Integrand size = 24, antiderivative size = 220 \[ \int \frac {\sinh ^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=-\frac {\left (2 \sqrt {a}-\sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{5/4} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} \sqrt {b} d}+\frac {\left (2 \sqrt {a}+\sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{5/4} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} \sqrt {b} d}+\frac {\tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 a (a-b) d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )} \]
-1/8*arctanh((a^(1/2)-b^(1/2))^(1/2)*tanh(d*x+c)/a^(1/4))*(2*a^(1/2)-b^(1/ 2))/a^(5/4)/d/(a^(1/2)-b^(1/2))^(3/2)/b^(1/2)+1/8*arctanh((a^(1/2)+b^(1/2) )^(1/2)*tanh(d*x+c)/a^(1/4))*(2*a^(1/2)+b^(1/2))/a^(5/4)/d/b^(1/2)/(a^(1/2 )+b^(1/2))^(3/2)+1/4*tanh(d*x+c)*(a-(a+b)*tanh(d*x+c)^2)/a/(a-b)/d/(a-2*a* tanh(d*x+c)^2+(a-b)*tanh(d*x+c)^4)
Time = 1.87 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.15 \[ \int \frac {\sinh ^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\frac {\frac {\sqrt {a} \left (2 a+\sqrt {a} \sqrt {b}-b\right ) \arctan \left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}} \sqrt {b}}+\frac {\sqrt {a} \left (2 a-\sqrt {a} \sqrt {b}-b\right ) \text {arctanh}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}} \sqrt {b}}+\frac {4 \sqrt {a} (2 a+b-b \cosh (2 (c+d x))) \sinh (2 (c+d x))}{8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))}}{8 a^{3/2} (a-b) d} \]
((Sqrt[a]*(2*a + Sqrt[a]*Sqrt[b] - b)*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(Sqrt[-a + Sqrt[a]*Sqrt[b]]*Sqrt[b]) + (Sqrt[a]*(2*a - Sqrt[a]*Sqrt[b] - b)*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a + Sqrt[a]*Sqrt[b]]*Sqrt[b]) + (4*Sqrt[a]*(2*a + b - b*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)]))/(8*a^(3/2)*(a - b)*d)
Time = 0.48 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.17, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {3042, 25, 3696, 1672, 27, 1480, 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.
| \(\displaystyle \int \frac {\sinh ^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sin (i c+i d x)^2}{\left (a-b \sin (i c+i d x)^4\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\sin (i c+i d x)^2}{\left (a-b \sin (i c+i d x)^4\right )^2}dx\) |
| \(\Big \downarrow \) 3696 |
| \(\displaystyle \frac {\int \frac {\tanh ^2(c+d x) \left (1-\tanh ^2(c+d x)\right )^2}{\left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1672 |
| \(\displaystyle \frac {\frac {\tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 a (a-b) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac {\int \frac {2 a b \left (a-(3 a-b) \tanh ^2(c+d x)\right )}{(a-b) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{8 a^2 b}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 a (a-b) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac {\int \frac {a-(3 a-b) \tanh ^2(c+d x)}{(a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a}d\tanh (c+d x)}{4 a (a-b)}}{d}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {\tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 a (a-b) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac {-\frac {1}{2} \left (-\frac {2 a^{3/2}}{\sqrt {b}}+3 a-b\right ) \int \frac {1}{(a-b) \tanh ^2(c+d x)-\sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )}d\tanh (c+d x)-\frac {1}{2} \left (\frac {2 a^{3/2}}{\sqrt {b}}+3 a-b\right ) \int \frac {1}{(a-b) \tanh ^2(c+d x)-\sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )}d\tanh (c+d x)}{4 a (a-b)}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 a (a-b) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac {\frac {\left (\frac {2 a^{3/2}}{\sqrt {b}}+3 a-b\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} \left (\sqrt {a}+\sqrt {b}\right )}+\frac {\left (-\frac {2 a^{3/2}}{\sqrt {b}}+3 a-b\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {b}\right ) \sqrt {\sqrt {a}+\sqrt {b}}}}{4 a (a-b)}}{d}\) |
(-1/4*(((3*a + (2*a^(3/2))/Sqrt[b] - b)*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*T anh[c + d*x])/a^(1/4)])/(2*a^(1/4)*Sqrt[Sqrt[a] - Sqrt[b]]*(Sqrt[a] + Sqrt [b])) + ((3*a - (2*a^(3/2))/Sqrt[b] - b)*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]* Tanh[c + d*x])/a^(1/4)])/(2*a^(1/4)*(Sqrt[a] - Sqrt[b])*Sqrt[Sqrt[a] + Sqr t[b]]))/(a*(a - b)) + (Tanh[c + d*x]*(a - (a + b)*Tanh[c + d*x]^2))/(4*a*( a - b)*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4)))/d
3.3.50.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^ 4)^(p_), x_Symbol] :> With[{f = Coeff[PolynomialRemainder[x^m*(d + e*x^2)^q , a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[x^m*(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)* Simp[ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[x^m*(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], x]] /; FreeQ[{a, b, c, d, e}, x ] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] && IGtQ[m/2, 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff^(m + 1 )/f Subst[Int[x^m*((a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2) ^(m/2 + 2*p + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] & & IntegerQ[m/2] && IntegerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.72 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.35
| method | result | size |
| derivativedivides | \(\frac {-\frac {8 \left (-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{16 \left (a -b \right )}+\frac {\left (a +4 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{16 \left (a -b \right ) a}+\frac {\left (a +4 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{16 \left (a -b \right ) a}-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 \left (a -b \right )}\right )}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a -4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a +6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -16 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (-a \,\textit {\_R}^{6}+\left (11 a -4 b \right ) \textit {\_R}^{4}+\left (-11 a +4 b \right ) \textit {\_R}^{2}+a \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}}{16 \left (a -b \right ) a}}{d}\) | \(297\) |
| default | \(\frac {-\frac {8 \left (-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{16 \left (a -b \right )}+\frac {\left (a +4 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{16 \left (a -b \right ) a}+\frac {\left (a +4 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{16 \left (a -b \right ) a}-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 \left (a -b \right )}\right )}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a -4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a +6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -16 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (-a \,\textit {\_R}^{6}+\left (11 a -4 b \right ) \textit {\_R}^{4}+\left (-11 a +4 b \right ) \textit {\_R}^{2}+a \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}}{16 \left (a -b \right ) a}}{d}\) | \(297\) |
| risch | \(\frac {2 \,{\mathrm e}^{6 d x +6 c} a -b \,{\mathrm e}^{6 d x +6 c}-8 \,{\mathrm e}^{4 d x +4 c} a +3 b \,{\mathrm e}^{4 d x +4 c}-2 a \,{\mathrm e}^{2 d x +2 c}-3 b \,{\mathrm e}^{2 d x +2 c}+b}{2 a d \left (a -b \right ) \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^{8} b^{2} d^{4}-196608 a^{7} b^{3} d^{4}+196608 a^{6} b^{4} d^{4}-65536 a^{5} b^{5} d^{4}\right ) \textit {\_Z}^{4}+\left (-2048 a^{5} b \,d^{2}-512 a^{4} b^{2} d^{2}+512 a^{3} b^{3} d^{2}\right ) \textit {\_Z}^{2}+16 a^{2}-8 a b +b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (-\frac {24576 a^{8} b^{2} d^{3}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}+\frac {81920 a^{7} b^{3} d^{3}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}-\frac {98304 a^{6} b^{4} d^{3}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}+\frac {49152 a^{5} b^{5} d^{3}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}-\frac {8192 a^{4} b^{6} d^{3}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}\right ) \textit {\_R}^{3}+\left (\frac {2048 d^{2} a^{7} b}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}-\frac {6656 d^{2} b^{2} a^{6}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}+\frac {7680 d^{2} b^{3} a^{5}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}-\frac {3584 d^{2} b^{4} a^{4}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}+\frac {512 d^{2} b^{5} a^{3}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}\right ) \textit {\_R}^{2}+\left (\frac {896 a^{5} d b}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}-\frac {352 a^{4} b^{2} d}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}-\frac {64 a^{3} b^{3} d}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}+\frac {32 a^{2} b^{4} d}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}\right ) \textit {\_R} -\frac {32 a^{4}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}-\frac {32 a^{3} b}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}+\frac {38 a^{2} b^{2}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}-\frac {11 a \,b^{3}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}+\frac {b^{4}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}\right )\right )\) | \(988\) |
1/d*(-8*(-1/16/(a-b)*tanh(1/2*d*x+1/2*c)^7+1/16*(a+4*b)/(a-b)/a*tanh(1/2*d *x+1/2*c)^5+1/16*(a+4*b)/(a-b)/a*tanh(1/2*d*x+1/2*c)^3-1/16/(a-b)*tanh(1/2 *d*x+1/2*c))/(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/ 16/(a-b)/a*sum((-a*_R^6+(11*a-4*b)*_R^4+(-11*a+4*b)*_R^2+a)/(_R^7*a-3*_R^5 *a+3*_R^3*a-8*_R^3*b-_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R=RootOf(a*_Z^8-4*a *_Z^6+(6*a-16*b)*_Z^4-4*a*_Z^2+a)))
Leaf count of result is larger than twice the leaf count of optimal. 6525 vs. \(2 (171) = 342\).
Time = 0.54 (sec) , antiderivative size = 6525, normalized size of antiderivative = 29.66 \[ \int \frac {\sinh ^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\sinh ^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {\sinh ^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int { \frac {\sinh \left (d x + c\right )^{2}}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \]
-1/2*((2*a*e^(6*c) - b*e^(6*c))*e^(6*d*x) - (8*a*e^(4*c) - 3*b*e^(4*c))*e^ (4*d*x) - (2*a*e^(2*c) + 3*b*e^(2*c))*e^(2*d*x) + b)/(a^2*b*d - a*b^2*d + (a^2*b*d*e^(8*c) - a*b^2*d*e^(8*c))*e^(8*d*x) - 4*(a^2*b*d*e^(6*c) - a*b^2 *d*e^(6*c))*e^(6*d*x) - 2*(8*a^3*d*e^(4*c) - 11*a^2*b*d*e^(4*c) + 3*a*b^2* d*e^(4*c))*e^(4*d*x) - 4*(a^2*b*d*e^(2*c) - a*b^2*d*e^(2*c))*e^(2*d*x)) - 1/4*integrate(4*((2*a*e^(6*c) - b*e^(6*c))*e^(6*d*x) - 2*(4*a*e^(4*c) - b* e^(4*c))*e^(4*d*x) + (2*a*e^(2*c) - b*e^(2*c))*e^(2*d*x))/(a^2*b - a*b^2 + (a^2*b*e^(8*c) - a*b^2*e^(8*c))*e^(8*d*x) - 4*(a^2*b*e^(6*c) - a*b^2*e^(6 *c))*e^(6*d*x) - 2*(8*a^3*e^(4*c) - 11*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c))*e^ (4*d*x) - 4*(a^2*b*e^(2*c) - a*b^2*e^(2*c))*e^(2*d*x)), x)
\[ \int \frac {\sinh ^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int { \frac {\sinh \left (d x + c\right )^{2}}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\sinh ^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^2}{{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^2} \,d x \]