Integrand size = 24, antiderivative size = 217 \[ \int \frac {\sinh ^5(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=-\frac {\left (\sqrt {a}-2 \sqrt {b}\right ) \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 \sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} b^{5/4} d}-\frac {\left (\sqrt {a}+2 \sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 \sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} b^{5/4} d}+\frac {\cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{4 (a-b) b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )} \] Output:
-1/8*(a^(1/2)-2*b^(1/2))*arctan(b^(1/4)*cosh(d*x+c)/(a^(1/2)-b^(1/2))^(1/2 ))/a^(1/2)/(a^(1/2)-b^(1/2))^(3/2)/b^(5/4)/d-1/8*(a^(1/2)+2*b^(1/2))*arcta nh(b^(1/4)*cosh(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))/a^(1/2)/(a^(1/2)+b^(1/2))^ (3/2)/b^(5/4)/d+1/4*cosh(d*x+c)*(a+b-b*cosh(d*x+c)^2)/(a-b)/b/d/(a-b+2*b*c osh(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 = 4.02 (sec) , antiderivative size = 597, normalized size of antiderivative = 2.75 \[ \int \frac {\sinh ^5(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\frac {\frac {32 \cosh (c+d x) (2 a+b-b \cosh (2 (c+d x)))}{8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))}+\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 {-b c-b d x-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 )-4 a c \text {$\#$1}^2+11 b c \text {$\#$1}^2-4 a d x \text {$\#$1}^2+11 b d x \text {$\#$1}^2-8 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+22 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+4 a c \text {$\#$1}^4-11 b c \text {$\#$1}^4+4 a d x \text {$\#$1}^4-11 b d x \text {$\#$1}^4+8 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}^4-22 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+b c \text {$\#$1}^6+b d x \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 ]}{32 (a-b) b d} \] Input:
Integrate[Sinh[c + d*x]^5/(a - b*Sinh[c + d*x]^4)^2,x]
Output:
((32*Cosh[c + d*x]*(2*a + b - b*Cosh[2*(c + d*x)]))/(8*a - 3*b + 4*b*Cosh[ 2*(c + d*x)] - b*Cosh[4*(c + d*x)]) + RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6 *b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-(b*c) - b*d*x - 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] - 4* a*c*#1^2 + 11*b*c*#1^2 - 4*a*d*x*#1^2 + 11*b*d*x*#1^2 - 8*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 + 22*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 + 4*a*c*#1^4 - 11*b*c*#1^4 + 4*a*d*x*#1 ^4 - 11*b*d*x*#1^4 + 8*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh [(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 22*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 + b*c*#1^6 + b*d*x*#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) & ])/(32*(a - b)*b*d)
Time = 0.52 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 26, 3694, 1517, 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.
| \(\displaystyle \int \frac {\sinh ^5(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \sin (i c+i d x)^5}{\left (a-b \sin (i c+i d x)^4\right )^2}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\sin (i c+i d x)^5}{\left (a-b \sin (i c+i d x)^4\right )^2}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle \frac {\int \frac {\left (1-\cosh ^2(c+d x)\right )^2}{\left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )^2}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1517 |
| \(\displaystyle \frac {\frac {\cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{4 b (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac {\int \frac {2 a \left (b \cosh ^2(c+d x)+a-3 b\right )}{-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b}d\cosh (c+d x)}{8 a b (a-b)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{4 b (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac {\int \frac {b \cosh ^2(c+d x)+a-3 b}{-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b}d\cosh (c+d x)}{4 b (a-b)}}{d}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {\cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{4 b (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac {\frac {1}{2} \sqrt {b} \left (\frac {a-2 b}{\sqrt {a}}+\sqrt {b}\right ) \int \frac {1}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}-b \cosh ^2(c+d x)}d\cosh (c+d x)-\frac {1}{2} \sqrt {b} \left (\frac {a-2 b}{\sqrt {a}}-\sqrt {b}\right ) \int \frac {1}{-b \cosh ^2(c+d x)-\left (\sqrt {a}-\sqrt {b}\right ) \sqrt {b}}d\cosh (c+d x)}{4 b (a-b)}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{4 b (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac {\frac {1}{2} \sqrt {b} \left (\frac {a-2 b}{\sqrt {a}}+\sqrt {b}\right ) \int \frac {1}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}-b \cosh ^2(c+d x)}d\cosh (c+d x)+\frac {\left (\frac {a-2 b}{\sqrt {a}}-\sqrt {b}\right ) \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt [4]{b} \sqrt {\sqrt {a}-\sqrt {b}}}}{4 b (a-b)}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{4 b (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac {\frac {\left (\frac {a-2 b}{\sqrt {a}}-\sqrt {b}\right ) \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt [4]{b} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\left (\frac {a-2 b}{\sqrt {a}}+\sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt [4]{b} \sqrt {\sqrt {a}+\sqrt {b}}}}{4 b (a-b)}}{d}\) |
Input:
Int[Sinh[c + d*x]^5/(a - b*Sinh[c + d*x]^4)^2,x]
Output:
(-1/4*((((a - 2*b)/Sqrt[a] - Sqrt[b])*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[ Sqrt[a] - Sqrt[b]]])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(1/4)) + (((a - 2*b)/Sqr t[a] + Sqrt[b])*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/ (2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(1/4)))/((a - b)*b) + (Cosh[c + d*x]*(a + b - b*Cosh[c + d*x]^2))/(4*(a - b)*b*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4)))/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)^(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. \(346\) vs. \(2(167)=334\).
Time = 7.22 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.60
| method | result | size |
| derivativedivides | \(\frac {\frac {-\frac {\left (a -2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2 b \left (a -b \right )}+\frac {\left (3 a -8 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 b \left (a -b \right )}-\frac {\left (3 a +2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 b \left (a -b \right )}+\frac {a}{2 b \left (a -b \right )}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a -4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a +6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -16 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a}+\frac {a \left (-\frac {\left (-\sqrt {a b}-a +2 b \right ) \arctan \left (\frac {-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}+2 a}{4 \sqrt {-a b -\sqrt {a b}\, a}}\right )}{4 a \sqrt {-a b -\sqrt {a b}\, a}}+\frac {\left (\sqrt {a b}-a +2 b \right ) \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}-2 a}{4 \sqrt {-a b +\sqrt {a b}\, a}}\right )}{4 a \sqrt {-a b +\sqrt {a b}\, a}}\right )}{2 b \left (a -b \right )}}{d}\) | \(347\) |
| default | \(\frac {\frac {-\frac {\left (a -2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2 b \left (a -b \right )}+\frac {\left (3 a -8 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 b \left (a -b \right )}-\frac {\left (3 a +2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 b \left (a -b \right )}+\frac {a}{2 b \left (a -b \right )}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a -4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a +6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -16 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a}+\frac {a \left (-\frac {\left (-\sqrt {a b}-a +2 b \right ) \arctan \left (\frac {-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}+2 a}{4 \sqrt {-a b -\sqrt {a b}\, a}}\right )}{4 a \sqrt {-a b -\sqrt {a b}\, a}}+\frac {\left (\sqrt {a b}-a +2 b \right ) \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}-2 a}{4 \sqrt {-a b +\sqrt {a b}\, a}}\right )}{4 a \sqrt {-a b +\sqrt {a b}\, a}}\right )}{2 b \left (a -b \right )}}{d}\) | \(347\) |
| risch | \(\frac {{\mathrm e}^{d x +c} \left (-{\mathrm e}^{6 d x +6 c} b +4 \,{\mathrm e}^{4 d x +4 c} a +b \,{\mathrm e}^{4 d x +4 c}+4 \,{\mathrm e}^{2 d x +2 c} a +{\mathrm e}^{2 d x +2 c} b -b \right )}{2 b \left (a -b \right ) d \left (-{\mathrm e}^{8 d x +8 c} b +4 \,{\mathrm e}^{6 d x +6 c} b +16 \,{\mathrm e}^{4 d x +4 c} a -6 b \,{\mathrm e}^{4 d x +4 c}+4 \,{\mathrm e}^{2 d x +2 c} b -b \right )}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (65536 a^{5} b^{5} d^{4}-196608 a^{4} b^{6} d^{4}+196608 a^{3} b^{7} d^{4}-65536 a^{2} b^{8} d^{4}\right ) \textit {\_Z}^{4}+\left (-512 a^{3} b^{3} d^{2}+512 a^{2} b^{4} d^{2}+2048 a \,b^{5} d^{2}\right ) \textit {\_Z}^{2}-a^{2}+8 a b -16 b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\left (\frac {16384 a^{4} b^{5} d^{3}}{a^{3}-9 a^{2} b +28 b^{2} a -32 b^{3}}-\frac {49152 a^{3} b^{6} d^{3}}{a^{3}-9 a^{2} b +28 b^{2} a -32 b^{3}}+\frac {49152 a^{2} b^{7} d^{3}}{a^{3}-9 a^{2} b +28 b^{2} a -32 b^{3}}-\frac {16384 a \,b^{8} d^{3}}{a^{3}-9 a^{2} b +28 b^{2} a -32 b^{3}}\right ) \textit {\_R}^{3}+\left (-\frac {32 a^{4} b d}{a^{3}-9 a^{2} b +28 b^{2} a -32 b^{3}}+\frac {256 a^{3} b^{2} d}{a^{3}-9 a^{2} b +28 b^{2} a -32 b^{3}}-\frac {800 a^{2} b^{3} d}{a^{3}-9 a^{2} b +28 b^{2} a -32 b^{3}}+\frac {832 a \,b^{4} d}{a^{3}-9 a^{2} b +28 b^{2} a -32 b^{3}}+\frac {256 b^{5} d}{a^{3}-9 a^{2} b +28 b^{2} a -32 b^{3}}\right ) \textit {\_R} \right ) {\mathrm e}^{d x +c}+1\right )\right )\) | \(572\) |
Input:
int(sinh(d*x+c)^5/(a-b*sinh(d*x+c)^4)^2,x,method=_RETURNVERBOSE)
Output:
1/d*(32*(-1/64*(a-2*b)/b/(a-b)*tanh(1/2*d*x+1/2*c)^6+1/64*(3*a-8*b)/b/(a-b )*tanh(1/2*d*x+1/2*c)^4-1/64*(3*a+2*b)/b/(a-b)*tanh(1/2*d*x+1/2*c)^2+1/64/ b/(a-b)*a)/(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)+1/2/ b/(a-b)*a*(-1/4*(-(a*b)^(1/2)-a+2*b)/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*((a*b)^(1/2)-a+2*b)/a/(-a*b+(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(2*tan h(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. 6250 vs. \(2 (169) = 338\).
Time = 0.25 (sec) , antiderivative size = 6250, normalized size of antiderivative = 28.80 \[ \int \frac {\sinh ^5(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(sinh(d*x+c)^5/(a-b*sinh(d*x+c)^4)^2,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\sinh ^5(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\text {Timed out} \] Input:
integrate(sinh(d*x+c)**5/(a-b*sinh(d*x+c)**4)**2,x)
Output:
Timed out
\[ \int \frac {\sinh ^5(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int { \frac {\sinh \left (d x + c\right )^{5}}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \] Input:
integrate(sinh(d*x+c)^5/(a-b*sinh(d*x+c)^4)^2,x, algorithm="maxima")
Output:
-1/2*((4*a*e^(5*c) + b*e^(5*c))*e^(5*d*x) + (4*a*e^(3*c) + b*e^(3*c))*e^(3 *d*x) - b*e^(7*d*x + 7*c) - b*e^(d*x + c))/(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)) + 1/32*integrate(16* ((4*a*e^(5*c) - 11*b*e^(5*c))*e^(5*d*x) - (4*a*e^(3*c) - 11*b*e^(3*c))*e^( 3*d*x) + b*e^(7*d*x + 7*c) - b*e^(d*x + 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)
\[ \int \frac {\sinh ^5(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int { \frac {\sinh \left (d x + c\right )^{5}}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \] Input:
integrate(sinh(d*x+c)^5/(a-b*sinh(d*x+c)^4)^2,x, algorithm="giac")
Output:
sage0*x
Timed out. \[ \int \frac {\sinh ^5(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^5}{{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^2} \,d x \] Input:
int(sinh(c + d*x)^5/(a - b*sinh(c + d*x)^4)^2,x)
Output:
int(sinh(c + d*x)^5/(a - b*sinh(c + d*x)^4)^2, x)
\[ \int \frac {\sinh ^5(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\text {too large to display} \] Input:
int(sinh(d*x+c)^5/(a-b*sinh(d*x+c)^4)^2,x)
Output:
(8*e**c*( - 2832*e**(14*c + 8*d*x)*int(e**(7*d*x)/(e**(16*c + 16*d*x)*b**2 - 8*e**(14*c + 14*d*x)*b**2 - 32*e**(12*c + 12*d*x)*a*b + 28*e**(12*c + 1 2*d*x)*b**2 + 128*e**(10*c + 10*d*x)*a*b - 56*e**(10*c + 10*d*x)*b**2 + 25 6*e**(8*c + 8*d*x)*a**2 - 192*e**(8*c + 8*d*x)*a*b + 70*e**(8*c + 8*d*x)*b **2 + 128*e**(6*c + 6*d*x)*a*b - 56*e**(6*c + 6*d*x)*b**2 - 32*e**(4*c + 4 *d*x)*a*b + 28*e**(4*c + 4*d*x)*b**2 - 8*e**(2*c + 2*d*x)*b**2 + b**2),x)* a*b**2*d - 48*e**(14*c + 8*d*x)*int(e**(7*d*x)/(e**(16*c + 16*d*x)*b**2 - 8*e**(14*c + 14*d*x)*b**2 - 32*e**(12*c + 12*d*x)*a*b + 28*e**(12*c + 12*d *x)*b**2 + 128*e**(10*c + 10*d*x)*a*b - 56*e**(10*c + 10*d*x)*b**2 + 256*e **(8*c + 8*d*x)*a**2 - 192*e**(8*c + 8*d*x)*a*b + 70*e**(8*c + 8*d*x)*b**2 + 128*e**(6*c + 6*d*x)*a*b - 56*e**(6*c + 6*d*x)*b**2 - 32*e**(4*c + 4*d* x)*a*b + 28*e**(4*c + 4*d*x)*b**2 - 8*e**(2*c + 2*d*x)*b**2 + b**2),x)*b** 3*d - 3840*e**(12*c + 8*d*x)*int(e**(5*d*x)/(e**(16*c + 16*d*x)*b**2 - 8*e **(14*c + 14*d*x)*b**2 - 32*e**(12*c + 12*d*x)*a*b + 28*e**(12*c + 12*d*x) *b**2 + 128*e**(10*c + 10*d*x)*a*b - 56*e**(10*c + 10*d*x)*b**2 + 256*e**( 8*c + 8*d*x)*a**2 - 192*e**(8*c + 8*d*x)*a*b + 70*e**(8*c + 8*d*x)*b**2 + 128*e**(6*c + 6*d*x)*a*b - 56*e**(6*c + 6*d*x)*b**2 - 32*e**(4*c + 4*d*x)* a*b + 28*e**(4*c + 4*d*x)*b**2 - 8*e**(2*c + 2*d*x)*b**2 + b**2),x)*a**2*b *d + 3744*e**(12*c + 8*d*x)*int(e**(5*d*x)/(e**(16*c + 16*d*x)*b**2 - 8*e* *(14*c + 14*d*x)*b**2 - 32*e**(12*c + 12*d*x)*a*b + 28*e**(12*c + 12*d*...