-\frac {\left (a^2+3 a b+3 b^2\right ) \coth (x)}{(a+b)^3}+\frac {(2 a+3 b) \coth ^3(x)}{3 (a+b)^2}-\frac {\coth ^5(x)}{5 (a+b)} \]
-(a^2+3*a*b+3*b^2)*coth(x)/(a+b)^3+1/3*(2*a+3*b)*coth(x)^3/(a+b)^2-1/5*cot h(x)^5/(a+b)-b^3*arctanh(a^(1/2)*tanh(x)/(a+b)^(1/2))/(a+b)^(7/2)/a^(1/2)
Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.03 \[ \int \frac {\text {csch}^6(x)}{a+b \cosh ^2(x)} \, dx=-\frac {b^3 \text {arctanh}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{\sqrt {a} (a+b)^{7/2}}-\frac {\coth (x) \left (8 a^2+26 a b+33 b^2-\left (4 a^2+13 a b+9 b^2\right ) \text {csch}^2(x)+3 (a+b)^2 \text {csch}^4(x)\right )}{15 (a+b)^3} \]
-((b^3*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(7/2))) - (Coth[x]*(8*a^2 + 26*a*b + 33*b^2 - (4*a^2 + 13*a*b + 9*b^2)*Csch[x]^2 + 3 *(a + b)^2*Csch[x]^4))/(15*(a + b)^3)
Time = 0.33 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 25, 3670, 300, 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.
| \(\displaystyle \int \frac {\text {csch}^6(x)}{a+b \cosh ^2(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\cos \left (\frac {\pi }{2}+i x\right )^6 \left (a+b \sin \left (\frac {\pi }{2}+i x\right )^2\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\cos \left (i x+\frac {\pi }{2}\right )^6 \left (b \sin \left (i x+\frac {\pi }{2}\right )^2+a\right )}dx\) |
| \(\Big \downarrow \) 3670 |
| \(\displaystyle -\int \frac {\left (1-\coth ^2(x)\right )^3}{a-(a+b) \coth ^2(x)}d\coth (x)\) |
| \(\Big \downarrow \) 300 |
| \(\displaystyle -\int \left (\frac {\coth ^4(x)}{a+b}-\frac {(2 a+3 b) \coth ^2(x)}{(a+b)^2}+\frac {a^2+3 b a+3 b^2}{(a+b)^3}+\frac {b^3}{(a+b)^3 \left (a-(a+b) \coth ^2(x)\right )}\right )d\coth (x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (a^2+3 a b+3 b^2\right ) \coth (x)}{(a+b)^3}-\frac {b^3 \text {arctanh}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^{7/2}}-\frac {\coth ^5(x)}{5 (a+b)}+\frac {(2 a+3 b) \coth ^3(x)}{3 (a+b)^2}\) |
-((b^3*ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a]])/(Sqrt[a]*(a + b)^(7/2))) - ((a^2 + 3*a*b + 3*b^2)*Coth[x])/(a + b)^3 + ((2*a + 3*b)*Coth[x]^3)/(3*(a + b)^2) - Coth[x]^5/(5*(a + b))
3.1.18.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int [PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c , d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Su bst[Int[(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(247\) vs. \(2(77)=154\).
Time = 54.96 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.79
| method | result | size |
| default | \(-\frac {\frac {a^{2} \tanh \left (\frac {x}{2}\right )^{5}}{5}+\frac {2 a b \tanh \left (\frac {x}{2}\right )^{5}}{5}+\frac {b^{2} \tanh \left (\frac {x}{2}\right )^{5}}{5}-\frac {5 a^{2} \tanh \left (\frac {x}{2}\right )^{3}}{3}-\frac {14 a b \tanh \left (\frac {x}{2}\right )^{3}}{3}-3 b^{2} \tanh \left (\frac {x}{2}\right )^{3}+10 a^{2} \tanh \left (\frac {x}{2}\right )+32 a b \tanh \left (\frac {x}{2}\right )+38 b^{2} \tanh \left (\frac {x}{2}\right )}{32 \left (a +b \right )^{3}}-\frac {1}{160 \left (a +b \right ) \tanh \left (\frac {x}{2}\right )^{5}}-\frac {-5 a -9 b}{96 \left (a +b \right )^{2} \tanh \left (\frac {x}{2}\right )^{3}}-\frac {10 a^{2}+32 a b +38 b^{2}}{32 \left (a +b \right )^{3} \tanh \left (\frac {x}{2}\right )}+\frac {2 b^{3} \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )^{2}+2 \tanh \left (\frac {x}{2}\right ) \sqrt {a}+\sqrt {a +b}\right )}{4 \sqrt {a}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )^{2}-2 \tanh \left (\frac {x}{2}\right ) \sqrt {a}+\sqrt {a +b}\right )}{4 \sqrt {a}\, \sqrt {a +b}}\right )}{\left (a +b \right )^{3}}\) | \(248\) |
| risch | \(-\frac {2 \left (15 b^{2} {\mathrm e}^{8 x}-30 a b \,{\mathrm e}^{6 x}-90 b^{2} {\mathrm e}^{6 x}+80 a^{2} {\mathrm e}^{4 x}+230 a b \,{\mathrm e}^{4 x}+240 b^{2} {\mathrm e}^{4 x}-40 a^{2} {\mathrm e}^{2 x}-130 b \,{\mathrm e}^{2 x} a -150 b^{2} {\mathrm e}^{2 x}+8 a^{2}+26 a b +33 b^{2}\right )}{15 \left ({\mathrm e}^{2 x}-1\right )^{5} \left (a +b \right )^{3}}+\frac {b^{3} \ln \left ({\mathrm e}^{2 x}+\frac {2 a \sqrt {a^{2}+a b}+b \sqrt {a^{2}+a b}+2 a^{2}+2 a b}{b \sqrt {a^{2}+a b}}\right )}{2 \sqrt {a^{2}+a b}\, \left (a +b \right )^{3}}-\frac {b^{3} \ln \left ({\mathrm e}^{2 x}+\frac {2 a \sqrt {a^{2}+a b}+b \sqrt {a^{2}+a b}-2 a^{2}-2 a b}{b \sqrt {a^{2}+a b}}\right )}{2 \sqrt {a^{2}+a b}\, \left (a +b \right )^{3}}\) | \(252\) |
-1/32/(a+b)^3*(1/5*a^2*tanh(1/2*x)^5+2/5*a*b*tanh(1/2*x)^5+1/5*b^2*tanh(1/ 2*x)^5-5/3*a^2*tanh(1/2*x)^3-14/3*a*b*tanh(1/2*x)^3-3*b^2*tanh(1/2*x)^3+10 *a^2*tanh(1/2*x)+32*a*b*tanh(1/2*x)+38*b^2*tanh(1/2*x))-1/160/(a+b)/tanh(1 /2*x)^5-1/96*(-5*a-9*b)/(a+b)^2/tanh(1/2*x)^3-1/32/(a+b)^3*(10*a^2+32*a*b+ 38*b^2)/tanh(1/2*x)+2*b^3/(a+b)^3*(-1/4/a^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2) *tanh(1/2*x)^2+2*tanh(1/2*x)*a^(1/2)+(a+b)^(1/2))+1/4/a^(1/2)/(a+b)^(1/2)* ln((a+b)^(1/2)*tanh(1/2*x)^2-2*tanh(1/2*x)*a^(1/2)+(a+b)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 2408 vs. \(2 (77) = 154\).
Time = 0.31 (sec) , antiderivative size = 4977, normalized size of antiderivative = 55.92 \[ \int \frac {\text {csch}^6(x)}{a+b \cosh ^2(x)} \, dx=\text {Too large to display} \]
[-1/30*(60*(a^2*b^2 + a*b^3)*cosh(x)^8 + 480*(a^2*b^2 + a*b^3)*cosh(x)*sin h(x)^7 + 60*(a^2*b^2 + a*b^3)*sinh(x)^8 - 120*(a^3*b + 4*a^2*b^2 + 3*a*b^3 )*cosh(x)^6 - 120*(a^3*b + 4*a^2*b^2 + 3*a*b^3 - 14*(a^2*b^2 + a*b^3)*cosh (x)^2)*sinh(x)^6 + 240*(14*(a^2*b^2 + a*b^3)*cosh(x)^3 - 3*(a^3*b + 4*a^2* b^2 + 3*a*b^3)*cosh(x))*sinh(x)^5 + 40*(8*a^4 + 31*a^3*b + 47*a^2*b^2 + 24 *a*b^3)*cosh(x)^4 + 40*(105*(a^2*b^2 + a*b^3)*cosh(x)^4 + 8*a^4 + 31*a^3*b + 47*a^2*b^2 + 24*a*b^3 - 45*(a^3*b + 4*a^2*b^2 + 3*a*b^3)*cosh(x)^2)*sin h(x)^4 + 32*a^4 + 136*a^3*b + 236*a^2*b^2 + 132*a*b^3 + 160*(21*(a^2*b^2 + a*b^3)*cosh(x)^5 - 15*(a^3*b + 4*a^2*b^2 + 3*a*b^3)*cosh(x)^3 + (8*a^4 + 31*a^3*b + 47*a^2*b^2 + 24*a*b^3)*cosh(x))*sinh(x)^3 - 40*(4*a^4 + 17*a^3* b + 28*a^2*b^2 + 15*a*b^3)*cosh(x)^2 + 40*(42*(a^2*b^2 + a*b^3)*cosh(x)^6 - 45*(a^3*b + 4*a^2*b^2 + 3*a*b^3)*cosh(x)^4 - 4*a^4 - 17*a^3*b - 28*a^2*b ^2 - 15*a*b^3 + 6*(8*a^4 + 31*a^3*b + 47*a^2*b^2 + 24*a*b^3)*cosh(x)^2)*si nh(x)^2 - 15*(b^3*cosh(x)^10 + 10*b^3*cosh(x)*sinh(x)^9 + b^3*sinh(x)^10 - 5*b^3*cosh(x)^8 + 10*b^3*cosh(x)^6 + 5*(9*b^3*cosh(x)^2 - b^3)*sinh(x)^8 + 40*(3*b^3*cosh(x)^3 - b^3*cosh(x))*sinh(x)^7 - 10*b^3*cosh(x)^4 + 10*(21 *b^3*cosh(x)^4 - 14*b^3*cosh(x)^2 + b^3)*sinh(x)^6 + 4*(63*b^3*cosh(x)^5 - 70*b^3*cosh(x)^3 + 15*b^3*cosh(x))*sinh(x)^5 + 5*b^3*cosh(x)^2 + 10*(21*b ^3*cosh(x)^6 - 35*b^3*cosh(x)^4 + 15*b^3*cosh(x)^2 - b^3)*sinh(x)^4 + 40*( 3*b^3*cosh(x)^7 - 7*b^3*cosh(x)^5 + 5*b^3*cosh(x)^3 - b^3*cosh(x))*sinh...
Timed out. \[ \int \frac {\text {csch}^6(x)}{a+b \cosh ^2(x)} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 307 vs. \(2 (77) = 154\).
Time = 0.31 (sec) , antiderivative size = 307, normalized size of antiderivative = 3.45 \[ \int \frac {\text {csch}^6(x)}{a+b \cosh ^2(x)} \, dx=\frac {b^{3} \log \left (\frac {b e^{\left (-2 \, x\right )} + 2 \, a + b - 2 \, \sqrt {{\left (a + b\right )} a}}{b e^{\left (-2 \, x\right )} + 2 \, a + b + 2 \, \sqrt {{\left (a + b\right )} a}}\right )}{2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {{\left (a + b\right )} a}} - \frac {2 \, {\left (15 \, b^{2} e^{\left (-8 \, x\right )} + 8 \, a^{2} + 26 \, a b + 33 \, b^{2} - 10 \, {\left (4 \, a^{2} + 13 \, a b + 15 \, b^{2}\right )} e^{\left (-2 \, x\right )} + 10 \, {\left (8 \, a^{2} + 23 \, a b + 24 \, b^{2}\right )} e^{\left (-4 \, x\right )} - 30 \, {\left (a b + 3 \, b^{2}\right )} e^{\left (-6 \, x\right )}\right )}}{15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3} - 5 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} e^{\left (-2 \, x\right )} + 10 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} e^{\left (-4 \, x\right )} - 10 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} e^{\left (-6 \, x\right )} + 5 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} e^{\left (-8 \, x\right )} - {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} e^{\left (-10 \, x\right )}\right )}} \]
1/2*b^3*log((b*e^(-2*x) + 2*a + b - 2*sqrt((a + b)*a))/(b*e^(-2*x) + 2*a + b + 2*sqrt((a + b)*a)))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sqrt((a + b)*a)) - 2/15*(15*b^2*e^(-8*x) + 8*a^2 + 26*a*b + 33*b^2 - 10*(4*a^2 + 13*a*b + 15*b^2)*e^(-2*x) + 10*(8*a^2 + 23*a*b + 24*b^2)*e^(-4*x) - 30*(a*b + 3*b^2 )*e^(-6*x))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3 - 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*e^(-2*x) + 10*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*e^(-4*x) - 10*(a^3 + 3* a^2*b + 3*a*b^2 + b^3)*e^(-6*x) + 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*e^(-8* x) - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*e^(-10*x))
\[ \int \frac {\text {csch}^6(x)}{a+b \cosh ^2(x)} \, dx=\int { \frac {\operatorname {csch}\left (x\right )^{6}}{b \cosh \left (x\right )^{2} + a} \,d x } \]
Time = 2.50 (sec) , antiderivative size = 333, normalized size of antiderivative = 3.74 \[ \int \frac {\text {csch}^6(x)}{a+b \cosh ^2(x)} \, dx=\frac {4\,\left (b^2+a\,b\right )}{{\left (a+b\right )}^3\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}-\frac {16}{\left (a+b\right )\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}-\frac {2\,b^2}{{\left (a+b\right )}^3\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {32}{5\,\left (a+b\right )\,\left (5\,{\mathrm {e}}^{2\,x}-10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}-5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}-1\right )}-\frac {8\,\left (4\,a+3\,b\right )}{3\,{\left (a+b\right )}^2\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}+\frac {b^3\,\ln \left (\frac {4\,b^4\,\left (2\,a\,b+8\,a^2\,{\mathrm {e}}^{2\,x}+b^2\,{\mathrm {e}}^{2\,x}+b^2+8\,a\,b\,{\mathrm {e}}^{2\,x}\right )}{a\,{\left (a+b\right )}^7}-\frac {8\,b^4\,\left (b+4\,a\,{\mathrm {e}}^{2\,x}+2\,b\,{\mathrm {e}}^{2\,x}\right )}{\sqrt {a}\,{\left (a+b\right )}^{13/2}}\right )}{2\,\sqrt {a}\,{\left (a+b\right )}^{7/2}}-\frac {b^3\,\ln \left (\frac {8\,b^4\,\left (b+4\,a\,{\mathrm {e}}^{2\,x}+2\,b\,{\mathrm {e}}^{2\,x}\right )}{\sqrt {a}\,{\left (a+b\right )}^{13/2}}+\frac {4\,b^4\,\left (2\,a\,b+8\,a^2\,{\mathrm {e}}^{2\,x}+b^2\,{\mathrm {e}}^{2\,x}+b^2+8\,a\,b\,{\mathrm {e}}^{2\,x}\right )}{a\,{\left (a+b\right )}^7}\right )}{2\,\sqrt {a}\,{\left (a+b\right )}^{7/2}} \]
(4*(a*b + b^2))/((a + b)^3*(exp(4*x) - 2*exp(2*x) + 1)) - 16/((a + b)*(6*e xp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1)) - (2*b^2)/((a + b)^3*(e xp(2*x) - 1)) - 32/(5*(a + b)*(5*exp(2*x) - 10*exp(4*x) + 10*exp(6*x) - 5* exp(8*x) + exp(10*x) - 1)) - (8*(4*a + 3*b))/(3*(a + b)^2*(3*exp(2*x) - 3* exp(4*x) + exp(6*x) - 1)) + (b^3*log((4*b^4*(2*a*b + 8*a^2*exp(2*x) + b^2* exp(2*x) + b^2 + 8*a*b*exp(2*x)))/(a*(a + b)^7) - (8*b^4*(b + 4*a*exp(2*x) + 2*b*exp(2*x)))/(a^(1/2)*(a + b)^(13/2))))/(2*a^(1/2)*(a + b)^(7/2)) - ( b^3*log((8*b^4*(b + 4*a*exp(2*x) + 2*b*exp(2*x)))/(a^(1/2)*(a + b)^(13/2)) + (4*b^4*(2*a*b + 8*a^2*exp(2*x) + b^2*exp(2*x) + b^2 + 8*a*b*exp(2*x)))/ (a*(a + b)^7)))/(2*a^(1/2)*(a + b)^(7/2))