Integrand size = 15, antiderivative size = 210 \[ \int \frac {1}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\frac {\left (4 \sqrt {a}-3 \sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}+\frac {\left (4 \sqrt {a}+3 \sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}-\frac {b \tanh (c+d x) \left (1-2 \tanh ^2(c+d x)\right )}{4 a (a-b) d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )} \] Output:
1/8*(4*a^(1/2)-3*b^(1/2))*arctanh((a^(1/2)-b^(1/2))^(1/2)*tanh(d*x+c)/a^(1 /4))/a^(7/4)/(a^(1/2)-b^(1/2))^(3/2)/d+1/8*(4*a^(1/2)+3*b^(1/2))*arctanh(( a^(1/2)+b^(1/2))^(1/2)*tanh(d*x+c)/a^(1/4))/a^(7/4)/(a^(1/2)+b^(1/2))^(3/2 )/d-1/4*b*tanh(d*x+c)*(1-2*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 = 7.02 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\frac {-\frac {\left (4 a+\sqrt {a} \sqrt {b}-3 b\right ) \arctan \left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}}}+\frac {\left (4 a-\sqrt {a} \sqrt {b}-3 b\right ) \text {arctanh}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}}}+\frac {2 \sqrt {a} b (-6 \sinh (2 (c+d x))+\sinh (4 (c+d x)))}{8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))}}{8 a^{3/2} (a-b) d} \] Input:
Integrate[(a - b*Sinh[c + d*x]^4)^(-2),x]
Output:
(-(((4*a + Sqrt[a]*Sqrt[b] - 3*b)*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x ])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]]) + ((4*a - Sqrt [a]*Sqrt[b] - 3*b)*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sq rt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] + (2*Sqrt[a]*b*(-6*Sinh[2*(c + d*x)] + Sinh[4*(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 = 0.53 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.24, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3688, 1517, 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.
| \(\displaystyle \int \frac {1}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a-b \sin (i c+i d x)^4\right )^2}dx\) |
| \(\Big \downarrow \) 3688 |
| \(\displaystyle \frac {\int \frac {\left (1-\tanh ^2(c+d x)\right )^3}{\left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1517 |
| \(\displaystyle \frac {-\frac {\int -\frac {2 a b \left (-2 (2 a-b) \tanh ^2(c+d x)+4 a-3 b\right )}{(a-b) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{8 a^2 b}-\frac {b \tanh (c+d x) \left (1-2 \tanh ^2(c+d x)\right )}{4 a (a-b) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {-2 (2 a-b) \tanh ^2(c+d x)+4 a-3 b}{(a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a}d\tanh (c+d x)}{4 a (a-b)}-\frac {b \tanh (c+d x) \left (1-2 \tanh ^2(c+d x)\right )}{4 a (a-b) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}}{d}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {-\frac {\left (4 \sqrt {a}-3 \sqrt {b}\right ) \left (\sqrt {a}+\sqrt {b}\right )^2 \int \frac {1}{(a-b) \tanh ^2(c+d x)-\sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )}d\tanh (c+d x)}{2 \sqrt {a}}-\frac {\left (4 \sqrt {a}+3 \sqrt {b}\right ) \left (-2 \sqrt {a} \sqrt {b}+a+b\right ) \int \frac {1}{(a-b) \tanh ^2(c+d x)-\sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )}d\tanh (c+d x)}{2 \sqrt {a}}}{4 a (a-b)}-\frac {b \tanh (c+d x) \left (1-2 \tanh ^2(c+d x)\right )}{4 a (a-b) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {\left (4 \sqrt {a}-3 \sqrt {b}\right ) \left (\sqrt {a}+\sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\left (4 \sqrt {a}+3 \sqrt {b}\right ) \left (-2 \sqrt {a} \sqrt {b}+a+b\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt {a}-\sqrt {b}\right ) \sqrt {\sqrt {a}+\sqrt {b}}}}{4 a (a-b)}-\frac {b \tanh (c+d x) \left (1-2 \tanh ^2(c+d x)\right )}{4 a (a-b) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}}{d}\) |
Input:
Int[(a - b*Sinh[c + d*x]^4)^(-2),x]
Output:
((((4*Sqrt[a] - 3*Sqrt[b])*(Sqrt[a] + Sqrt[b])*ArcTanh[(Sqrt[Sqrt[a] - Sqr t[b]]*Tanh[c + d*x])/a^(1/4)])/(2*a^(3/4)*Sqrt[Sqrt[a] - Sqrt[b]]) + ((4*S qrt[a] + 3*Sqrt[b])*(a - 2*Sqrt[a]*Sqrt[b] + b)*ArcTanh[(Sqrt[Sqrt[a] + Sq rt[b]]*Tanh[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] - Sqrt[b])*Sqrt[Sqrt[a ] + Sqrt[b]]))/(4*a*(a - b)) - (b*Tanh[c + d*x]*(1 - 2*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
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[((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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[(a + 2*a*ff^2*x^2 + ( a + b)*ff^4*x^4)^p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff], x]] / ; FreeQ[{a, b, e, f}, x] && IntegerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.15 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.46
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 a \left (a -b \right )}-\frac {5 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 a \left (a -b \right )}-\frac {5 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4 a \left (a -b \right )}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a \left (a -b \right )}\right )}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a -4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a +6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -16 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\left (4 a -3 b \right ) \textit {\_R}^{6}+\left (-12 a +5 b \right ) \textit {\_R}^{4}+\left (12 a -5 b \right ) \textit {\_R}^{2}-4 a +3 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}}{16 a \left (a -b \right )}}{d}\) | \(307\) |
| default | \(\frac {-\frac {2 \left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 a \left (a -b \right )}-\frac {5 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 a \left (a -b \right )}-\frac {5 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4 a \left (a -b \right )}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a \left (a -b \right )}\right )}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a -4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a +6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -16 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\left (4 a -3 b \right ) \textit {\_R}^{6}+\left (-12 a +5 b \right ) \textit {\_R}^{4}+\left (12 a -5 b \right ) \textit {\_R}^{2}-4 a +3 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}}{16 a \left (a -b \right )}}{d}\) | \(307\) |
| risch | \(\frac {-{\mathrm e}^{6 d x +6 c} b +8 \,{\mathrm e}^{4 d x +4 c} a -3 b \,{\mathrm e}^{4 d x +4 c}+5 \,{\mathrm e}^{2 d x +2 c} b -b}{2 a d \left (a -b \right ) \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^{10} d^{4}-196608 a^{9} b \,d^{4}+196608 a^{8} b^{2} d^{4}-65536 a^{7} b^{3} d^{4}\right ) \textit {\_Z}^{4}+\left (-8192 a^{6} d^{2}+7680 a^{5} b \,d^{2}-1536 a^{4} b^{2} d^{2}\right ) \textit {\_Z}^{2}+256 a^{2}-288 a b +81 b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (-\frac {32768 d^{3} a^{10}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {114688 a^{9} b \,d^{3}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {147456 a^{8} b^{2} d^{3}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {81920 a^{7} b^{3} d^{3}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {16384 a^{6} b^{4} d^{3}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}\right ) \textit {\_R}^{3}+\left (\frac {8192 d^{2} a^{8}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {29184 a^{7} b \,d^{2}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {38400 a^{6} b^{2} d^{2}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {22016 a^{5} b^{3} d^{2}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {4608 a^{4} b^{4} d^{2}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}\right ) \textit {\_R}^{2}+\left (\frac {2048 d \,a^{6}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {896 a^{5} b d}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {5600 a^{4} b^{2} d}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {4032 a^{3} b^{3} d}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {864 d \,b^{4} a^{2}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}\right ) \textit {\_R} -\frac {512 a^{4}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {384 a^{3} b}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {314 a^{2} b^{2}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {351 a \,b^{3}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {81 b^{4}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}\right )\right )\) | \(987\) |
Input:
int(1/(a-b*sinh(d*x+c)^4)^2,x,method=_RETURNVERBOSE)
Output:
1/d*(-2*(1/4*b/a/(a-b)*tanh(1/2*d*x+1/2*c)^7-5/4*b/a/(a-b)*tanh(1/2*d*x+1/ 2*c)^5-5/4*b/a/(a-b)*tanh(1/2*d*x+1/2*c)^3+1/4*b/a/(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/(a-b )*sum(((4*a-3*b)*_R^6+(-12*a+5*b)*_R^4+(12*a-5*b)*_R^2-4*a+3*b)/(_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. 6522 vs. \(2 (164) = 328\).
Time = 0.40 (sec) , antiderivative size = 6522, normalized size of antiderivative = 31.06 \[ \int \frac {1}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(a-b*sinh(d*x+c)^4)^2,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {1}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(a-b*sinh(d*x+c)**4)**2,x)
Output:
Timed out
\[ \int \frac {1}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int { \frac {1}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \] Input:
integrate(1/(a-b*sinh(d*x+c)^4)^2,x, algorithm="maxima")
Output:
-1/2*((8*a*e^(4*c) - 3*b*e^(4*c))*e^(4*d*x) - b*e^(6*d*x + 6*c) + 5*b*e^(2 *d*x + 2*c) - 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)) + integrate(-(2*(8*a*e^(4*c) - 5*b*e^( 4*c))*e^(4*d*x) - b*e^(6*d*x + 6*c) - b*e^(2*d*x + 2*c))/(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)
\[ \int \frac {1}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int { \frac {1}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \] Input:
integrate(1/(a-b*sinh(d*x+c)^4)^2,x, algorithm="giac")
Output:
sage0*x
Timed out. \[ \int \frac {1}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int \frac {1}{{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^2} \,d x \] Input:
int(1/(a - b*sinh(c + d*x)^4)^2,x)
Output:
int(1/(a - b*sinh(c + d*x)^4)^2, x)
\[ \int \frac {1}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\text {too large to display} \] Input:
int(1/(a-b*sinh(d*x+c)^4)^2,x)
Output:
(32*(94371840*e**(12*c + 8*d*x)*int(e**(4*d*x)/(1920*e**(16*c + 16*d*x)*a* *2*b**2 + 80*e**(16*c + 16*d*x)*a*b**3 - 3*e**(16*c + 16*d*x)*b**4 - 15360 *e**(14*c + 14*d*x)*a**2*b**2 - 640*e**(14*c + 14*d*x)*a*b**3 + 24*e**(14* c + 14*d*x)*b**4 - 61440*e**(12*c + 12*d*x)*a**3*b + 51200*e**(12*c + 12*d *x)*a**2*b**2 + 2336*e**(12*c + 12*d*x)*a*b**3 - 84*e**(12*c + 12*d*x)*b** 4 + 245760*e**(10*c + 10*d*x)*a**3*b - 97280*e**(10*c + 10*d*x)*a**2*b**2 - 4864*e**(10*c + 10*d*x)*a*b**3 + 168*e**(10*c + 10*d*x)*b**4 + 491520*e* *(8*c + 8*d*x)*a**4 - 348160*e**(8*c + 8*d*x)*a**3*b + 118272*e**(8*c + 8* d*x)*a**2*b**2 + 6176*e**(8*c + 8*d*x)*a*b**3 - 210*e**(8*c + 8*d*x)*b**4 + 245760*e**(6*c + 6*d*x)*a**3*b - 97280*e**(6*c + 6*d*x)*a**2*b**2 - 4864 *e**(6*c + 6*d*x)*a*b**3 + 168*e**(6*c + 6*d*x)*b**4 - 61440*e**(4*c + 4*d *x)*a**3*b + 51200*e**(4*c + 4*d*x)*a**2*b**2 + 2336*e**(4*c + 4*d*x)*a*b* *3 - 84*e**(4*c + 4*d*x)*b**4 - 15360*e**(2*c + 2*d*x)*a**2*b**2 - 640*e** (2*c + 2*d*x)*a*b**3 + 24*e**(2*c + 2*d*x)*b**4 + 1920*a**2*b**2 + 80*a*b* *3 - 3*b**4),x)*a**5*b*d - 57016320*e**(12*c + 8*d*x)*int(e**(4*d*x)/(1920 *e**(16*c + 16*d*x)*a**2*b**2 + 80*e**(16*c + 16*d*x)*a*b**3 - 3*e**(16*c + 16*d*x)*b**4 - 15360*e**(14*c + 14*d*x)*a**2*b**2 - 640*e**(14*c + 14*d* x)*a*b**3 + 24*e**(14*c + 14*d*x)*b**4 - 61440*e**(12*c + 12*d*x)*a**3*b + 51200*e**(12*c + 12*d*x)*a**2*b**2 + 2336*e**(12*c + 12*d*x)*a*b**3 - 84* e**(12*c + 12*d*x)*b**4 + 245760*e**(10*c + 10*d*x)*a**3*b - 97280*e**(...