Integrand size = 24, antiderivative size = 235 \[ \int \frac {\sinh ^9(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=-\frac {\sqrt {a} \left (5 \sqrt {a}-6 \sqrt {b}\right ) \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 \left (\sqrt {a}-\sqrt {b}\right )^{3/2} b^{9/4} d}-\frac {\sqrt {a} \left (5 \sqrt {a}+6 \sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 \left (\sqrt {a}+\sqrt {b}\right )^{3/2} b^{9/4} d}+\frac {\cosh (c+d x)}{b^2 d}+\frac {a \cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{4 (a-b) b^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )} \]
cosh(d*x+c)/b^2/d+1/4*a*cosh(d*x+c)*(a+b-b*cosh(d*x+c)^2)/(a-b)/b^2/d/(a-b +2*b*cosh(d*x+c)^2-b*cosh(d*x+c)^4)-1/8*arctan(b^(1/4)*cosh(d*x+c)/(a^(1/2 )-b^(1/2))^(1/2))*a^(1/2)*(5*a^(1/2)-6*b^(1/2))/b^(9/4)/d/(a^(1/2)-b^(1/2) )^(3/2)-1/8*arctanh(b^(1/4)*cosh(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))*a^(1/2)*( 5*a^(1/2)+6*b^(1/2))/b^(9/4)/d/(a^(1/2)+b^(1/2))^(3/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 3.45 (sec) , antiderivative size = 615, normalized size of antiderivative = 2.62 \[ \int \frac {\sinh ^9(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\frac {32 \cosh (c+d x)+\frac {32 a \cosh (c+d x) (2 a+b-b \cosh (2 (c+d x)))}{(a-b) (8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x)))}+\frac {a \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-b c-b d x-2 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )-20 a c \text {$\#$1}^2+27 b c \text {$\#$1}^2-20 a d x \text {$\#$1}^2+27 b d x \text {$\#$1}^2-40 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+54 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+20 a c \text {$\#$1}^4-27 b c \text {$\#$1}^4+20 a d x \text {$\#$1}^4-27 b d x \text {$\#$1}^4+40 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-54 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+b c \text {$\#$1}^6+b d x \text {$\#$1}^6+2 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{a-b}}{32 b^2 d} \]
(32*Cosh[c + d*x] + (32*a*Cosh[c + d*x]*(2*a + b - b*Cosh[2*(c + d*x)]))/( (a - b)*(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)])) + (a*Ro otSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-(b*c) - b*d*x - 2*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/ 2]*#1 - Sinh[(c + d*x)/2]*#1] - 20*a*c*#1^2 + 27*b*c*#1^2 - 20*a*d*x*#1^2 + 27*b*d*x*#1^2 - 40*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[( c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + 54*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + 20*a*c*#1^4 - 27*b*c*#1^4 + 20*a*d*x*#1^4 - 27*b*d*x*#1^4 + 40*a*Log[-Cosh [(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/ 2]*#1]*#1^4 - 54*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + b*c*#1^6 + b*d*x*#1^6 + 2*b*Log[ -Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ]) /(a - b))/(32*b^2*d)
Time = 0.64 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {3042, 26, 3694, 1517, 27, 2205, 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.
| \(\displaystyle \int \frac {\sinh ^9(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \sin (i c+i d x)^9}{\left (a-b \sin (i c+i d x)^4\right )^2}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\sin (i c+i d x)^9}{\left (a-b \sin (i c+i d x)^4\right )^2}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle \frac {\int \frac {\left (1-\cosh ^2(c+d x)\right )^4}{\left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )^2}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1517 |
| \(\displaystyle \frac {\frac {a \cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{4 b^2 (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac {\int \frac {2 \left (4 a (a-b) \cosh ^4(c+d x)-a (7 a-8 b) \cosh ^2(c+d x)+a \left (\frac {a^2}{b}+a-4 b\right )\right )}{-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b}d\cosh (c+d x)}{8 a b (a-b)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {a \cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{4 b^2 (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac {\int \frac {4 a (a-b) \cosh ^4(c+d x)-a (7 a-8 b) \cosh ^2(c+d x)+a \left (\frac {a^2}{b}+a-4 b\right )}{-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b}d\cosh (c+d x)}{4 a b (a-b)}}{d}\) |
| \(\Big \downarrow \) 2205 |
| \(\displaystyle \frac {\frac {a \cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{4 b^2 (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac {\int \left (\frac {b \cosh ^2(c+d x) a^2+(5 a-7 b) a^2}{b \left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )}-\frac {4 a (a-b)}{b}\right )d\cosh (c+d x)}{4 a b (a-b)}}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a \cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{4 b^2 (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac {\frac {a^{3/2} \left (-\sqrt {a} \sqrt {b}+5 a-6 b\right ) \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{5/4} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {a^{3/2} \left (\sqrt {a} \sqrt {b}+5 a-6 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 b^{5/4} \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {4 a (a-b) \cosh (c+d x)}{b}}{4 a b (a-b)}}{d}\) |
(-1/4*((a^(3/2)*(5*a - Sqrt[a]*Sqrt[b] - 6*b)*ArcTan[(b^(1/4)*Cosh[c + d*x ])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(5/4)) + (a^(3/2 )*(5*a + Sqrt[a]*Sqrt[b] - 6*b)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[ a] + Sqrt[b]]])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(5/4)) - (4*a*(a - b)*Cosh[c + d*x])/b)/(a*(a - b)*b) + (a*Cosh[c + d*x]*(a + b - b*Cosh[c + d*x]^2))/( 4*(a - b)*b^2*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4)))/d
3.3.41.3.1 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[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x _Symbol] :> With[{f = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a*b*g - f*(b^2 - 2* a*c) - c*(b*f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a* (p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)* x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ [c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1] && LtQ[p, -1]
Int[(Px_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandInte grand[Px/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x^ 2] && Expon[Px, x^2] > 1
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Leaf count of result is larger than twice the leaf count of optimal. \(372\) vs. \(2(185)=370\).
Time = 5.28 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.59
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 a \left (\frac {\frac {\left (a -2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4 a -4 b}-\frac {\left (3 a -8 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4 \left (a -b \right )}+\frac {\left (3 a +2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{4 a -4 b}-\frac {a}{4 \left (a -b \right )}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a -4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a +6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -16 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a}+\frac {a \left (-\frac {\left (\sqrt {a b}+5 a -6 b \right ) \arctan \left (\frac {-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}+2 a}{4 \sqrt {-a b -\sqrt {a b}\, a}}\right )}{4 a \sqrt {-a b -\sqrt {a b}\, a}}+\frac {\left (-\sqrt {a b}+5 a -6 b \right ) \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}-2 a}{4 \sqrt {-a b +\sqrt {a b}\, a}}\right )}{4 a \sqrt {-a b +\sqrt {a b}\, a}}\right )}{4 a -4 b}\right )}{b^{2}}+\frac {1}{b^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {1}{b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(373\) |
| default | \(\frac {-\frac {2 a \left (\frac {\frac {\left (a -2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4 a -4 b}-\frac {\left (3 a -8 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4 \left (a -b \right )}+\frac {\left (3 a +2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{4 a -4 b}-\frac {a}{4 \left (a -b \right )}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a -4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a +6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -16 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a}+\frac {a \left (-\frac {\left (\sqrt {a b}+5 a -6 b \right ) \arctan \left (\frac {-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}+2 a}{4 \sqrt {-a b -\sqrt {a b}\, a}}\right )}{4 a \sqrt {-a b -\sqrt {a b}\, a}}+\frac {\left (-\sqrt {a b}+5 a -6 b \right ) \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}-2 a}{4 \sqrt {-a b +\sqrt {a b}\, a}}\right )}{4 a \sqrt {-a b +\sqrt {a b}\, a}}\right )}{4 a -4 b}\right )}{b^{2}}+\frac {1}{b^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {1}{b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(373\) |
| risch | \(\frac {{\mathrm e}^{d x +c}}{2 b^{2} d}+\frac {{\mathrm e}^{-d x -c}}{2 b^{2} d}+\frac {a \,{\mathrm e}^{d x +c} \left (-b \,{\mathrm e}^{6 d x +6 c}+4 \,{\mathrm e}^{4 d x +4 c} a +b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}+b \,{\mathrm e}^{2 d x +2 c}-b \right )}{2 b^{2} \left (a -b \right ) d \left (-b \,{\mathrm e}^{8 d x +8 c}+4 b \,{\mathrm e}^{6 d x +6 c}+16 \,{\mathrm e}^{4 d x +4 c} a -6 b \,{\mathrm e}^{4 d x +4 c}+4 b \,{\mathrm e}^{2 d x +2 c}-b \right )}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (65536 a^{3} b^{9} d^{4}-196608 a^{2} b^{10} d^{4}+196608 a \,b^{11} d^{4}-65536 b^{12} d^{4}\right ) \textit {\_Z}^{4}+\left (7680 a^{3} b^{5} d^{2}-24064 a^{2} b^{6} d^{2}+18432 a \,b^{7} d^{2}\right ) \textit {\_Z}^{2}-625 a^{4}+1800 a^{3} b -1296 a^{2} b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\left (-\frac {32768 a^{4} b^{7} d^{3}}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}+\frac {147456 a^{3} b^{8} d^{3}}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}-\frac {245760 a^{2} b^{9} d^{3}}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}+\frac {180224 a \,b^{10} d^{3}}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}-\frac {49152 b^{11} d^{3}}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}\right ) \textit {\_R}^{3}+\left (-\frac {4000 a^{5} b^{2} d}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}+\frac {14720 a^{4} b^{3} d}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}-\frac {14240 a^{3} b^{4} d}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}-\frac {2880 a^{2} b^{5} d}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}+\frac {6912 a \,b^{6} d}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}\right ) \textit {\_R} \right ) {\mathrm e}^{d x +c}+1\right )\right )\) | \(710\) |
1/d*(-2*a/b^2*((1/4*(a-2*b)/(a-b)*tanh(1/2*d*x+1/2*c)^6-1/4*(3*a-8*b)/(a-b )*tanh(1/2*d*x+1/2*c)^4+1/4*(3*a+2*b)/(a-b)*tanh(1/2*d*x+1/2*c)^2-1/4*a/(a -b))/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2 *c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)+1/4/(a-b)* a*(-1/4*((a*b)^(1/2)+5*a-6*b)/a/(-a*b-(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(-2* tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)+2*a)/(-a*b-(a*b)^(1/2)*a)^(1/2))+1/4 *(-(a*b)^(1/2)+5*a-6*b)/a/(-a*b+(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(2*tanh(1/ 2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)-2*a)/(-a*b+(a*b)^(1/2)*a)^(1/2))))+1/b^2/(1 +tanh(1/2*d*x+1/2*c))-1/b^2/(tanh(1/2*d*x+1/2*c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 7664 vs. \(2 (187) = 374\).
Time = 0.49 (sec) , antiderivative size = 7664, normalized size of antiderivative = 32.61 \[ \int \frac {\sinh ^9(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\sinh ^9(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {\sinh ^9(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int { \frac {\sinh \left (d x + c\right )^{9}}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \]
1/2*(a*b - b^2 + (a*b*e^(10*c) - b^2*e^(10*c))*e^(10*d*x) - (2*a*b*e^(8*c) - 3*b^2*e^(8*c))*e^(8*d*x) - (20*a^2*e^(6*c) - 17*a*b*e^(6*c) + 2*b^2*e^( 6*c))*e^(6*d*x) - (20*a^2*e^(4*c) - 17*a*b*e^(4*c) + 2*b^2*e^(4*c))*e^(4*d *x) - (2*a*b*e^(2*c) - 3*b^2*e^(2*c))*e^(2*d*x))/((a*b^3*d*e^(9*c) - b^4*d *e^(9*c))*e^(9*d*x) - 4*(a*b^3*d*e^(7*c) - b^4*d*e^(7*c))*e^(7*d*x) - 2*(8 *a^2*b^2*d*e^(5*c) - 11*a*b^3*d*e^(5*c) + 3*b^4*d*e^(5*c))*e^(5*d*x) - 4*( a*b^3*d*e^(3*c) - b^4*d*e^(3*c))*e^(3*d*x) + (a*b^3*d*e^c - b^4*d*e^c)*e^( d*x)) + 1/512*integrate(256*(a*b*e^(7*d*x + 7*c) - a*b*e^(d*x + c) + (20*a ^2*e^(5*c) - 27*a*b*e^(5*c))*e^(5*d*x) - (20*a^2*e^(3*c) - 27*a*b*e^(3*c)) *e^(3*d*x))/(a*b^3 - b^4 + (a*b^3*e^(8*c) - b^4*e^(8*c))*e^(8*d*x) - 4*(a* b^3*e^(6*c) - b^4*e^(6*c))*e^(6*d*x) - 2*(8*a^2*b^2*e^(4*c) - 11*a*b^3*e^( 4*c) + 3*b^4*e^(4*c))*e^(4*d*x) - 4*(a*b^3*e^(2*c) - b^4*e^(2*c))*e^(2*d*x )), x)
\[ \int \frac {\sinh ^9(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int { \frac {\sinh \left (d x + c\right )^{9}}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\sinh ^9(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^9}{{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^2} \,d x \]