Integrand size = 24, antiderivative size = 235 \[ \int \frac {\sinh ^9(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=-\frac {\sqrt {a} \left (5 \sqrt {a}-6 \sqrt {b}\right ) \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 \left (\sqrt {a}-\sqrt {b}\right )^{3/2} b^{9/4} d}-\frac {\sqrt {a} \left (5 \sqrt {a}+6 \sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 \left (\sqrt {a}+\sqrt {b}\right )^{3/2} b^{9/4} d}+\frac {\cosh (c+d x)}{b^2 d}+\frac {a \cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{4 (a-b) b^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )} \] Output:
-1/8*a^(1/2)*(5*a^(1/2)-6*b^(1/2))*arctan(b^(1/4)*cosh(d*x+c)/(a^(1/2)-b^( 1/2))^(1/2))/(a^(1/2)-b^(1/2))^(3/2)/b^(9/4)/d-1/8*a^(1/2)*(5*a^(1/2)+6*b^ (1/2))*arctanh(b^(1/4)*cosh(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))/(a^(1/2)+b^(1/ 2))^(3/2)/b^(9/4)/d+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)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 3.34 (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} \] Input:
Integrate[Sinh[c + d*x]^9/(a - b*Sinh[c + d*x]^4)^2,x]
Output:
(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 = 0.72 (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 |
\(\displaystyle \frac {\frac {a \cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{4 b^2 (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac {\frac {a^{3/2} \left (-\sqrt {a} \sqrt {b}+5 a-6 b\right ) \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{5/4} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {a^{3/2} \left (\sqrt {a} \sqrt {b}+5 a-6 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 b^{5/4} \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {4 a (a-b) \cosh (c+d x)}{b}}{4 a b (a-b)}}{d}\) |
Input:
Int[Sinh[c + d*x]^9/(a - b*Sinh[c + d*x]^4)^2,x]
Output:
(-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
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 = 12.63 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.59
method | result | size |
derivativedivides | \(\frac {\frac {1}{b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {2 a \left (\frac {\frac {\left (a -2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4 a -4 b}-\frac {\left (3 a -8 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4 \left (a -b \right )}+\frac {\left (3 a +2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{4 a -4 b}-\frac {a}{4 \left (a -b \right )}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a -4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a +6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -16 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a}+\frac {a \left (-\frac {\left (\sqrt {a b}+5 a -6 b \right ) \arctan \left (\frac {-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}+2 a}{4 \sqrt {-a b -\sqrt {a b}\, a}}\right )}{4 a \sqrt {-a b -\sqrt {a b}\, a}}+\frac {\left (-\sqrt {a b}+5 a -6 b \right ) \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}-2 a}{4 \sqrt {-a b +\sqrt {a b}\, a}}\right )}{4 a \sqrt {-a b +\sqrt {a b}\, a}}\right )}{4 a -4 b}\right )}{b^{2}}-\frac {1}{b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(373\) |
default | \(\frac {\frac {1}{b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {2 a \left (\frac {\frac {\left (a -2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4 a -4 b}-\frac {\left (3 a -8 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4 \left (a -b \right )}+\frac {\left (3 a +2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{4 a -4 b}-\frac {a}{4 \left (a -b \right )}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a -4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a +6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -16 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a}+\frac {a \left (-\frac {\left (\sqrt {a b}+5 a -6 b \right ) \arctan \left (\frac {-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}+2 a}{4 \sqrt {-a b -\sqrt {a b}\, a}}\right )}{4 a \sqrt {-a b -\sqrt {a b}\, a}}+\frac {\left (-\sqrt {a b}+5 a -6 b \right ) \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}-2 a}{4 \sqrt {-a b +\sqrt {a b}\, a}}\right )}{4 a \sqrt {-a b +\sqrt {a b}\, a}}\right )}{4 a -4 b}\right )}{b^{2}}-\frac {1}{b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(373\) |
risch | \(\frac {{\mathrm e}^{d x +c}}{2 b^{2} d}+\frac {{\mathrm e}^{-d x -c}}{2 b^{2} d}+\frac {{\mathrm e}^{d x +c} a \left (-{\mathrm e}^{6 d x +6 c} b +4 \,{\mathrm e}^{4 d x +4 c} a +b \,{\mathrm e}^{4 d x +4 c}+4 \,{\mathrm e}^{2 d x +2 c} a +{\mathrm e}^{2 d x +2 c} b -b \right )}{2 b^{2} \left (a -b \right ) d \left (-{\mathrm e}^{8 d x +8 c} b +4 \,{\mathrm e}^{6 d x +6 c} b +16 \,{\mathrm e}^{4 d x +4 c} a -6 b \,{\mathrm e}^{4 d x +4 c}+4 \,{\mathrm e}^{2 d x +2 c} b -b \right )}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (65536 a^{3} b^{9} d^{4}-196608 a^{2} b^{10} d^{4}+196608 a \,b^{11} d^{4}-65536 b^{12} d^{4}\right ) \textit {\_Z}^{4}+\left (7680 a^{3} b^{5} d^{2}-24064 a^{2} b^{6} d^{2}+18432 a \,b^{7} d^{2}\right ) \textit {\_Z}^{2}-625 a^{4}+1800 a^{3} b -1296 a^{2} b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\left (-\frac {32768 a^{4} b^{7} d^{3}}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}+\frac {147456 a^{3} b^{8} d^{3}}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}-\frac {245760 a^{2} b^{9} d^{3}}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}+\frac {180224 a \,b^{10} d^{3}}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}-\frac {49152 b^{11} d^{3}}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}\right ) \textit {\_R}^{3}+\left (-\frac {4000 a^{5} b^{2} d}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}+\frac {14720 a^{4} b^{3} d}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}-\frac {14240 a^{3} b^{4} d}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}-\frac {2880 a^{2} b^{5} d}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}+\frac {6912 a \,b^{6} d}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}\right ) \textit {\_R} \right ) {\mathrm e}^{d x +c}+1\right )\right )\) | \(710\) |
Input:
int(sinh(d*x+c)^9/(a-b*sinh(d*x+c)^4)^2,x,method=_RETURNVERBOSE)
Output:
1/d*(1/b^2/(tanh(1/2*d*x+1/2*c)+1)-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)*t anh(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/(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 = 0.33 (sec) , antiderivative size = 7664, normalized size of antiderivative = 32.61 \[ \int \frac {\sinh ^9(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(sinh(d*x+c)^9/(a-b*sinh(d*x+c)^4)^2,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\sinh ^9(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\text {Timed out} \] Input:
integrate(sinh(d*x+c)**9/(a-b*sinh(d*x+c)**4)**2,x)
Output:
Timed out
\[ \int \frac {\sinh ^9(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int { \frac {\sinh \left (d x + c\right )^{9}}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \] Input:
integrate(sinh(d*x+c)^9/(a-b*sinh(d*x+c)^4)^2,x, algorithm="maxima")
Output:
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)
\[ \int \frac {\sinh ^9(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int { \frac {\sinh \left (d x + c\right )^{9}}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \] Input:
integrate(sinh(d*x+c)^9/(a-b*sinh(d*x+c)^4)^2,x, algorithm="giac")
Output:
sage0*x
Timed out. \[ \int \frac {\sinh ^9(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^9}{{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^2} \,d x \] Input:
int(sinh(c + d*x)^9/(a - b*sinh(c + d*x)^4)^2,x)
Output:
int(sinh(c + d*x)^9/(a - b*sinh(c + d*x)^4)^2, x)
\[ \int \frac {\sinh ^9(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\text {too large to display} \] Input:
int(sinh(d*x+c)^9/(a-b*sinh(d*x+c)^4)^2,x)
Output:
(315*e**(10*c + 10*d*x)*b**2 - 921600*e**(14*c + 9*d*x)*int(e**(5*d*x)/(e* *(16*c + 16*d*x)*b**2 - 8*e**(14*c + 14*d*x)*b**2 - 32*e**(12*c + 12*d*x)* a*b + 28*e**(12*c + 12*d*x)*b**2 + 128*e**(10*c + 10*d*x)*a*b - 56*e**(10* c + 10*d*x)*b**2 + 256*e**(8*c + 8*d*x)*a**2 - 192*e**(8*c + 8*d*x)*a*b + 70*e**(8*c + 8*d*x)*b**2 + 128*e**(6*c + 6*d*x)*a*b - 56*e**(6*c + 6*d*x)* b**2 - 32*e**(4*c + 4*d*x)*a*b + 28*e**(4*c + 4*d*x)*b**2 - 8*e**(2*c + 2* d*x)*b**2 + b**2),x)*a**3*b*d - 744960*e**(14*c + 9*d*x)*int(e**(5*d*x)/(e **(16*c + 16*d*x)*b**2 - 8*e**(14*c + 14*d*x)*b**2 - 32*e**(12*c + 12*d*x) *a*b + 28*e**(12*c + 12*d*x)*b**2 + 128*e**(10*c + 10*d*x)*a*b - 56*e**(10 *c + 10*d*x)*b**2 + 256*e**(8*c + 8*d*x)*a**2 - 192*e**(8*c + 8*d*x)*a*b + 70*e**(8*c + 8*d*x)*b**2 + 128*e**(6*c + 6*d*x)*a*b - 56*e**(6*c + 6*d*x) *b**2 - 32*e**(4*c + 4*d*x)*a*b + 28*e**(4*c + 4*d*x)*b**2 - 8*e**(2*c + 2 *d*x)*b**2 + b**2),x)*a**2*b**2*d - 512*e**(14*c + 9*d*x)*int(e**(5*d*x)/( e**(16*c + 16*d*x)*b**2 - 8*e**(14*c + 14*d*x)*b**2 - 32*e**(12*c + 12*d*x )*a*b + 28*e**(12*c + 12*d*x)*b**2 + 128*e**(10*c + 10*d*x)*a*b - 56*e**(1 0*c + 10*d*x)*b**2 + 256*e**(8*c + 8*d*x)*a**2 - 192*e**(8*c + 8*d*x)*a*b + 70*e**(8*c + 8*d*x)*b**2 + 128*e**(6*c + 6*d*x)*a*b - 56*e**(6*c + 6*d*x )*b**2 - 32*e**(4*c + 4*d*x)*a*b + 28*e**(4*c + 4*d*x)*b**2 - 8*e**(2*c + 2*d*x)*b**2 + b**2),x)*a*b**3*d - 3747840*e**(12*c + 9*d*x)*int(e**(3*d*x) /(e**(16*c + 16*d*x)*b**2 - 8*e**(14*c + 14*d*x)*b**2 - 32*e**(12*c + 1...