Integrand size = 24, antiderivative size = 288 \[ \int \frac {\sinh ^3(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=-\frac {\left (5 \sqrt {a}-2 \sqrt {b}\right ) \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{3/4} d}+\frac {\left (5 \sqrt {a}+2 \sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{3/4} d}-\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac {\cosh (c+d x) \left (11 a+b-(5 a+b) \cosh ^2(c+d x)\right )}{32 a (a-b)^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )} \]
-1/8*cosh(d*x+c)*(2-cosh(d*x+c)^2)/(a-b)/d/(a-b+2*b*cosh(d*x+c)^2-b*cosh(d *x+c)^4)^2-1/32*cosh(d*x+c)*(11*a+b-(5*a+b)*cosh(d*x+c)^2)/a/(a-b)^2/d/(a- b+2*b*cosh(d*x+c)^2-b*cosh(d*x+c)^4)-1/64*arctan(b^(1/4)*cosh(d*x+c)/(a^(1 /2)-b^(1/2))^(1/2))*(5*a^(1/2)-2*b^(1/2))/a^(3/2)/b^(3/4)/d/(a^(1/2)-b^(1/ 2))^(5/2)+1/64*arctanh(b^(1/4)*cosh(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))*(5*a^( 1/2)+2*b^(1/2))/a^(3/2)/b^(3/4)/d/(a^(1/2)+b^(1/2))^(5/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.52 (sec) , antiderivative size = 802, normalized size of antiderivative = 2.78 \[ \int \frac {\sinh ^3(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\frac {\frac {32 \cosh (c+d x) (-17 a-b+(5 a+b) \cosh (2 (c+d x)))}{a (8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x)))}+\frac {512 (a-b) (-5 \cosh (c+d x)+\cosh (3 (c+d x)))}{(-8 a+3 b-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x)))^2}+\frac {\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 {5 a c+b c+5 a d x+b d x+10 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 )+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 )-47 a c \text {$\#$1}^2+5 b c \text {$\#$1}^2-47 a d x \text {$\#$1}^2+5 b d x \text {$\#$1}^2-94 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+10 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+47 a c \text {$\#$1}^4-5 b c \text {$\#$1}^4+47 a d x \text {$\#$1}^4-5 b d x \text {$\#$1}^4+94 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}^4-10 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}^4-5 a c \text {$\#$1}^6-b c \text {$\#$1}^6-5 a d x \text {$\#$1}^6-b d x \text {$\#$1}^6-10 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}^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}}{256 (a-b)^2 d} \]
((32*Cosh[c + d*x]*(-17*a - b + (5*a + b)*Cosh[2*(c + d*x)]))/(a*(8*a - 3* b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)])) + (512*(a - b)*(-5*Cosh[ c + d*x] + Cosh[3*(c + d*x)]))/(-8*a + 3*b - 4*b*Cosh[2*(c + d*x)] + b*Cos h[4*(c + d*x)])^2 + RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (5*a*c + b*c + 5*a*d*x + b*d*x + 10*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 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] - 47*a*c*#1^2 + 5*b*c*#1^2 - 47*a*d*x*#1^2 + 5*b*d*x*#1^2 - 94*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 + 10*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 + 47*a*c*#1^4 - 5* b*c*#1^4 + 47*a*d*x*#1^4 - 5*b*d*x*#1^4 + 94*a*Log[-Cosh[(c + d*x)/2] - Si nh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 10*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 - 5*a*c*#1^6 - b*c*#1^6 - 5*a*d*x*#1^6 - b*d*x*#1^6 - 10*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6 - 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)/(256*(a - b)^2*d)
Time = 0.56 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3042, 26, 3694, 1492, 27, 1492, 27, 1480, 218, 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.
| \(\displaystyle \int \frac {\sinh ^3(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \sin (i c+i d x)^3}{\left (a-b \sin (i c+i d x)^4\right )^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\sin (i c+i d x)^3}{\left (a-b \sin (i c+i d x)^4\right )^3}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle -\frac {\int \frac {1-\cosh ^2(c+d x)}{\left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )^3}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle -\frac {\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}-\frac {\int -\frac {2 a b \left (6-5 \cosh ^2(c+d x)\right )}{\left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )^2}d\cosh (c+d x)}{16 a b (a-b)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {6-5 \cosh ^2(c+d x)}{\left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )^2}d\cosh (c+d x)}{8 (a-b)}+\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle -\frac {\frac {\frac {\cosh (c+d x) \left (-\left ((5 a+b) \cosh ^2(c+d x)\right )+11 a+b\right )}{4 a (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac {\int -\frac {2 b \left (-\left ((5 a+b) \cosh ^2(c+d x)\right )+13 a-b\right )}{-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b}d\cosh (c+d x)}{8 a b (a-b)}}{8 (a-b)}+\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {-\left ((5 a+b) \cosh ^2(c+d x)\right )+13 a-b}{-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b}d\cosh (c+d x)}{4 a (a-b)}+\frac {\cosh (c+d x) \left (-\left ((5 a+b) \cosh ^2(c+d x)\right )+11 a+b\right )}{4 a (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}}{8 (a-b)}+\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {\frac {\frac {-\frac {1}{2} \left (\frac {2 \sqrt {b} (4 a-b)}{\sqrt {a}}+5 a+b\right ) \int \frac {1}{-b \cosh ^2(c+d x)-\left (\sqrt {a}-\sqrt {b}\right ) \sqrt {b}}d\cosh (c+d x)-\frac {1}{2} \left (-\frac {2 \sqrt {b} (4 a-b)}{\sqrt {a}}+5 a+b\right ) \int \frac {1}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}-b \cosh ^2(c+d x)}d\cosh (c+d x)}{4 a (a-b)}+\frac {\cosh (c+d x) \left (-\left ((5 a+b) \cosh ^2(c+d x)\right )+11 a+b\right )}{4 a (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}}{8 (a-b)}+\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\left (\frac {2 \sqrt {b} (4 a-b)}{\sqrt {a}}+5 a+b\right ) \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{3/4} \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {1}{2} \left (-\frac {2 \sqrt {b} (4 a-b)}{\sqrt {a}}+5 a+b\right ) \int \frac {1}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}-b \cosh ^2(c+d x)}d\cosh (c+d x)}{4 a (a-b)}+\frac {\cosh (c+d x) \left (-\left ((5 a+b) \cosh ^2(c+d x)\right )+11 a+b\right )}{4 a (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}}{8 (a-b)}+\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\left (\frac {2 \sqrt {b} (4 a-b)}{\sqrt {a}}+5 a+b\right ) \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{3/4} \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\left (-\frac {2 \sqrt {b} (4 a-b)}{\sqrt {a}}+5 a+b\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 b^{3/4} \sqrt {\sqrt {a}+\sqrt {b}}}}{4 a (a-b)}+\frac {\cosh (c+d x) \left (-\left ((5 a+b) \cosh ^2(c+d x)\right )+11 a+b\right )}{4 a (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}}{8 (a-b)}+\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}}{d}\) |
-(((Cosh[c + d*x]*(2 - Cosh[c + d*x]^2))/(8*(a - b)*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4)^2) + ((((5*a + (2*(4*a - b)*Sqrt[b])/Sqrt[a] + b)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*Sqrt[Sqrt[ a] - Sqrt[b]]*b^(3/4)) - ((5*a - (2*(4*a - b)*Sqrt[b])/Sqrt[a] + b)*ArcTan h[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*Sqrt[Sqrt[a] + Sqrt [b]]*b^(3/4)))/(4*a*(a - b)) + (Cosh[c + d*x]*(11*a + b - (5*a + b)*Cosh[c + d*x]^2))/(4*a*(a - b)*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4) ))/(8*(a - b)))/d)
3.3.56.3.1 Defintions of rubi rules used
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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), 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] && LtQ[p, -1] && IntegerQ[2*p]
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. \(592\) vs. \(2(236)=472\).
Time = 11.49 (sec) , antiderivative size = 593, normalized size of antiderivative = 2.06
| method | result | size |
| derivativedivides | \(\frac {\frac {-\frac {\left (4 a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{8 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (a^{2}+58 a b -32 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{8 a \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (20 a^{2}-73 a b +48 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8 a \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (175 a^{3}-550 a^{2} b +832 a \,b^{2}-256 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8 a^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (220 a^{2}-533 a b +112 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{8 a \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (141 a^{2}-158 a b +32 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 a \left (a^{2}-2 a b +b^{2}\right )}+\frac {8 \left (44 a -17 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{64 a^{2}-128 a b +64 b^{2}}-\frac {5 a -2 b}{8 \left (a^{2}-2 a b +b^{2}\right )}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a -4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a +6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -16 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a \right )^{2}}+\frac {-\frac {\left (5 \sqrt {a b}\, a +\sqrt {a b}\, b -8 a b +2 b^{2}\right ) \arctan \left (\frac {-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}+2 a}{4 \sqrt {-a b -\sqrt {a b}\, a}}\right )}{8 a b \sqrt {-a b -\sqrt {a b}\, a}}+\frac {\left (-5 \sqrt {a b}\, a -\sqrt {a b}\, b -8 a b +2 b^{2}\right ) \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}-2 a}{4 \sqrt {-a b +\sqrt {a b}\, a}}\right )}{8 a b \sqrt {-a b +\sqrt {a b}\, a}}}{8 a^{2}-16 a b +8 b^{2}}}{d}\) | \(593\) |
| default | \(\frac {\frac {-\frac {\left (4 a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{8 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (a^{2}+58 a b -32 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{8 a \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (20 a^{2}-73 a b +48 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8 a \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (175 a^{3}-550 a^{2} b +832 a \,b^{2}-256 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8 a^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (220 a^{2}-533 a b +112 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{8 a \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (141 a^{2}-158 a b +32 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 a \left (a^{2}-2 a b +b^{2}\right )}+\frac {8 \left (44 a -17 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{64 a^{2}-128 a b +64 b^{2}}-\frac {5 a -2 b}{8 \left (a^{2}-2 a b +b^{2}\right )}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a -4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a +6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -16 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a \right )^{2}}+\frac {-\frac {\left (5 \sqrt {a b}\, a +\sqrt {a b}\, b -8 a b +2 b^{2}\right ) \arctan \left (\frac {-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}+2 a}{4 \sqrt {-a b -\sqrt {a b}\, a}}\right )}{8 a b \sqrt {-a b -\sqrt {a b}\, a}}+\frac {\left (-5 \sqrt {a b}\, a -\sqrt {a b}\, b -8 a b +2 b^{2}\right ) \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}-2 a}{4 \sqrt {-a b +\sqrt {a b}\, a}}\right )}{8 a b \sqrt {-a b +\sqrt {a b}\, a}}}{8 a^{2}-16 a b +8 b^{2}}}{d}\) | \(593\) |
| risch | \(\frac {{\mathrm e}^{d x +c} \left (-5 a b \,{\mathrm e}^{14 d x +14 c}-b^{2} {\mathrm e}^{14 d x +14 c}+49 a b \,{\mathrm e}^{12 d x +12 c}+5 b^{2} {\mathrm e}^{12 d x +12 c}+144 a^{2} {\mathrm e}^{10 d x +10 c}-165 a b \,{\mathrm e}^{10 d x +10 c}-9 b^{2} {\mathrm e}^{10 d x +10 c}-784 \,{\mathrm e}^{8 d x +8 c} a^{2}+377 \,{\mathrm e}^{8 d x +8 c} a b +5 b^{2} {\mathrm e}^{8 d x +8 c}-784 a^{2} {\mathrm e}^{6 d x +6 c}+377 \,{\mathrm e}^{6 d x +6 c} a b +5 b^{2} {\mathrm e}^{6 d x +6 c}+144 \,{\mathrm e}^{4 d x +4 c} a^{2}-165 \,{\mathrm e}^{4 d x +4 c} a b -9 b^{2} {\mathrm e}^{4 d x +4 c}+49 \,{\mathrm e}^{2 d x +2 c} b a +5 b^{2} {\mathrm e}^{2 d x +2 c}-5 a b -b^{2}\right )}{16 a \left (a -b \right )^{2} d \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 )^{2}}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (268435456 a^{11} b^{3} d^{4}-1342177280 a^{10} b^{4} d^{4}+2684354560 a^{9} b^{5} d^{4}-2684354560 a^{8} b^{6} d^{4}+1342177280 a^{7} b^{7} d^{4}-268435456 a^{6} b^{8} d^{4}\right ) \textit {\_Z}^{4}+\left (3440640 a^{6} b^{2} d^{2}+2293760 a^{5} b^{3} d^{2}-1146880 a^{4} b^{4} d^{2}+131072 a^{3} b^{5} d^{2}\right ) \textit {\_Z}^{2}-625 a^{2}+200 a b -16 b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\left (\frac {20971520 a^{10} b^{2} d^{3}}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}-\frac {67108864 a^{9} b^{3} d^{3}}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}+\frac {12582912 a^{8} b^{4} d^{3}}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}+\frac {209715200 a^{7} b^{5} d^{3}}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}-\frac {356515840 a^{6} b^{6} d^{3}}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}+\frac {251658240 a^{5} b^{7} d^{3}}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}-\frac {79691776 a^{4} b^{8} d^{3}}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}+\frac {8388608 a^{3} b^{9} d^{3}}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}\right ) \textit {\_R}^{3}+\left (\frac {217600 a^{5} b d}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}+\frac {837632 a^{4} b^{2} d}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}-\frac {93184 a^{3} b^{3} d}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}-\frac {102400 a^{2} b^{4} d}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}+\frac {27136 a \,b^{5} d}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}-\frac {2048 b^{6} d}{625 a^{3}+3750 a^{2} b -1491 a \,b^{2}+140 b^{3}}\right ) \textit {\_R} \right ) {\mathrm e}^{d x +c}+1\right )\right )\) | \(998\) |
1/d*(8*(-1/64*(4*a-b)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^14+1/64*(a^2+58* a*b-32*b^2)/a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^12+3/64/a*(20*a^2-73*a*b +48*b^2)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^10-1/64/a^2*(175*a^3-550*a^2* b+832*a*b^2-256*b^3)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^8+1/64/a*(220*a^2 -533*a*b+112*b^2)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^6-1/64*(141*a^2-158* a*b+32*b^2)/a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4+1/64*(44*a-17*b)/(a^2- 2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^2-1/64*(5*a-2*b)/(a^2-2*a*b+b^2))/(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*t anh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2+1/8/(a^2-2*a*b+b^2)*(- 1/8*(5*(a*b)^(1/2)*a+(a*b)^(1/2)*b-8*a*b+2*b^2)/a/b/(-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))+1/8*(-5*(a*b)^(1/2)*a-(a*b)^(1/2)*b-8*a*b+2*b^2)/a/b/(-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))))
Leaf count of result is larger than twice the leaf count of optimal. 20961 vs. \(2 (233) = 466\).
Time = 0.66 (sec) , antiderivative size = 20961, normalized size of antiderivative = 72.78 \[ \int \frac {\sinh ^3(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\sinh ^3(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {\sinh ^3(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\int { -\frac {\sinh \left (d x + c\right )^{3}}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \]
-1/16*((5*a*b*e^(15*c) + b^2*e^(15*c))*e^(15*d*x) - (49*a*b*e^(13*c) + 5*b ^2*e^(13*c))*e^(13*d*x) - 3*(48*a^2*e^(11*c) - 55*a*b*e^(11*c) - 3*b^2*e^( 11*c))*e^(11*d*x) + (784*a^2*e^(9*c) - 377*a*b*e^(9*c) - 5*b^2*e^(9*c))*e^ (9*d*x) + (784*a^2*e^(7*c) - 377*a*b*e^(7*c) - 5*b^2*e^(7*c))*e^(7*d*x) - 3*(48*a^2*e^(5*c) - 55*a*b*e^(5*c) - 3*b^2*e^(5*c))*e^(5*d*x) - (49*a*b*e^ (3*c) + 5*b^2*e^(3*c))*e^(3*d*x) + (5*a*b*e^c + b^2*e^c)*e^(d*x))/(a^3*b^2 *d - 2*a^2*b^3*d + a*b^4*d + (a^3*b^2*d*e^(16*c) - 2*a^2*b^3*d*e^(16*c) + a*b^4*d*e^(16*c))*e^(16*d*x) - 8*(a^3*b^2*d*e^(14*c) - 2*a^2*b^3*d*e^(14*c ) + a*b^4*d*e^(14*c))*e^(14*d*x) - 4*(8*a^4*b*d*e^(12*c) - 23*a^3*b^2*d*e^ (12*c) + 22*a^2*b^3*d*e^(12*c) - 7*a*b^4*d*e^(12*c))*e^(12*d*x) + 8*(16*a^ 4*b*d*e^(10*c) - 39*a^3*b^2*d*e^(10*c) + 30*a^2*b^3*d*e^(10*c) - 7*a*b^4*d *e^(10*c))*e^(10*d*x) + 2*(128*a^5*d*e^(8*c) - 352*a^4*b*d*e^(8*c) + 355*a ^3*b^2*d*e^(8*c) - 166*a^2*b^3*d*e^(8*c) + 35*a*b^4*d*e^(8*c))*e^(8*d*x) + 8*(16*a^4*b*d*e^(6*c) - 39*a^3*b^2*d*e^(6*c) + 30*a^2*b^3*d*e^(6*c) - 7*a *b^4*d*e^(6*c))*e^(6*d*x) - 4*(8*a^4*b*d*e^(4*c) - 23*a^3*b^2*d*e^(4*c) + 22*a^2*b^3*d*e^(4*c) - 7*a*b^4*d*e^(4*c))*e^(4*d*x) - 8*(a^3*b^2*d*e^(2*c) - 2*a^2*b^3*d*e^(2*c) + a*b^4*d*e^(2*c))*e^(2*d*x)) - 1/8*integrate(1/2*( (5*a*e^(7*c) + b*e^(7*c))*e^(7*d*x) - (47*a*e^(5*c) - 5*b*e^(5*c))*e^(5*d* x) + (47*a*e^(3*c) - 5*b*e^(3*c))*e^(3*d*x) - (5*a*e^c + b*e^c)*e^(d*x))/( a^3*b - 2*a^2*b^2 + a*b^3 + (a^3*b*e^(8*c) - 2*a^2*b^2*e^(8*c) + a*b^3*...
\[ \int \frac {\sinh ^3(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\int { -\frac {\sinh \left (d x + c\right )^{3}}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\sinh ^3(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^3}{{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^3} \,d x \]