Integrand size = 22, antiderivative size = 313 \[ \int \frac {\sinh (c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\frac {3 \left (7 a-10 \sqrt {a} \sqrt {b}+4 b\right ) \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} \sqrt [4]{b} d}+\frac {3 \left (7 a+10 \sqrt {a} \sqrt {b}+4 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} \sqrt [4]{b} d}+\frac {\cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}+\frac {\cosh (c+d x) \left ((7 a-3 b) (a+2 b)-6 (2 a-b) b \cosh ^2(c+d x)\right )}{32 a^2 (a-b)^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )} \] Output:
3/64*(7*a-10*a^(1/2)*b^(1/2)+4*b)*arctan(b^(1/4)*cosh(d*x+c)/(a^(1/2)-b^(1 /2))^(1/2))/a^(5/2)/(a^(1/2)-b^(1/2))^(5/2)/b^(1/4)/d+3/64*(7*a+10*a^(1/2) *b^(1/2)+4*b)*arctanh(b^(1/4)*cosh(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))/a^(5/2) /(a^(1/2)+b^(1/2))^(5/2)/b^(1/4)/d+1/8*cosh(d*x+c)*(a+b-b*cosh(d*x+c)^2)/a /(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)*((7*a- 3*b)*(a+2*b)-6*(2*a-b)*b*cosh(d*x+c)^2)/a^2/(a-b)^2/d/(a-b+2*b*cosh(d*x+c) ^2-b*cosh(d*x+c)^4)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.27 (sec) , antiderivative size = 1018, normalized size of antiderivative = 3.25 \[ \int \frac {\sinh (c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:
Integrate[Sinh[c + d*x]/(a - b*Sinh[c + d*x]^4)^3,x]
Output:
((32*Cosh[c + d*x]*(7*a^2 + 5*a*b - 3*b^2 + 3*b*(-2*a + b)*Cosh[2*(c + d*x )]))/(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)]) + (512*a*(a - b)*Cosh[c + d*x]*(2*a + b - b*Cosh[2*(c + d*x)]))/(-8*a + 3*b - 4*b*Cos h[2*(c + d*x)] + b*Cosh[4*(c + d*x)])^2 + 3*RootSum[b - 4*b*#1^2 - 16*a*#1 ^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-2*a*b*c + b^2*c - 2*a*b*d*x + b^2* d*x - 4*a*b*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^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x )/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 14*a^2*c*#1^2 - 12*a *b*c*#1^2 + 5*b^2*c*#1^2 + 14*a^2*d*x*#1^2 - 12*a*b*d*x*#1^2 + 5*b^2*d*x*# 1^2 + 28*a^2*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 - 24*a*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 + 10*b^2*L og[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 14*a^2*c*#1^4 + 12*a*b*c*#1^4 - 5*b^2*c*#1^4 - 14*a^ 2*d*x*#1^4 + 12*a*b*d*x*#1^4 - 5*b^2*d*x*#1^4 - 28*a^2*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 + 24*a*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 - 10*b^2*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 + 2*a*b*c*#1 ^6 - b^2*c*#1^6 + 2*a*b*d*x*#1^6 - b^2*d*x*#1^6 + 4*a*b*Log[-Cosh[(c + ...
Time = 0.67 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.16, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {3042, 26, 3694, 1405, 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.
| \(\displaystyle \int \frac {\sinh (c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \sin (i c+i d x)}{\left (a-b \sin (i c+i d x)^4\right )^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\sin (i c+i d x)}{\left (a-b \sin (i c+i d x)^4\right )^3}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle \frac {\int \frac {1}{\left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )^3}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle \frac {\frac {\cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{8 a (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}-\frac {\int -\frac {2 b \left (-5 b \cosh ^2(c+d x)+7 a-b\right )}{\left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )^2}d\cosh (c+d x)}{16 a b (a-b)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {-5 b \cosh ^2(c+d x)+7 a-b}{\left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )^2}d\cosh (c+d x)}{8 a (a-b)}+\frac {\cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{8 a (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle \frac {\frac {\frac {\cosh (c+d x) \left ((7 a-3 b) (a+2 b)-6 b (2 a-b) \cosh ^2(c+d x)\right )}{4 a (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac {\int -\frac {6 b \left (7 a^2-5 b a+2 b^2-2 (2 a-b) b \cosh ^2(c+d x)\right )}{-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b}d\cosh (c+d x)}{8 a b (a-b)}}{8 a (a-b)}+\frac {\cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{8 a (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {7 a^2-5 b a+2 b^2-2 (2 a-b) b \cosh ^2(c+d x)}{-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b}d\cosh (c+d x)}{4 a (a-b)}+\frac {\cosh (c+d x) \left ((7 a-3 b) (a+2 b)-6 b (2 a-b) \cosh ^2(c+d x)\right )}{4 a (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}}{8 a (a-b)}+\frac {\cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{8 a (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\sqrt {b} \left (-2 \sqrt {a} \sqrt {b}+a+b\right ) \left (10 \sqrt {a} \sqrt {b}+7 a+4 b\right ) \int \frac {1}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}-b \cosh ^2(c+d x)}d\cosh (c+d x)}{2 \sqrt {a}}-\frac {\sqrt {b} \left (\sqrt {a}+\sqrt {b}\right )^2 \left (-10 \sqrt {a} \sqrt {b}+7 a+4 b\right ) \int \frac {1}{-b \cosh ^2(c+d x)-\left (\sqrt {a}-\sqrt {b}\right ) \sqrt {b}}d\cosh (c+d x)}{2 \sqrt {a}}\right )}{4 a (a-b)}+\frac {\cosh (c+d x) \left ((7 a-3 b) (a+2 b)-6 b (2 a-b) \cosh ^2(c+d x)\right )}{4 a (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}}{8 a (a-b)}+\frac {\cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{8 a (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\sqrt {b} \left (-2 \sqrt {a} \sqrt {b}+a+b\right ) \left (10 \sqrt {a} \sqrt {b}+7 a+4 b\right ) \int \frac {1}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}-b \cosh ^2(c+d x)}d\cosh (c+d x)}{2 \sqrt {a}}+\frac {\left (-10 \sqrt {a} \sqrt {b}+7 a+4 b\right ) \left (\sqrt {a}+\sqrt {b}\right )^2 \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a}-\sqrt {b}}}\right )}{4 a (a-b)}+\frac {\cosh (c+d x) \left ((7 a-3 b) (a+2 b)-6 b (2 a-b) \cosh ^2(c+d x)\right )}{4 a (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}}{8 a (a-b)}+\frac {\cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{8 a (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\left (-10 \sqrt {a} \sqrt {b}+7 a+4 b\right ) \left (\sqrt {a}+\sqrt {b}\right )^2 \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\left (-2 \sqrt {a} \sqrt {b}+a+b\right ) \left (10 \sqrt {a} \sqrt {b}+7 a+4 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a}+\sqrt {b}}}\right )}{4 a (a-b)}+\frac {\cosh (c+d x) \left ((7 a-3 b) (a+2 b)-6 b (2 a-b) \cosh ^2(c+d x)\right )}{4 a (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}}{8 a (a-b)}+\frac {\cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{8 a (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}}{d}\) |
Input:
Int[Sinh[c + d*x]/(a - b*Sinh[c + d*x]^4)^3,x]
Output:
((Cosh[c + d*x]*(a + b - b*Cosh[c + d*x]^2))/(8*a*(a - b)*(a - b + 2*b*Cos h[c + d*x]^2 - b*Cosh[c + d*x]^4)^2) + ((3*(((Sqrt[a] + Sqrt[b])^2*(7*a - 10*Sqrt[a]*Sqrt[b] + 4*b)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sq rt[b]]])/(2*Sqrt[a]*Sqrt[Sqrt[a] - Sqrt[b]]*b^(1/4)) + ((a - 2*Sqrt[a]*Sqr t[b] + b)*(7*a + 10*Sqrt[a]*Sqrt[b] + 4*b)*ArcTanh[(b^(1/4)*Cosh[c + d*x]) /Sqrt[Sqrt[a] + Sqrt[b]]])/(2*Sqrt[a]*Sqrt[Sqrt[a] + Sqrt[b]]*b^(1/4))))/( 4*a*(a - b)) + (Cosh[c + d*x]*((7*a - 3*b)*(a + 2*b) - 6*(2*a - b)*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*(a - b)))/d
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[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 - 2*a*c + b*c*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[(b^2 - 2*a*c + 2*(p + 1)*( b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
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. \(640\) vs. \(2(261)=522\).
Time = 19.66 (sec) , antiderivative size = 641, normalized size of antiderivative = 2.05
| method | result | size |
| derivativedivides | \(\frac {\frac {-\frac {\left (11 a^{2}-37 a b +20 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{16 a \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (77 a^{2}-283 a b +152 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{16 a \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (231 a^{3}-857 a^{2} b +788 b^{2} a -192 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{16 a^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (385 a^{3}-1231 a^{2} b +1888 b^{2} a -640 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{16 a^{2} \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (385 a^{3}-831 a^{2} b -148 b^{2} a +192 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{16 a^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (231 a^{2}-209 a b +8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{16 a \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (77 a^{2}-3 a b -20 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{16 a \left (a^{2}-2 a b +b^{2}\right )}+\frac {2 \left (11 a -5 b \right )}{32 a^{2}-64 a b +32 b^{2}}}{\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 {3 \left (-4 \sqrt {a b}\, a +2 \sqrt {a b}\, b +7 a^{2}-9 a b +4 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 )}{64 a \sqrt {-a b -\sqrt {a b}\, a}}+\frac {3 \left (4 \sqrt {a b}\, a -2 \sqrt {a b}\, b +7 a^{2}-9 a b +4 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 )}{64 a \sqrt {-a b +\sqrt {a b}\, a}}}{a \left (a^{2}-2 a b +b^{2}\right )}}{d}\) | \(641\) |
| default | \(\frac {\frac {-\frac {\left (11 a^{2}-37 a b +20 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{16 a \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (77 a^{2}-283 a b +152 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{16 a \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (231 a^{3}-857 a^{2} b +788 b^{2} a -192 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{16 a^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (385 a^{3}-1231 a^{2} b +1888 b^{2} a -640 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{16 a^{2} \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (385 a^{3}-831 a^{2} b -148 b^{2} a +192 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{16 a^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (231 a^{2}-209 a b +8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{16 a \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (77 a^{2}-3 a b -20 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{16 a \left (a^{2}-2 a b +b^{2}\right )}+\frac {2 \left (11 a -5 b \right )}{32 a^{2}-64 a b +32 b^{2}}}{\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 {3 \left (-4 \sqrt {a b}\, a +2 \sqrt {a b}\, b +7 a^{2}-9 a b +4 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 )}{64 a \sqrt {-a b -\sqrt {a b}\, a}}+\frac {3 \left (4 \sqrt {a b}\, a -2 \sqrt {a b}\, b +7 a^{2}-9 a b +4 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 )}{64 a \sqrt {-a b +\sqrt {a b}\, a}}}{a \left (a^{2}-2 a b +b^{2}\right )}}{d}\) | \(641\) |
| risch | \(\text {Expression too large to display}\) | \(1290\) |
Input:
int(sinh(d*x+c)/(a-b*sinh(d*x+c)^4)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(2*(-1/32*(11*a^2-37*a*b+20*b^2)/a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c) ^14+1/32/a*(77*a^2-283*a*b+152*b^2)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^12 -1/32/a^2*(231*a^3-857*a^2*b+788*a*b^2-192*b^3)/(a^2-2*a*b+b^2)*tanh(1/2*d *x+1/2*c)^10+1/32/a^2*(385*a^3-1231*a^2*b+1888*a*b^2-640*b^3)/(a^2-2*a*b+b ^2)*tanh(1/2*d*x+1/2*c)^8-1/32/a^2*(385*a^3-831*a^2*b-148*a*b^2+192*b^3)/( a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^6+1/32/a*(231*a^2-209*a*b+8*b^2)/(a^2-2 *a*b+b^2)*tanh(1/2*d*x+1/2*c)^4-1/32*(77*a^2-3*a*b-20*b^2)/a/(a^2-2*a*b+b^ 2)*tanh(1/2*d*x+1/2*c)^2+1/32*(11*a-5*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*tanh(1/2 *d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2+3/16/a/(a^2-2*a*b+b^2)*(-1/4* (-4*(a*b)^(1/2)*a+2*(a*b)^(1/2)*b+7*a^2-9*a*b+4*b^2)/a/(-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))+1/4*(4*(a*b)^(1/2)*a-2*(a*b)^(1/2)*b+7*a^2-9*a*b+4*b^2 )/a/(-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. 22332 vs. \(2 (263) = 526\).
Time = 0.61 (sec) , antiderivative size = 22332, normalized size of antiderivative = 71.35 \[ \int \frac {\sinh (c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(sinh(d*x+c)/(a-b*sinh(d*x+c)^4)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\sinh (c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\text {Timed out} \] Input:
integrate(sinh(d*x+c)/(a-b*sinh(d*x+c)**4)**3,x)
Output:
Timed out
\[ \int \frac {\sinh (c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\int { -\frac {\sinh \left (d x + c\right )}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \] Input:
integrate(sinh(d*x+c)/(a-b*sinh(d*x+c)^4)^3,x, algorithm="maxima")
Output:
1/8*(3*(2*a*b^2*e^(15*c) - b^3*e^(15*c))*e^(15*d*x) - (14*a^2*b*e^(13*c) + 28*a*b^2*e^(13*c) - 15*b^3*e^(13*c))*e^(13*d*x) - (86*a^2*b*e^(11*c) - 12 8*a*b^2*e^(11*c) + 27*b^3*e^(11*c))*e^(11*d*x) + (352*a^3*e^(9*c) - 60*a^2 *b*e^(9*c) - 106*a*b^2*e^(9*c) + 15*b^3*e^(9*c))*e^(9*d*x) + (352*a^3*e^(7 *c) - 60*a^2*b*e^(7*c) - 106*a*b^2*e^(7*c) + 15*b^3*e^(7*c))*e^(7*d*x) - ( 86*a^2*b*e^(5*c) - 128*a*b^2*e^(5*c) + 27*b^3*e^(5*c))*e^(5*d*x) - (14*a^2 *b*e^(3*c) + 28*a*b^2*e^(3*c) - 15*b^3*e^(3*c))*e^(3*d*x) + 3*(2*a*b^2*e^c - b^3*e^c)*e^(d*x))/(a^4*b^2*d - 2*a^3*b^3*d + a^2*b^4*d + (a^4*b^2*d*e^( 16*c) - 2*a^3*b^3*d*e^(16*c) + a^2*b^4*d*e^(16*c))*e^(16*d*x) - 8*(a^4*b^2 *d*e^(14*c) - 2*a^3*b^3*d*e^(14*c) + a^2*b^4*d*e^(14*c))*e^(14*d*x) - 4*(8 *a^5*b*d*e^(12*c) - 23*a^4*b^2*d*e^(12*c) + 22*a^3*b^3*d*e^(12*c) - 7*a^2* b^4*d*e^(12*c))*e^(12*d*x) + 8*(16*a^5*b*d*e^(10*c) - 39*a^4*b^2*d*e^(10*c ) + 30*a^3*b^3*d*e^(10*c) - 7*a^2*b^4*d*e^(10*c))*e^(10*d*x) + 2*(128*a^6* d*e^(8*c) - 352*a^5*b*d*e^(8*c) + 355*a^4*b^2*d*e^(8*c) - 166*a^3*b^3*d*e^ (8*c) + 35*a^2*b^4*d*e^(8*c))*e^(8*d*x) + 8*(16*a^5*b*d*e^(6*c) - 39*a^4*b ^2*d*e^(6*c) + 30*a^3*b^3*d*e^(6*c) - 7*a^2*b^4*d*e^(6*c))*e^(6*d*x) - 4*( 8*a^5*b*d*e^(4*c) - 23*a^4*b^2*d*e^(4*c) + 22*a^3*b^3*d*e^(4*c) - 7*a^2*b^ 4*d*e^(4*c))*e^(4*d*x) - 8*(a^4*b^2*d*e^(2*c) - 2*a^3*b^3*d*e^(2*c) + a^2* b^4*d*e^(2*c))*e^(2*d*x)) + 1/2*integrate(3/4*((2*a*b*e^(7*c) - b^2*e^(7*c ))*e^(7*d*x) - (14*a^2*e^(5*c) - 12*a*b*e^(5*c) + 5*b^2*e^(5*c))*e^(5*d...
\[ \int \frac {\sinh (c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\int { -\frac {\sinh \left (d x + c\right )}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \] Input:
integrate(sinh(d*x+c)/(a-b*sinh(d*x+c)^4)^3,x, algorithm="giac")
Output:
sage0*x
Timed out. \[ \int \frac {\sinh (c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\int \frac {\mathrm {sinh}\left (c+d\,x\right )}{{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^3} \,d x \] Input:
int(sinh(c + d*x)/(a - b*sinh(c + d*x)^4)^3,x)
Output:
int(sinh(c + d*x)/(a - b*sinh(c + d*x)^4)^3, x)
\[ \int \frac {\sinh (c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\text {too large to display} \] Input:
int(sinh(d*x+c)/(a-b*sinh(d*x+c)^4)^3,x)
Output:
(2048*e**c*( - 15952*e**(22*c + 16*d*x)*int(e**(7*d*x)/(e**(24*c + 24*d*x) *b**3 - 12*e**(22*c + 22*d*x)*b**3 - 48*e**(20*c + 20*d*x)*a*b**2 + 66*e** (20*c + 20*d*x)*b**3 + 384*e**(18*c + 18*d*x)*a*b**2 - 220*e**(18*c + 18*d *x)*b**3 + 768*e**(16*c + 16*d*x)*a**2*b - 1344*e**(16*c + 16*d*x)*a*b**2 + 495*e**(16*c + 16*d*x)*b**3 - 3072*e**(14*c + 14*d*x)*a**2*b + 2688*e**( 14*c + 14*d*x)*a*b**2 - 792*e**(14*c + 14*d*x)*b**3 - 4096*e**(12*c + 12*d *x)*a**3 + 4608*e**(12*c + 12*d*x)*a**2*b - 3360*e**(12*c + 12*d*x)*a*b**2 + 924*e**(12*c + 12*d*x)*b**3 - 3072*e**(10*c + 10*d*x)*a**2*b + 2688*e** (10*c + 10*d*x)*a*b**2 - 792*e**(10*c + 10*d*x)*b**3 + 768*e**(8*c + 8*d*x )*a**2*b - 1344*e**(8*c + 8*d*x)*a*b**2 + 495*e**(8*c + 8*d*x)*b**3 + 384* e**(6*c + 6*d*x)*a*b**2 - 220*e**(6*c + 6*d*x)*b**3 - 48*e**(4*c + 4*d*x)* a*b**2 + 66*e**(4*c + 4*d*x)*b**3 - 12*e**(2*c + 2*d*x)*b**3 + b**3),x)*a* b**3*d - 2734*e**(22*c + 16*d*x)*int(e**(7*d*x)/(e**(24*c + 24*d*x)*b**3 - 12*e**(22*c + 22*d*x)*b**3 - 48*e**(20*c + 20*d*x)*a*b**2 + 66*e**(20*c + 20*d*x)*b**3 + 384*e**(18*c + 18*d*x)*a*b**2 - 220*e**(18*c + 18*d*x)*b** 3 + 768*e**(16*c + 16*d*x)*a**2*b - 1344*e**(16*c + 16*d*x)*a*b**2 + 495*e **(16*c + 16*d*x)*b**3 - 3072*e**(14*c + 14*d*x)*a**2*b + 2688*e**(14*c + 14*d*x)*a*b**2 - 792*e**(14*c + 14*d*x)*b**3 - 4096*e**(12*c + 12*d*x)*a** 3 + 4608*e**(12*c + 12*d*x)*a**2*b - 3360*e**(12*c + 12*d*x)*a*b**2 + 924* e**(12*c + 12*d*x)*b**3 - 3072*e**(10*c + 10*d*x)*a**2*b + 2688*e**(10*...