Integrand size = 24, antiderivative size = 345 \[ \int \frac {\sinh ^6(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\frac {\left (4 a-10 \sqrt {a} \sqrt {b}+3 b\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{5/4} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{3/2} d}-\frac {\left (4 a+10 \sqrt {a} \sqrt {b}+3 b\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{5/4} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{3/2} d}+\frac {\tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 (a-b)^3 d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )^2}+\frac {\tanh (c+d x) \left (\frac {2 a \left (a^2-a b-8 b^2\right )}{(a-b)^3}-\frac {\left (2 a^2+15 a b+3 b^2\right ) \tanh ^2(c+d x)}{(a-b)^2}\right )}{32 a b d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )} \]
1/64*arctanh((a^(1/2)-b^(1/2))^(1/2)*tanh(d*x+c)/a^(1/4))*(4*a+3*b-10*a^(1 /2)*b^(1/2))/a^(5/4)/b^(3/2)/d/(a^(1/2)-b^(1/2))^(5/2)-1/64*arctanh((a^(1/ 2)+b^(1/2))^(1/2)*tanh(d*x+c)/a^(1/4))*(4*a+3*b+10*a^(1/2)*b^(1/2))/a^(5/4 )/b^(3/2)/d/(a^(1/2)+b^(1/2))^(5/2)+1/8*tanh(d*x+c)*(a*(a+3*b)-(a^2+6*a*b+ b^2)*tanh(d*x+c)^2)/(a-b)^3/d/(a-2*a*tanh(d*x+c)^2+(a-b)*tanh(d*x+c)^4)^2+ 1/32*tanh(d*x+c)*(2*a*(a^2-a*b-8*b^2)/(a-b)^3-(2*a^2+15*a*b+3*b^2)*tanh(d* x+c)^2/(a-b)^2)/a/b/d/(a-2*a*tanh(d*x+c)^2+(a-b)*tanh(d*x+c)^4)
Time = 7.92 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.02 \[ \int \frac {\sinh ^6(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=-\frac {\frac {\left (\sqrt {a}+\sqrt {b}\right )^2 \left (4 a-10 \sqrt {a} \sqrt {b}+3 b\right ) \arctan \left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{a \sqrt {-a+\sqrt {a} \sqrt {b}} b^{3/2}}+\frac {\left (\sqrt {a}-\sqrt {b}\right )^2 \left (4 a+10 \sqrt {a} \sqrt {b}+3 b\right ) \text {arctanh}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{a \sqrt {a+\sqrt {a} \sqrt {b}} b^{3/2}}+\frac {4 \left (4 a^2-19 a b-3 b^2+3 b (a+b) \cosh (2 (c+d x))\right ) \sinh (2 (c+d x))}{a b (8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x)))}-\frac {128 (a-b) (2 a+b-b \cosh (2 (c+d x))) \sinh (2 (c+d x))}{b (-8 a+3 b-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x)))^2}}{64 (a-b)^2 d} \]
-1/64*(((Sqrt[a] + Sqrt[b])^2*(4*a - 10*Sqrt[a]*Sqrt[b] + 3*b)*ArcTan[((Sq rt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(a*Sqrt[-a + Sqrt[a]*Sqrt[b]]*b^(3/2)) + ((Sqrt[a] - Sqrt[b])^2*(4*a + 10*Sqrt[a]*Sqrt[ b] + 3*b)*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqr t[b]]])/(a*Sqrt[a + Sqrt[a]*Sqrt[b]]*b^(3/2)) + (4*(4*a^2 - 19*a*b - 3*b^2 + 3*b*(a + b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/(a*b*(8*a - 3*b + 4*b *Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)])) - (128*(a - b)*(2*a + b - b*Cos h[2*(c + d*x)])*Sinh[2*(c + d*x)])/(b*(-8*a + 3*b - 4*b*Cosh[2*(c + d*x)] + b*Cosh[4*(c + d*x)])^2))/((a - b)^2*d)
Time = 0.88 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.11, 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, 1672, 27, 2206, 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.
| \(\displaystyle \int \frac {\sinh ^6(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sin (i c+i d x)^6}{\left (a-b \sin (i c+i d x)^4\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\sin (i c+i d x)^6}{\left (a-b \sin (i c+i d x)^4\right )^3}dx\) |
| \(\Big \downarrow \) 3696 |
| \(\displaystyle \frac {\int \frac {\tanh ^6(c+d x) \left (1-\tanh ^2(c+d x)\right )^2}{\left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1672 |
| \(\displaystyle \frac {\frac {\tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 (a-b)^3 \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}-\frac {\int \frac {2 \left (-\frac {8 a^2 b \tanh ^6(c+d x)}{a-b}-\frac {16 a^2 b^2 \tanh ^4(c+d x)}{(a-b)^2}+\frac {a^2 b \left (5 a^2+6 b a-3 b^2\right ) \tanh ^2(c+d x)}{(a-b)^3}+\frac {a^3 b (a+3 b)}{(a-b)^3}\right )}{\left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{16 a^2 b}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 (a-b)^3 \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}-\frac {\int \frac {-\frac {8 a^2 b \tanh ^6(c+d x)}{a-b}-\frac {16 a^2 b^2 \tanh ^4(c+d x)}{(a-b)^2}+\frac {a^2 b \left (5 a^2+6 b a-3 b^2\right ) \tanh ^2(c+d x)}{(a-b)^3}+\frac {a^3 b (a+3 b)}{(a-b)^3}}{\left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{8 a^2 b}}{d}\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle \frac {\frac {\tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 (a-b)^3 \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}-\frac {-\frac {\int -\frac {2 a^3 b \left (\left (2 a^2-17 b a+3 b^2\right ) \tanh ^2(c+d x)+2 a (a+2 b)\right )}{(a-b)^2 \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{8 a^2 b}-\frac {a \tanh (c+d x) \left (\frac {2 a \left (a^2-a b-8 b^2\right )}{(a-b)^3}-\frac {\left (2 a^2+15 a b+3 b^2\right ) \tanh ^2(c+d x)}{(a-b)^2}\right )}{4 \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}}{8 a^2 b}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 (a-b)^3 \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}-\frac {\frac {a \int \frac {\left (2 a^2-17 b a+3 b^2\right ) \tanh ^2(c+d x)+2 a (a+2 b)}{(a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a}d\tanh (c+d x)}{4 (a-b)^2}-\frac {a \tanh (c+d x) \left (\frac {2 a \left (a^2-a b-8 b^2\right )}{(a-b)^3}-\frac {\left (2 a^2+15 a b+3 b^2\right ) \tanh ^2(c+d x)}{(a-b)^2}\right )}{4 \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}}{8 a^2 b}}{d}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {\tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 (a-b)^3 \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}-\frac {\frac {a \left (\frac {\left (\sqrt {a}+\sqrt {b}\right )^3 \left (-10 \sqrt {a} \sqrt {b}+4 a+3 b\right ) \int \frac {1}{(a-b) \tanh ^2(c+d x)-\sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )}d\tanh (c+d x)}{2 \sqrt {b}}-\frac {\left (\sqrt {a}-\sqrt {b}\right )^3 \left (10 \sqrt {a} \sqrt {b}+4 a+3 b\right ) \int \frac {1}{(a-b) \tanh ^2(c+d x)-\sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )}d\tanh (c+d x)}{2 \sqrt {b}}\right )}{4 (a-b)^2}-\frac {a \tanh (c+d x) \left (\frac {2 a \left (a^2-a b-8 b^2\right )}{(a-b)^3}-\frac {\left (2 a^2+15 a b+3 b^2\right ) \tanh ^2(c+d x)}{(a-b)^2}\right )}{4 \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}}{8 a^2 b}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 (a-b)^3 \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}-\frac {\frac {a \left (\frac {\left (\sqrt {a}-\sqrt {b}\right )^2 \left (10 \sqrt {a} \sqrt {b}+4 a+3 b\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \sqrt {b} \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\left (\sqrt {a}+\sqrt {b}\right )^2 \left (-10 \sqrt {a} \sqrt {b}+4 a+3 b\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \sqrt {b} \sqrt {\sqrt {a}-\sqrt {b}}}\right )}{4 (a-b)^2}-\frac {a \tanh (c+d x) \left (\frac {2 a \left (a^2-a b-8 b^2\right )}{(a-b)^3}-\frac {\left (2 a^2+15 a b+3 b^2\right ) \tanh ^2(c+d x)}{(a-b)^2}\right )}{4 \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}}{8 a^2 b}}{d}\) |
((Tanh[c + d*x]*(a*(a + 3*b) - (a^2 + 6*a*b + b^2)*Tanh[c + d*x]^2))/(8*(a - b)^3*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4)^2) - ((a*(-1/2 *((Sqrt[a] + Sqrt[b])^2*(4*a - 10*Sqrt[a]*Sqrt[b] + 3*b)*ArcTanh[(Sqrt[Sqr t[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(a^(1/4)*Sqrt[Sqrt[a] - Sqrt[b]]* Sqrt[b]) + ((Sqrt[a] - Sqrt[b])^2*(4*a + 10*Sqrt[a]*Sqrt[b] + 3*b)*ArcTanh [(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(2*a^(1/4)*Sqrt[Sqrt[a] + Sqrt[b]]*Sqrt[b])))/(4*(a - b)^2) - (a*Tanh[c + d*x]*((2*a*(a^2 - a*b - 8*b^2))/(a - b)^3 - ((2*a^2 + 15*a*b + 3*b^2)*Tanh[c + d*x]^2)/(a - b)^2) )/(4*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4)))/(8*a^2*b))/d
3.3.60.3.1 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[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*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[Px, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x ^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
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 = 5.74 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.72
| method | result | size |
| derivativedivides | \(\frac {-\frac {128 \left (-\frac {\left (a +2 b \right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{1024 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (5 a^{2}+24 a b -2 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{1024 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (9 a^{2}+76 a b -70 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{1024 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (5 a^{3}+54 a^{2} b -164 a \,b^{2}-96 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{1024 a b \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (5 a^{3}+54 a^{2} b -164 a \,b^{2}-96 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{1024 a b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (9 a^{2}+76 a b -70 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{1024 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (5 a^{2}+24 a b -2 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{1024 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (a +2 b \right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{1024 b \left (a^{2}-2 a b +b^{2}\right )}\right )}{\left (\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 \right )^{2}}-\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 \left (-a -2 b \right ) \textit {\_R}^{6}+\left (-5 a^{2}+32 a b -6 b^{2}\right ) \textit {\_R}^{4}+\left (5 a^{2}-32 a b +6 b^{2}\right ) \textit {\_R}^{2}+a^{2}+2 a 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}}{64 a b \left (a^{2}-2 a b +b^{2}\right )}}{d}\) | \(594\) |
| default | \(\frac {-\frac {128 \left (-\frac {\left (a +2 b \right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{1024 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (5 a^{2}+24 a b -2 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{1024 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (9 a^{2}+76 a b -70 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{1024 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (5 a^{3}+54 a^{2} b -164 a \,b^{2}-96 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{1024 a b \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (5 a^{3}+54 a^{2} b -164 a \,b^{2}-96 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{1024 a b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (9 a^{2}+76 a b -70 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{1024 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (5 a^{2}+24 a b -2 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{1024 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (a +2 b \right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{1024 b \left (a^{2}-2 a b +b^{2}\right )}\right )}{\left (\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 \right )^{2}}-\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 \left (-a -2 b \right ) \textit {\_R}^{6}+\left (-5 a^{2}+32 a b -6 b^{2}\right ) \textit {\_R}^{4}+\left (5 a^{2}-32 a b +6 b^{2}\right ) \textit {\_R}^{2}+a^{2}+2 a 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}}{64 a b \left (a^{2}-2 a b +b^{2}\right )}}{d}\) | \(594\) |
| risch | \(\text {Expression too large to display}\) | \(2359\) |
1/d*(-128*(-1/1024*(a+2*b)*a/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)+1/1024* (5*a^2+24*a*b-2*b^2)/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3-1/1024/b*(9*a ^2+76*a*b-70*b^2)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^5+1/1024*(5*a^3+54*a ^2*b-164*a*b^2-96*b^3)/a/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^7+1/1024*(5 *a^3+54*a^2*b-164*a*b^2-96*b^3)/a/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^9- 1/1024/b*(9*a^2+76*a*b-70*b^2)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^11+1/10 24*(5*a^2+24*a*b-2*b^2)/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^13-1/1024*(a +2*b)*a/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^15)/(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)^2-1/64/a/b/(a^2-2*a*b+b^2)*sum((a*(-a-2 *b)*_R^6+(-5*a^2+32*a*b-6*b^2)*_R^4+(5*a^2-32*a*b+6*b^2)*_R^2+a^2+2*a*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=Root Of(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. 22729 vs. \(2 (292) = 584\).
Time = 1.12 (sec) , antiderivative size = 22729, normalized size of antiderivative = 65.88 \[ \int \frac {\sinh ^6(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\sinh ^6(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {\sinh ^6(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\int { -\frac {\sinh \left (d x + c\right )^{6}}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \]
-1/16*(3*a*b^2 + 3*b^3 - (4*a^2*b*e^(14*c) - 13*a*b^2*e^(14*c) + 3*b^3*e^( 14*c))*e^(14*d*x) + 3*(8*a^2*b*e^(12*c) - 33*a*b^2*e^(12*c) + 7*b^3*e^(12* c))*e^(12*d*x) - (64*a^3*e^(10*c) + 68*a^2*b*e^(10*c) - 225*a*b^2*e^(10*c) + 63*b^3*e^(10*c))*e^(10*d*x) + 3*(128*a^3*e^(8*c) + 32*a^2*b*e^(8*c) - 6 1*a*b^2*e^(8*c) + 35*b^3*e^(8*c))*e^(8*d*x) + (64*a^3*e^(6*c) + 452*a^2*b* e^(6*c) - 9*a*b^2*e^(6*c) - 105*b^3*e^(6*c))*e^(6*d*x) - 3*(40*a^2*b*e^(4* c) - 29*a*b^2*e^(4*c) - 21*b^3*e^(4*c))*e^(4*d*x) + (4*a^2*b*e^(2*c) - 37* a*b^2*e^(2*c) - 21*b^3*e^(2*c))*e^(2*d*x))/(a^3*b^3*d - 2*a^2*b^4*d + a*b^ 5*d + (a^3*b^3*d*e^(16*c) - 2*a^2*b^4*d*e^(16*c) + a*b^5*d*e^(16*c))*e^(16 *d*x) - 8*(a^3*b^3*d*e^(14*c) - 2*a^2*b^4*d*e^(14*c) + a*b^5*d*e^(14*c))*e ^(14*d*x) - 4*(8*a^4*b^2*d*e^(12*c) - 23*a^3*b^3*d*e^(12*c) + 22*a^2*b^4*d *e^(12*c) - 7*a*b^5*d*e^(12*c))*e^(12*d*x) + 8*(16*a^4*b^2*d*e^(10*c) - 39 *a^3*b^3*d*e^(10*c) + 30*a^2*b^4*d*e^(10*c) - 7*a*b^5*d*e^(10*c))*e^(10*d* x) + 2*(128*a^5*b*d*e^(8*c) - 352*a^4*b^2*d*e^(8*c) + 355*a^3*b^3*d*e^(8*c ) - 166*a^2*b^4*d*e^(8*c) + 35*a*b^5*d*e^(8*c))*e^(8*d*x) + 8*(16*a^4*b^2* d*e^(6*c) - 39*a^3*b^3*d*e^(6*c) + 30*a^2*b^4*d*e^(6*c) - 7*a*b^5*d*e^(6*c ))*e^(6*d*x) - 4*(8*a^4*b^2*d*e^(4*c) - 23*a^3*b^3*d*e^(4*c) + 22*a^2*b^4* d*e^(4*c) - 7*a*b^5*d*e^(4*c))*e^(4*d*x) - 8*(a^3*b^3*d*e^(2*c) - 2*a^2*b^ 4*d*e^(2*c) + a*b^5*d*e^(2*c))*e^(2*d*x)) + 1/64*integrate(8*((4*a^2*e^(6* c) - 13*a*b*e^(6*c) + 3*b^2*e^(6*c))*e^(6*d*x) + 6*(7*a*b*e^(4*c) - b^2...
\[ \int \frac {\sinh ^6(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\int { -\frac {\sinh \left (d x + c\right )^{6}}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\sinh ^6(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^6}{{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^3} \,d x \]