Integrand size = 24, antiderivative size = 222 \[ \int \frac {\sinh ^6(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\frac {\left (2 \sqrt {a}-3 \sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} b^{3/2} d}-\frac {\left (2 \sqrt {a}+3 \sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} b^{3/2} d}+\frac {\tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 (a-b) b d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )} \] Output:
1/8*(2*a^(1/2)-3*b^(1/2))*arctanh((a^(1/2)-b^(1/2))^(1/2)*tanh(d*x+c)/a^(1 /4))/a^(1/4)/(a^(1/2)-b^(1/2))^(3/2)/b^(3/2)/d-1/8*(2*a^(1/2)+3*b^(1/2))*a rctanh((a^(1/2)+b^(1/2))^(1/2)*tanh(d*x+c)/a^(1/4))/a^(1/4)/(a^(1/2)+b^(1/ 2))^(3/2)/b^(3/2)/d+1/4*tanh(d*x+c)*(a-(a+b)*tanh(d*x+c)^2)/(a-b)/b/d/(a-2 *a*tanh(d*x+c)^2+(a-b)*tanh(d*x+c)^4)
Time = 4.82 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.07 \[ \int \frac {\sinh ^6(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\frac {\frac {\sqrt {b} \left (-2 a+\sqrt {a} \sqrt {b}+3 b\right ) \arctan \left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}}}+\frac {\sqrt {b} \left (-2 a-\sqrt {a} \sqrt {b}+3 b\right ) \text {arctanh}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}}}-\frac {4 b (-2 a-b+b \cosh (2 (c+d x))) \sinh (2 (c+d x))}{8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))}}{8 (a-b) b^2 d} \] Input:
Integrate[Sinh[c + d*x]^6/(a - b*Sinh[c + d*x]^4)^2,x]
Output:
((Sqrt[b]*(-2*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]] + (Sqrt[ b]*(-2*a - Sqrt[a]*Sqrt[b] + 3*b)*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d* x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] - (4*b*(-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 - b)*b^2*d)
Time = 0.52 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.23, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3042, 25, 3696, 1440, 27, 1602, 25, 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.
| \(\displaystyle \int \frac {\sinh ^6(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sin (i c+i d x)^6}{\left (a-b \sin (i c+i d x)^4\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\sin (i c+i d x)^6}{\left (a-b \sin (i c+i d x)^4\right )^2}dx\) |
| \(\Big \downarrow \) 3696 |
| \(\displaystyle \frac {\int \frac {\tanh ^6(c+d x)}{\left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1440 |
| \(\displaystyle \frac {\frac {\tanh ^3(c+d x) \left (1-\tanh ^2(c+d x)\right )}{4 b \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac {\int \frac {2 a \tanh ^2(c+d x) \left (3-\tanh ^2(c+d x)\right )}{(a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a}d\tanh (c+d x)}{8 a b}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\tanh ^3(c+d x) \left (1-\tanh ^2(c+d x)\right )}{4 b \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac {\int \frac {\tanh ^2(c+d x) \left (3-\tanh ^2(c+d x)\right )}{(a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a}d\tanh (c+d x)}{4 b}}{d}\) |
| \(\Big \downarrow \) 1602 |
| \(\displaystyle \frac {\frac {\tanh ^3(c+d x) \left (1-\tanh ^2(c+d x)\right )}{4 b \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac {-\frac {\int -\frac {(a-3 b) \tanh ^2(c+d x)+a}{(a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a}d\tanh (c+d x)}{a-b}-\frac {\tanh (c+d x)}{a-b}}{4 b}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\tanh ^3(c+d x) \left (1-\tanh ^2(c+d x)\right )}{4 b \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac {\frac {\int \frac {(a-3 b) \tanh ^2(c+d x)+a}{(a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a}d\tanh (c+d x)}{a-b}-\frac {\tanh (c+d x)}{a-b}}{4 b}}{d}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {\tanh ^3(c+d x) \left (1-\tanh ^2(c+d x)\right )}{4 b \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac {\frac {\frac {1}{2} \left (-\frac {2 \sqrt {a} (a-2 b)}{\sqrt {b}}+a-3 b\right ) \int \frac {1}{(a-b) \tanh ^2(c+d x)-\sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )}d\tanh (c+d x)+\frac {1}{2} \left (\frac {2 \sqrt {a} (a-2 b)}{\sqrt {b}}+a-3 b\right ) \int \frac {1}{(a-b) \tanh ^2(c+d x)-\sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )}d\tanh (c+d x)}{a-b}-\frac {\tanh (c+d x)}{a-b}}{4 b}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\tanh ^3(c+d x) \left (1-\tanh ^2(c+d x)\right )}{4 b \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac {\frac {-\frac {\left (\frac {2 \sqrt {a} (a-2 b)}{\sqrt {b}}+a-3 b\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} \left (\sqrt {a}+\sqrt {b}\right )}-\frac {\left (-\frac {2 \sqrt {a} (a-2 b)}{\sqrt {b}}+a-3 b\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {b}\right ) \sqrt {\sqrt {a}+\sqrt {b}}}}{a-b}-\frac {\tanh (c+d x)}{a-b}}{4 b}}{d}\) |
Input:
Int[Sinh[c + d*x]^6/(a - b*Sinh[c + d*x]^4)^2,x]
Output:
(-1/4*((-1/2*((a + (2*Sqrt[a]*(a - 2*b))/Sqrt[b] - 3*b)*ArcTanh[(Sqrt[Sqrt [a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(a^(1/4)*Sqrt[Sqrt[a] - Sqrt[b]]*( Sqrt[a] + Sqrt[b])) - ((a - (2*Sqrt[a]*(a - 2*b))/Sqrt[b] - 3*b)*ArcTanh[( Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(2*a^(1/4)*(Sqrt[a] - Sqr t[b])*Sqrt[Sqrt[a] + Sqrt[b]]))/(a - b) - Tanh[c + d*x]/(a - b))/b + (Tanh [c + d*x]^3*(1 - Tanh[c + d*x]^2))/(4*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_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-d^3)*(d*x)^(m - 3)*(2*a + b*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2* (p + 1)*(b^2 - 4*a*c))), x] + Simp[d^4/(2*(p + 1)*(b^2 - 4*a*c)) Int[(d*x )^(m - 4)*(2*a*(m - 3) + b*(m + 4*p + 3)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && Gt Q[m, 3] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
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[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( x_)^4)^(p_), x_Symbol] :> Simp[e*f*(f*x)^(m - 1)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 3))), x] - Simp[f^2/(c*(m + 4*p + 3)) Int[(f*x)^(m - 2)* (a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p + 3))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c , 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] | | IntegerQ[m])
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 = 3.86 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.37
| method | result | size |
| derivativedivides | \(\frac {-\frac {128 \left (-\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{256 b \left (a -b \right )}+\frac {\left (a +4 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{256 b \left (a -b \right )}+\frac {\left (a +4 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{256 b \left (a -b \right )}-\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{256 b \left (a -b \right )}\right )}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a -4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a +6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -16 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (-a \,\textit {\_R}^{6}+\left (-5 a +12 b \right ) \textit {\_R}^{4}+\left (5 a -12 b \right ) \textit {\_R}^{2}+a \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}}{16 b \left (a -b \right )}}{d}\) | \(305\) |
| default | \(\frac {-\frac {128 \left (-\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{256 b \left (a -b \right )}+\frac {\left (a +4 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{256 b \left (a -b \right )}+\frac {\left (a +4 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{256 b \left (a -b \right )}-\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{256 b \left (a -b \right )}\right )}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a -4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a +6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -16 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (-a \,\textit {\_R}^{6}+\left (-5 a +12 b \right ) \textit {\_R}^{4}+\left (5 a -12 b \right ) \textit {\_R}^{2}+a \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}}{16 b \left (a -b \right )}}{d}\) | \(305\) |
| risch | \(\frac {2 \,{\mathrm e}^{6 d x +6 c} a -{\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}-2 \,{\mathrm e}^{2 d x +2 c} a -3 \,{\mathrm e}^{2 d x +2 c} b +b}{2 b 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^{4} b^{6} d^{4}-196608 a^{3} b^{7} d^{4}+196608 a^{2} b^{8} d^{4}-65536 a \,b^{9} d^{4}\right ) \textit {\_Z}^{4}+\left (-2048 a^{3} b^{3} d^{2}+7680 a^{2} b^{4} d^{2}-7680 a \,b^{5} d^{2}\right ) \textit {\_Z}^{2}+16 a^{2}-72 a b +81 b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (-\frac {8192 a^{5} b^{5} d^{3}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}+\frac {49152 a^{4} b^{6} d^{3}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}-\frac {98304 a^{3} b^{7} d^{3}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}+\frac {81920 a^{2} b^{8} d^{3}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}-\frac {24576 a \,b^{9} d^{3}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}\right ) \textit {\_R}^{3}+\left (-\frac {2048 d^{2} b^{3} a^{5}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}+\frac {10752 d^{2} b^{4} a^{4}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}-\frac {19968 d^{2} b^{5} a^{3}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}+\frac {15872 d^{2} b^{6} a^{2}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}-\frac {4608 d^{2} b^{7} a}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}\right ) \textit {\_R}^{2}+\left (\frac {128 a^{4} b^{2} d}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}-\frac {1184 a^{3} b^{3} d}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}+\frac {3136 a^{2} b^{4} d}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}-\frac {2592 a \,b^{5} d}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}\right ) \textit {\_R} +\frac {32 a^{4}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}-\frac {192 a^{3} b}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}+\frac {370 a^{2} b^{2}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}-\frac {189 a \,b^{3}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}-\frac {81 b^{4}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}\right )\right )\) | \(873\) |
Input:
int(sinh(d*x+c)^6/(a-b*sinh(d*x+c)^4)^2,x,method=_RETURNVERBOSE)
Output:
1/d*(-128*(-1/256/b/(a-b)*a*tanh(1/2*d*x+1/2*c)^7+1/256/b*(a+4*b)/(a-b)*ta nh(1/2*d*x+1/2*c)^5+1/256/b*(a+4*b)/(a-b)*tanh(1/2*d*x+1/2*c)^3-1/256/b/(a -b)*a*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/b/(a-b)*sum((-a*_R^6+(-5*a+12*b)*_R^4+(5*a-12*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=Roo tOf(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. 6045 vs. \(2 (173) = 346\).
Time = 0.41 (sec) , antiderivative size = 6045, normalized size of antiderivative = 27.23 \[ \int \frac {\sinh ^6(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(sinh(d*x+c)^6/(a-b*sinh(d*x+c)^4)^2,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\sinh ^6(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\text {Timed out} \] Input:
integrate(sinh(d*x+c)**6/(a-b*sinh(d*x+c)**4)**2,x)
Output:
Timed out
\[ \int \frac {\sinh ^6(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int { \frac {\sinh \left (d x + c\right )^{6}}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \] Input:
integrate(sinh(d*x+c)^6/(a-b*sinh(d*x+c)^4)^2,x, algorithm="maxima")
Output:
-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*b^2*d - b^3*d + (a *b^2*d*e^(8*c) - b^3*d*e^(8*c))*e^(8*d*x) - 4*(a*b^2*d*e^(6*c) - b^3*d*e^( 6*c))*e^(6*d*x) - 2*(8*a^2*b*d*e^(4*c) - 11*a*b^2*d*e^(4*c) + 3*b^3*d*e^(4 *c))*e^(4*d*x) - 4*(a*b^2*d*e^(2*c) - b^3*d*e^(2*c))*e^(2*d*x)) + 1/64*int egrate(64*((2*a*e^(6*c) - 3*b*e^(6*c))*e^(6*d*x) + (2*a*e^(2*c) - 3*b*e^(2 *c))*e^(2*d*x) + 6*b*e^(4*d*x + 4*c))/(a*b^2 - b^3 + (a*b^2*e^(8*c) - b^3* e^(8*c))*e^(8*d*x) - 4*(a*b^2*e^(6*c) - b^3*e^(6*c))*e^(6*d*x) - 2*(8*a^2* b*e^(4*c) - 11*a*b^2*e^(4*c) + 3*b^3*e^(4*c))*e^(4*d*x) - 4*(a*b^2*e^(2*c) - b^3*e^(2*c))*e^(2*d*x)), x)
\[ \int \frac {\sinh ^6(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int { \frac {\sinh \left (d x + c\right )^{6}}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \] Input:
integrate(sinh(d*x+c)^6/(a-b*sinh(d*x+c)^4)^2,x, algorithm="giac")
Output:
sage0*x
Timed out. \[ \int \frac {\sinh ^6(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^6}{{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^2} \,d x \] Input:
int(sinh(c + d*x)^6/(a - b*sinh(c + d*x)^4)^2,x)
Output:
int(sinh(c + d*x)^6/(a - b*sinh(c + d*x)^4)^2, x)
\[ \int \frac {\sinh ^6(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\text {too large to display} \] Input:
int(sinh(d*x+c)^6/(a-b*sinh(d*x+c)^4)^2,x)
Output:
(4026531840*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**7*b*d - 8136949760*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**(...