Integrand size = 15, antiderivative size = 88 \[ \int \frac {\sinh ^6(x)}{a+b \cosh ^2(x)} \, dx=\frac {\left (8 a^2+20 a b+15 b^2\right ) x}{8 b^3}-\frac {(a+b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{\sqrt {a} b^3}-\frac {(4 a+7 b) \cosh (x) \sinh (x)}{8 b^2}+\frac {\cosh (x) \sinh ^3(x)}{4 b} \]
1/8*(8*a^2+20*a*b+15*b^2)*x/b^3-1/8*(4*a+7*b)*cosh(x)*sinh(x)/b^2+1/4*cosh (x)*sinh(x)^3/b-(a+b)^(5/2)*arctanh(a^(1/2)*tanh(x)/(a+b)^(1/2))/b^3/a^(1/ 2)
Time = 0.15 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.86 \[ \int \frac {\sinh ^6(x)}{a+b \cosh ^2(x)} \, dx=\frac {4 \left (8 a^2+20 a b+15 b^2\right ) x-\frac {32 (a+b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{\sqrt {a}}-8 b (a+2 b) \sinh (2 x)+b^2 \sinh (4 x)}{32 b^3} \]
(4*(8*a^2 + 20*a*b + 15*b^2)*x - (32*(a + b)^(5/2)*ArcTanh[(Sqrt[a]*Tanh[x ])/Sqrt[a + b]])/Sqrt[a] - 8*b*(a + 2*b)*Sinh[2*x] + b^2*Sinh[4*x])/(32*b^ 3)
Time = 0.38 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.34, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 25, 3670, 316, 25, 402, 25, 397, 219, 221}
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.
| \(\displaystyle \int \frac {\sinh ^6(x)}{a+b \cosh ^2(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\cos \left (\frac {\pi }{2}+i x\right )^6}{a+b \sin \left (\frac {\pi }{2}+i x\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\cos \left (i x+\frac {\pi }{2}\right )^6}{b \sin \left (i x+\frac {\pi }{2}\right )^2+a}dx\) |
| \(\Big \downarrow \) 3670 |
| \(\displaystyle -\int \frac {1}{\left (1-\coth ^2(x)\right )^3 \left (a-(a+b) \coth ^2(x)\right )}d\coth (x)\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\int -\frac {3 (a+b) \coth ^2(x)+a+4 b}{\left (1-\coth ^2(x)\right )^2 \left (a-(a+b) \coth ^2(x)\right )}d\coth (x)}{4 b}+\frac {\coth (x)}{4 b \left (1-\coth ^2(x)\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\coth (x)}{4 b \left (1-\coth ^2(x)\right )^2}-\frac {\int \frac {3 (a+b) \coth ^2(x)+a+4 b}{\left (1-\coth ^2(x)\right )^2 \left (a-(a+b) \coth ^2(x)\right )}d\coth (x)}{4 b}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\coth (x)}{4 b \left (1-\coth ^2(x)\right )^2}-\frac {-\frac {\int -\frac {4 a^2+9 b a+8 b^2+(a+b) (4 a+7 b) \coth ^2(x)}{\left (1-\coth ^2(x)\right ) \left (a-(a+b) \coth ^2(x)\right )}d\coth (x)}{2 b}-\frac {(4 a+7 b) \coth (x)}{2 b \left (1-\coth ^2(x)\right )}}{4 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\coth (x)}{4 b \left (1-\coth ^2(x)\right )^2}-\frac {\frac {\int \frac {4 a^2+9 b a+8 b^2+(a+b) (4 a+7 b) \coth ^2(x)}{\left (1-\coth ^2(x)\right ) \left (a-(a+b) \coth ^2(x)\right )}d\coth (x)}{2 b}-\frac {(4 a+7 b) \coth (x)}{2 b \left (1-\coth ^2(x)\right )}}{4 b}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\coth (x)}{4 b \left (1-\coth ^2(x)\right )^2}-\frac {\frac {\frac {8 (a+b)^3 \int \frac {1}{a-(a+b) \coth ^2(x)}d\coth (x)}{b}-\frac {\left (8 a^2+20 a b+15 b^2\right ) \int \frac {1}{1-\coth ^2(x)}d\coth (x)}{b}}{2 b}-\frac {(4 a+7 b) \coth (x)}{2 b \left (1-\coth ^2(x)\right )}}{4 b}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\coth (x)}{4 b \left (1-\coth ^2(x)\right )^2}-\frac {\frac {\frac {8 (a+b)^3 \int \frac {1}{a-(a+b) \coth ^2(x)}d\coth (x)}{b}-\frac {\left (8 a^2+20 a b+15 b^2\right ) \text {arctanh}(\coth (x))}{b}}{2 b}-\frac {(4 a+7 b) \coth (x)}{2 b \left (1-\coth ^2(x)\right )}}{4 b}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\coth (x)}{4 b \left (1-\coth ^2(x)\right )^2}-\frac {\frac {\frac {8 (a+b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a}}\right )}{\sqrt {a} b}-\frac {\left (8 a^2+20 a b+15 b^2\right ) \text {arctanh}(\coth (x))}{b}}{2 b}-\frac {(4 a+7 b) \coth (x)}{2 b \left (1-\coth ^2(x)\right )}}{4 b}\) |
Coth[x]/(4*b*(1 - Coth[x]^2)^2) - ((-(((8*a^2 + 20*a*b + 15*b^2)*ArcTanh[C oth[x]])/b) + (8*(a + b)^(5/2)*ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a]])/(Sq rt[a]*b))/(2*b) - ((4*a + 7*b)*Coth[x])/(2*b*(1 - Coth[x]^2)))/(4*b)
3.1.13.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
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. \(288\) vs. \(2(74)=148\).
Time = 0.08 (sec) , antiderivative size = 289, normalized size of antiderivative = 3.28
\[\frac {2 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )^{2}+2 \tanh \left (\frac {x}{2}\right ) \sqrt {a}+\sqrt {a +b}\right )}{4 \sqrt {a}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )^{2}-2 \tanh \left (\frac {x}{2}\right ) \sqrt {a}+\sqrt {a +b}\right )}{4 \sqrt {a}\, \sqrt {a +b}}\right )}{b^{3}}+\frac {1}{4 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {1}{2 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {4 a +7 b}{8 b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {5 b +4 a}{8 b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {\left (-8 a^{2}-20 a b -15 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8 b^{3}}-\frac {1}{4 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {1}{2 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {4 a +7 b}{8 b^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {-5 b -4 a}{8 b^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {\left (8 a^{2}+20 a b +15 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8 b^{3}}\]
2/b^3*(a^3+3*a^2*b+3*a*b^2+b^3)*(-1/4/a^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*t anh(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)))+1/4/b/(tanh (1/2*x)-1)^4+1/2/b/(tanh(1/2*x)-1)^3-1/8*(4*a+7*b)/b^2/(tanh(1/2*x)-1)-1/8 *(5*b+4*a)/b^2/(tanh(1/2*x)-1)^2+1/8/b^3*(-8*a^2-20*a*b-15*b^2)*ln(tanh(1/ 2*x)-1)-1/4/b/(tanh(1/2*x)+1)^4+1/2/b/(tanh(1/2*x)+1)^3-1/8*(4*a+7*b)/b^2/ (tanh(1/2*x)+1)-1/8*(-5*b-4*a)/b^2/(tanh(1/2*x)+1)^2+1/8*(8*a^2+20*a*b+15* b^2)/b^3*ln(tanh(1/2*x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 566 vs. \(2 (74) = 148\).
Time = 0.29 (sec) , antiderivative size = 1308, normalized size of antiderivative = 14.86 \[ \int \frac {\sinh ^6(x)}{a+b \cosh ^2(x)} \, dx=\text {Too large to display} \]
[1/64*(b^2*cosh(x)^8 + 8*b^2*cosh(x)*sinh(x)^7 + b^2*sinh(x)^8 - 8*(a*b + 2*b^2)*cosh(x)^6 + 4*(7*b^2*cosh(x)^2 - 2*a*b - 4*b^2)*sinh(x)^6 + 8*(8*a^ 2 + 20*a*b + 15*b^2)*x*cosh(x)^4 + 8*(7*b^2*cosh(x)^3 - 6*(a*b + 2*b^2)*co sh(x))*sinh(x)^5 + 2*(35*b^2*cosh(x)^4 - 60*(a*b + 2*b^2)*cosh(x)^2 + 4*(8 *a^2 + 20*a*b + 15*b^2)*x)*sinh(x)^4 + 8*(7*b^2*cosh(x)^5 - 20*(a*b + 2*b^ 2)*cosh(x)^3 + 4*(8*a^2 + 20*a*b + 15*b^2)*x*cosh(x))*sinh(x)^3 + 8*(a*b + 2*b^2)*cosh(x)^2 + 4*(7*b^2*cosh(x)^6 - 30*(a*b + 2*b^2)*cosh(x)^4 + 12*( 8*a^2 + 20*a*b + 15*b^2)*x*cosh(x)^2 + 2*a*b + 4*b^2)*sinh(x)^2 + 32*((a^2 + 2*a*b + b^2)*cosh(x)^4 + 4*(a^2 + 2*a*b + b^2)*cosh(x)^3*sinh(x) + 6*(a ^2 + 2*a*b + b^2)*cosh(x)^2*sinh(x)^2 + 4*(a^2 + 2*a*b + b^2)*cosh(x)*sinh (x)^3 + (a^2 + 2*a*b + b^2)*sinh(x)^4)*sqrt((a + b)/a)*log((b^2*cosh(x)^4 + 4*b^2*cosh(x)*sinh(x)^3 + b^2*sinh(x)^4 + 2*(2*a*b + b^2)*cosh(x)^2 + 2* (3*b^2*cosh(x)^2 + 2*a*b + b^2)*sinh(x)^2 + 8*a^2 + 8*a*b + b^2 + 4*(b^2*c osh(x)^3 + (2*a*b + b^2)*cosh(x))*sinh(x) + 4*(a*b*cosh(x)^2 + 2*a*b*cosh( x)*sinh(x) + a*b*sinh(x)^2 + 2*a^2 + a*b)*sqrt((a + b)/a))/(b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 + 2*(2*a + b)*cosh(x)^2 + 2*(3*b*cosh( x)^2 + 2*a + b)*sinh(x)^2 + 4*(b*cosh(x)^3 + (2*a + b)*cosh(x))*sinh(x) + b)) - b^2 + 8*(b^2*cosh(x)^7 - 6*(a*b + 2*b^2)*cosh(x)^5 + 4*(8*a^2 + 20*a *b + 15*b^2)*x*cosh(x)^3 + 2*(a*b + 2*b^2)*cosh(x))*sinh(x))/(b^3*cosh(x)^ 4 + 4*b^3*cosh(x)^3*sinh(x) + 6*b^3*cosh(x)^2*sinh(x)^2 + 4*b^3*cosh(x)...
Timed out. \[ \int \frac {\sinh ^6(x)}{a+b \cosh ^2(x)} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 651 vs. \(2 (74) = 148\).
Time = 0.31 (sec) , antiderivative size = 651, normalized size of antiderivative = 7.40 \[ \int \frac {\sinh ^6(x)}{a+b \cosh ^2(x)} \, dx=-\frac {15 \, {\left (2 \, a + b\right )} \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 )}{64 \, \sqrt {{\left (a + b\right )} a} b} + \frac {5 \, \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 )}{32 \, \sqrt {{\left (a + b\right )} a}} + \frac {3 \, {\left (2 \, a + b\right )} x}{2 \, b^{2}} + \frac {15 \, x}{16 \, b} - \frac {{\left (4 \, {\left (2 \, a + b\right )} e^{\left (-2 \, x\right )} - b\right )} e^{\left (4 \, x\right )}}{64 \, b^{2}} - \frac {3 \, e^{\left (2 \, x\right )}}{16 \, b} + \frac {3 \, e^{\left (-2 \, x\right )}}{16 \, b} + \frac {{\left (4 \, {\left (2 \, a + b\right )} e^{\left (2 \, x\right )} - b\right )} e^{\left (-4 \, x\right )}}{64 \, b^{2}} - \frac {3 \, {\left (2 \, a + b\right )} \log \left (b e^{\left (4 \, x\right )} + 2 \, {\left (2 \, a + b\right )} e^{\left (2 \, x\right )} + b\right )}{16 \, b^{2}} + \frac {3 \, {\left (2 \, a + b\right )} \log \left (2 \, {\left (2 \, a + b\right )} e^{\left (-2 \, x\right )} + b e^{\left (-4 \, x\right )} + b\right )}{16 \, b^{2}} - \frac {3 \, {\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \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 )}{64 \, \sqrt {{\left (a + b\right )} a} b^{2}} + \frac {3 \, {\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \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 )}{64 \, \sqrt {{\left (a + b\right )} a} b^{2}} + \frac {{\left (16 \, a^{2} + 16 \, a b + 3 \, b^{2}\right )} x}{8 \, b^{3}} - \frac {{\left (16 \, a^{2} + 16 \, a b + 3 \, b^{2}\right )} \log \left (b e^{\left (4 \, x\right )} + 2 \, {\left (2 \, a + b\right )} e^{\left (2 \, x\right )} + b\right )}{64 \, b^{3}} + \frac {{\left (16 \, a^{2} + 16 \, a b + 3 \, b^{2}\right )} \log \left (2 \, {\left (2 \, a + b\right )} e^{\left (-2 \, x\right )} + b e^{\left (-4 \, x\right )} + b\right )}{64 \, b^{3}} - \frac {{\left (32 \, a^{3} + 48 \, a^{2} b + 18 \, a b^{2} + b^{3}\right )} \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 )}{128 \, \sqrt {{\left (a + b\right )} a} b^{3}} + \frac {{\left (32 \, a^{3} + 48 \, a^{2} b + 18 \, a b^{2} + b^{3}\right )} \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 )}{128 \, \sqrt {{\left (a + b\right )} a} b^{3}} \]
-15/64*(2*a + b)*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)))/(sqrt((a + b)*a)*b) + 5/32*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)) )/sqrt((a + b)*a) + 3/2*(2*a + b)*x/b^2 + 15/16*x/b - 1/64*(4*(2*a + b)*e^ (-2*x) - b)*e^(4*x)/b^2 - 3/16*e^(2*x)/b + 3/16*e^(-2*x)/b + 1/64*(4*(2*a + b)*e^(2*x) - b)*e^(-4*x)/b^2 - 3/16*(2*a + b)*log(b*e^(4*x) + 2*(2*a + b )*e^(2*x) + b)/b^2 + 3/16*(2*a + b)*log(2*(2*a + b)*e^(-2*x) + b*e^(-4*x) + b)/b^2 - 3/64*(8*a^2 + 8*a*b + b^2)*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)))/(sqrt((a + b)*a)*b^2) + 3/64*(8*a^2 + 8*a*b + b^2)*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)))/(sqrt((a + b)*a)*b^2) + 1/8 *(16*a^2 + 16*a*b + 3*b^2)*x/b^3 - 1/64*(16*a^2 + 16*a*b + 3*b^2)*log(b*e^ (4*x) + 2*(2*a + b)*e^(2*x) + b)/b^3 + 1/64*(16*a^2 + 16*a*b + 3*b^2)*log( 2*(2*a + b)*e^(-2*x) + b*e^(-4*x) + b)/b^3 - 1/128*(32*a^3 + 48*a^2*b + 18 *a*b^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)))/(sqrt((a + b)*a)*b^3) + 1/128*(32*a^3 + 48*a^ 2*b + 18*a*b^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)))/(sqrt((a + b)*a)*b^3)
Leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (74) = 148\).
Time = 0.28 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.89 \[ \int \frac {\sinh ^6(x)}{a+b \cosh ^2(x)} \, dx=\frac {b e^{\left (4 \, x\right )} - 8 \, a e^{\left (2 \, x\right )} - 16 \, b e^{\left (2 \, x\right )}}{64 \, b^{2}} + \frac {{\left (8 \, a^{2} + 20 \, a b + 15 \, b^{2}\right )} x}{8 \, b^{3}} - \frac {{\left (48 \, a^{2} e^{\left (4 \, x\right )} + 120 \, a b e^{\left (4 \, x\right )} + 90 \, b^{2} e^{\left (4 \, x\right )} - 8 \, a b e^{\left (2 \, x\right )} - 16 \, b^{2} e^{\left (2 \, x\right )} + b^{2}\right )} e^{\left (-4 \, x\right )}}{64 \, b^{3}} - \frac {{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {b e^{\left (2 \, x\right )} + 2 \, a + b}{2 \, \sqrt {-a^{2} - a b}}\right )}{\sqrt {-a^{2} - a b} b^{3}} \]
1/64*(b*e^(4*x) - 8*a*e^(2*x) - 16*b*e^(2*x))/b^2 + 1/8*(8*a^2 + 20*a*b + 15*b^2)*x/b^3 - 1/64*(48*a^2*e^(4*x) + 120*a*b*e^(4*x) + 90*b^2*e^(4*x) - 8*a*b*e^(2*x) - 16*b^2*e^(2*x) + b^2)*e^(-4*x)/b^3 - (a^3 + 3*a^2*b + 3*a* b^2 + b^3)*arctan(1/2*(b*e^(2*x) + 2*a + b)/sqrt(-a^2 - a*b))/(sqrt(-a^2 - a*b)*b^3)
Time = 2.57 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.82 \[ \int \frac {\sinh ^6(x)}{a+b \cosh ^2(x)} \, dx=\frac {{\mathrm {e}}^{4\,x}}{64\,b}-\frac {{\mathrm {e}}^{-4\,x}}{64\,b}+\frac {x\,\left (8\,a^2+20\,a\,b+15\,b^2\right )}{8\,b^3}+\frac {{\mathrm {e}}^{-2\,x}\,\left (a+2\,b\right )}{8\,b^2}-\frac {{\mathrm {e}}^{2\,x}\,\left (a+2\,b\right )}{8\,b^2}+\frac {\ln \left (\frac {4\,{\left (a+b\right )}^5\,\left (2\,a\,b+8\,a^2\,{\mathrm {e}}^{2\,x}+b^2\,{\mathrm {e}}^{2\,x}+b^2+8\,a\,b\,{\mathrm {e}}^{2\,x}\right )}{a\,b^8}-\frac {8\,{\left (a+b\right )}^{11/2}\,\left (b+4\,a\,{\mathrm {e}}^{2\,x}+2\,b\,{\mathrm {e}}^{2\,x}\right )}{\sqrt {a}\,b^8}\right )\,{\left (a+b\right )}^{5/2}}{2\,\sqrt {a}\,b^3}-\frac {\ln \left (\frac {8\,{\left (a+b\right )}^{11/2}\,\left (b+4\,a\,{\mathrm {e}}^{2\,x}+2\,b\,{\mathrm {e}}^{2\,x}\right )}{\sqrt {a}\,b^8}+\frac {4\,{\left (a+b\right )}^5\,\left (2\,a\,b+8\,a^2\,{\mathrm {e}}^{2\,x}+b^2\,{\mathrm {e}}^{2\,x}+b^2+8\,a\,b\,{\mathrm {e}}^{2\,x}\right )}{a\,b^8}\right )\,{\left (a+b\right )}^{5/2}}{2\,\sqrt {a}\,b^3} \]
exp(4*x)/(64*b) - exp(-4*x)/(64*b) + (x*(20*a*b + 8*a^2 + 15*b^2))/(8*b^3) + (exp(-2*x)*(a + 2*b))/(8*b^2) - (exp(2*x)*(a + 2*b))/(8*b^2) + (log((4* (a + b)^5*(2*a*b + 8*a^2*exp(2*x) + b^2*exp(2*x) + b^2 + 8*a*b*exp(2*x)))/ (a*b^8) - (8*(a + b)^(11/2)*(b + 4*a*exp(2*x) + 2*b*exp(2*x)))/(a^(1/2)*b^ 8))*(a + b)^(5/2))/(2*a^(1/2)*b^3) - (log((8*(a + b)^(11/2)*(b + 4*a*exp(2 *x) + 2*b*exp(2*x)))/(a^(1/2)*b^8) + (4*(a + b)^5*(2*a*b + 8*a^2*exp(2*x) + b^2*exp(2*x) + b^2 + 8*a*b*exp(2*x)))/(a*b^8))*(a + b)^(5/2))/(2*a^(1/2) *b^3)