Integrand size = 24, antiderivative size = 313 \[ \int \frac {\sinh ^5(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=-\frac {\left (3 a-10 \sqrt {a} \sqrt {b}+4 b\right ) \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{5/4} d}-\frac {\left (3 a+10 \sqrt {a} \sqrt {b}+4 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{5/4} d}+\frac {\cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac {\cosh (c+d x) \left (a^2-11 a b-2 b^2+2 b (2 a+b) \cosh ^2(c+d x)\right )}{32 a (a-b)^2 b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )} \] Output:
-1/64*(3*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^(3/2)/(a^(1/2)-b^(1/2))^(5/2)/b^(5/4)/d-1/64*(3*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^(3/2 )/(a^(1/2)+b^(1/2))^(5/2)/b^(5/4)/d+1/8*cosh(d*x+c)*(a+b-b*cosh(d*x+c)^2)/ (a-b)/b/d/(a-b+2*b*cosh(d*x+c)^2-b*cosh(d*x+c)^4)^2-1/32*cosh(d*x+c)*(a^2- 11*a*b-2*b^2+2*b*(2*a+b)*cosh(d*x+c)^2)/a/(a-b)^2/b/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 = 6.89 (sec) , antiderivative size = 1019, normalized size of antiderivative = 3.26 \[ \int \frac {\sinh ^5(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:
Integrate[Sinh[c + d*x]^5/(a - b*Sinh[c + d*x]^4)^3,x]
Output:
-1/128*((32*Cosh[c + d*x]*(a^2 - 9*a*b - b^2 + b*(2*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)*Cosh[c + d*x]*(2*a + b - b*Cosh[2*(c + d*x)]))/(-8*a + 3*b - 4*b*C osh[2*(c + d*x)] + b*Cosh[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 & , (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] + 6*a^2*c*#1^2 - 32*a*b *c*#1^2 + 5*b^2*c*#1^2 + 6*a^2*d*x*#1^2 - 32*a*b*d*x*#1^2 + 5*b^2*d*x*#1^2 + 12*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 - 64*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*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 - 6*a^2*c*#1^4 + 32*a*b*c*#1^4 - 5*b^2*c*#1^4 - 6*a^2*d*x *#1^4 + 32*a*b*d*x*#1^4 - 5*b^2*d*x*#1^4 - 12*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 + 6 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]*#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 + d*x)/...
Time = 0.66 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.15, 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, 1517, 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.
| \(\displaystyle \int \frac {\sinh ^5(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \sin (i c+i d x)^5}{\left (a-b \sin (i c+i d x)^4\right )^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\sin (i c+i d x)^5}{\left (a-b \sin (i c+i d x)^4\right )^3}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle \frac {\int \frac {\left (1-\cosh ^2(c+d x)\right )^2}{\left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )^3}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1517 |
| \(\displaystyle \frac {\frac {\cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{8 b (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}-\frac {\int \frac {2 a \left (5 b \cosh ^2(c+d x)+a-7 b\right )}{\left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )^2}d\cosh (c+d x)}{16 a b (a-b)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{8 b (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}-\frac {\int \frac {5 b \cosh ^2(c+d x)+a-7 b}{\left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )^2}d\cosh (c+d x)}{8 b (a-b)}}{d}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle \frac {\frac {\cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{8 b (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}-\frac {\frac {\cosh (c+d x) \left (a^2+2 b (2 a+b) \cosh ^2(c+d x)-11 a b-2 b^2\right )}{4 a (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac {\int -\frac {2 b \left (3 a^2-17 b a+2 b^2+2 b (2 a+b) \cosh ^2(c+d x)\right )}{-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b}d\cosh (c+d x)}{8 a b (a-b)}}{8 b (a-b)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{8 b (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}-\frac {\frac {\int \frac {3 a^2-17 b a+2 b^2+2 b (2 a+b) \cosh ^2(c+d x)}{-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b}d\cosh (c+d x)}{4 a (a-b)}+\frac {\cosh (c+d x) \left (a^2+2 b (2 a+b) \cosh ^2(c+d x)-11 a b-2 b^2\right )}{4 a (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}}{8 b (a-b)}}{d}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {\cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{8 b (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}-\frac {\frac {\frac {\sqrt {b} \left (-2 \sqrt {a} \sqrt {b}+a+b\right ) \left (10 \sqrt {a} \sqrt {b}+3 a+4 b\right ) \int \frac {1}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}-b \cosh ^2(c+d x)}d\cosh (c+d x)}{2 \sqrt {a}}-\frac {\sqrt {b} \left (\sqrt {a}+\sqrt {b}\right )^2 \left (-10 \sqrt {a} \sqrt {b}+3 a+4 b\right ) \int \frac {1}{-b \cosh ^2(c+d x)-\left (\sqrt {a}-\sqrt {b}\right ) \sqrt {b}}d\cosh (c+d x)}{2 \sqrt {a}}}{4 a (a-b)}+\frac {\cosh (c+d x) \left (a^2+2 b (2 a+b) \cosh ^2(c+d x)-11 a b-2 b^2\right )}{4 a (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}}{8 b (a-b)}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{8 b (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}-\frac {\frac {\frac {\sqrt {b} \left (-2 \sqrt {a} \sqrt {b}+a+b\right ) \left (10 \sqrt {a} \sqrt {b}+3 a+4 b\right ) \int \frac {1}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}-b \cosh ^2(c+d x)}d\cosh (c+d x)}{2 \sqrt {a}}+\frac {\left (-10 \sqrt {a} \sqrt {b}+3 a+4 b\right ) \left (\sqrt {a}+\sqrt {b}\right )^2 \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a}-\sqrt {b}}}}{4 a (a-b)}+\frac {\cosh (c+d x) \left (a^2+2 b (2 a+b) \cosh ^2(c+d x)-11 a b-2 b^2\right )}{4 a (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}}{8 b (a-b)}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{8 b (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}-\frac {\frac {\cosh (c+d x) \left (a^2+2 b (2 a+b) \cosh ^2(c+d x)-11 a b-2 b^2\right )}{4 a (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}+\frac {\frac {\left (-10 \sqrt {a} \sqrt {b}+3 a+4 b\right ) \left (\sqrt {a}+\sqrt {b}\right )^2 \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\left (-2 \sqrt {a} \sqrt {b}+a+b\right ) \left (10 \sqrt {a} \sqrt {b}+3 a+4 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} \sqrt {\sqrt {a}+\sqrt {b}}}}{4 a (a-b)}}{8 b (a-b)}}{d}\) |
Input:
Int[Sinh[c + d*x]^5/(a - b*Sinh[c + d*x]^4)^3,x]
Output:
((Cosh[c + d*x]*(a + b - b*Cosh[c + d*x]^2))/(8*(a - b)*b*(a - b + 2*b*Cos h[c + d*x]^2 - b*Cosh[c + d*x]^4)^2) - ((((Sqrt[a] + Sqrt[b])^2*(3*a - 10* Sqrt[a]*Sqrt[b] + 4*b)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[ b]]])/(2*Sqrt[a]*Sqrt[Sqrt[a] - Sqrt[b]]*b^(1/4)) + ((a - 2*Sqrt[a]*Sqrt[b ] + b)*(3*a + 10*Sqrt[a]*Sqrt[b] + 4*b)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sq rt[Sqrt[a] + Sqrt[b]]])/(2*Sqrt[a]*Sqrt[Sqrt[a] + Sqrt[b]]*b^(1/4)))/(4*a* (a - b)) + (Cosh[c + d*x]*(a^2 - 11*a*b - 2*b^2 + 2*b*(2*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)*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[((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[((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[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. \(649\) vs. \(2(261)=522\).
Time = 20.54 (sec) , antiderivative size = 650, normalized size of antiderivative = 2.08
| method | result | size |
| derivativedivides | \(\frac {\frac {-\frac {\left (3 a^{2}-13 a b +4 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{16 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (7 a^{2}-33 a b +8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{16 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (63 a^{3}-225 a^{2} b +68 b^{2} a +64 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{16 b \left (a^{2}-2 a b +b^{2}\right ) a}+\frac {3 \left (35 a^{3}-61 a^{2} b +32 b^{2} a +128 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{16 a b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (105 a^{3}+9 a^{2} b -452 b^{2} a -64 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{16 a b \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (21 a^{2}+29 a b -40 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{16 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (21 a^{2}+37 a b -4 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{16 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 a \left (a +b \right )}{16 b \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 (-4 \sqrt {a b}\, a -2 \sqrt {a b}\, b -3 a^{2}+13 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 )}{4 a \sqrt {-a b -\sqrt {a b}\, a}}+\frac {\left (4 \sqrt {a b}\, a +2 \sqrt {a b}\, b -3 a^{2}+13 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 )}{4 a \sqrt {-a b +\sqrt {a b}\, a}}}{16 b \left (a^{2}-2 a b +b^{2}\right )}}{d}\) | \(650\) |
| default | \(\frac {\frac {-\frac {\left (3 a^{2}-13 a b +4 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{16 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (7 a^{2}-33 a b +8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{16 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (63 a^{3}-225 a^{2} b +68 b^{2} a +64 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{16 b \left (a^{2}-2 a b +b^{2}\right ) a}+\frac {3 \left (35 a^{3}-61 a^{2} b +32 b^{2} a +128 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{16 a b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (105 a^{3}+9 a^{2} b -452 b^{2} a -64 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{16 a b \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (21 a^{2}+29 a b -40 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{16 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (21 a^{2}+37 a b -4 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{16 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 a \left (a +b \right )}{16 b \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 (-4 \sqrt {a b}\, a -2 \sqrt {a b}\, b -3 a^{2}+13 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 )}{4 a \sqrt {-a b -\sqrt {a b}\, a}}+\frac {\left (4 \sqrt {a b}\, a +2 \sqrt {a b}\, b -3 a^{2}+13 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 )}{4 a \sqrt {-a b +\sqrt {a b}\, a}}}{16 b \left (a^{2}-2 a b +b^{2}\right )}}{d}\) | \(650\) |
| risch | \(\text {Expression too large to display}\) | \(1427\) |
Input:
int(sinh(d*x+c)^5/(a-b*sinh(d*x+c)^4)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(32*(-1/512*(3*a^2-13*a*b+4*b^2)/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c) ^14+3/512/b*(7*a^2-33*a*b+8*b^2)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^12-1/ 512/b*(63*a^3-225*a^2*b+68*a*b^2+64*b^3)/(a^2-2*a*b+b^2)/a*tanh(1/2*d*x+1/ 2*c)^10+3/512*(35*a^3-61*a^2*b+32*a*b^2+128*b^3)/a/b/(a^2-2*a*b+b^2)*tanh( 1/2*d*x+1/2*c)^8-1/512/a*(105*a^3+9*a^2*b-452*a*b^2-64*b^3)/b/(a^2-2*a*b+b ^2)*tanh(1/2*d*x+1/2*c)^6+3/512*(21*a^2+29*a*b-40*b^2)/b/(a^2-2*a*b+b^2)*t anh(1/2*d*x+1/2*c)^4-1/512*(21*a^2+37*a*b-4*b^2)/b/(a^2-2*a*b+b^2)*tanh(1/ 2*d*x+1/2*c)^2+3/512*a*(a+b)/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+1/16/b/(a^2-2*a*b+b^2)*(-1/4*(-4*(a*b)^ (1/2)*a-2*(a*b)^(1/2)*b-3*a^2+13*a*b-4*b^2)/a/(-a*b-(a*b)^(1/2)*a)^(1/2)*a rctan(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-3*a^2+13*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. 22506 vs. \(2 (262) = 524\).
Time = 0.61 (sec) , antiderivative size = 22506, normalized size of antiderivative = 71.90 \[ \int \frac {\sinh ^5(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(sinh(d*x+c)^5/(a-b*sinh(d*x+c)^4)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\sinh ^5(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\text {Timed out} \] Input:
integrate(sinh(d*x+c)**5/(a-b*sinh(d*x+c)**4)**3,x)
Output:
Timed out
\[ \int \frac {\sinh ^5(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\int { -\frac {\sinh \left (d x + c\right )^{5}}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \] Input:
integrate(sinh(d*x+c)^5/(a-b*sinh(d*x+c)^4)^3,x, algorithm="maxima")
Output:
1/8*((2*a*b^2*e^(15*c) + b^3*e^(15*c))*e^(15*d*x) + (2*a^2*b*e^(13*c) - 24 *a*b^2*e^(13*c) - 5*b^3*e^(13*c))*e^(13*d*x) - (70*a^2*b*e^(11*c) - 76*a*b ^2*e^(11*c) - 9*b^3*e^(11*c))*e^(11*d*x) + (96*a^3*e^(9*c) + 164*a^2*b*e^( 9*c) - 54*a*b^2*e^(9*c) - 5*b^3*e^(9*c))*e^(9*d*x) + (96*a^3*e^(7*c) + 164 *a^2*b*e^(7*c) - 54*a*b^2*e^(7*c) - 5*b^3*e^(7*c))*e^(7*d*x) - (70*a^2*b*e ^(5*c) - 76*a*b^2*e^(5*c) - 9*b^3*e^(5*c))*e^(5*d*x) + (2*a^2*b*e^(3*c) - 24*a*b^2*e^(3*c) - 5*b^3*e^(3*c))*e^(3*d*x) + (2*a*b^2*e^c + b^3*e^c)*e^(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)) + 1/32*integrate(4*((2*a*b*e^(7*c) + b^2*e^(7*c))*e^(7*d*x) + (6*a^2*e ^(5*c) - 32*a*b*e^(5*c) + 5*b^2*e^(5*c))*e^(5*d*x) - (6*a^2*e^(3*c) - 3...
\[ \int \frac {\sinh ^5(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\int { -\frac {\sinh \left (d x + c\right )^{5}}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \] Input:
integrate(sinh(d*x+c)^5/(a-b*sinh(d*x+c)^4)^3,x, algorithm="giac")
Output:
sage0*x
Timed out. \[ \int \frac {\sinh ^5(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^5}{{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^3} \,d x \] Input:
int(sinh(c + d*x)^5/(a - b*sinh(c + d*x)^4)^3,x)
Output:
int(sinh(c + d*x)^5/(a - b*sinh(c + d*x)^4)^3, x)
\[ \int \frac {\sinh ^5(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\text {too large to display} \] Input:
int(sinh(d*x+c)^5/(a-b*sinh(d*x+c)^4)^3,x)
Output:
(128*e**c*(4549888*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**(2 0*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**(1 0*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**2 *b**3*d + 920544*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 + 49 5*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 + 9 24*e**(12*c + 12*d*x)*b**3 - 3072*e**(10*c + 10*d*x)*a**2*b + 2688*e**(...