Integrand size = 24, antiderivative size = 290 \[ \int \frac {\sinh ^7(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\frac {3 \left (\sqrt {a}-2 \sqrt {b}\right ) \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 \sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{7/4} d}-\frac {3 \left (\sqrt {a}+2 \sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 \sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{7/4} d}-\frac {a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}+\frac {\cosh (c+d x) \left (5 a-17 b-3 (a-3 b) \cosh ^2(c+d x)\right )}{32 (a-b)^2 b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )} \] Output:
3/64*(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))^(5/2)/b^(7/4)/d-3/64*(a^(1/2)+2*b^(1/2))*arct anh(b^(1/4)*cosh(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))/a^(1/2)/(a^(1/2)+b^(1/2)) ^(5/2)/b^(7/4)/d-1/8*a*cosh(d*x+c)*(2-cosh(d*x+c)^2)/(a-b)/b/d/(a-b+2*b*co sh(d*x+c)^2-b*cosh(d*x+c)^4)^2+1/32*cosh(d*x+c)*(5*a-17*b-3*(a-3*b)*cosh(d *x+c)^2)/(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 = 6.46 (sec) , antiderivative size = 802, normalized size of antiderivative = 2.77 \[ \int \frac {\sinh ^7(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:
Integrate[Sinh[c + d*x]^7/(a - b*Sinh[c + d*x]^4)^3,x]
Output:
((-32*Cosh[c + d*x]*(-7*a + 25*b + 3*(a - 3*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)*(-5*Co sh[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 - 3*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b *#1^6 + b*#1^8 & , (a*c - 3*b*c + a*d*x - 3*b*d*x + 2*a*Log[-Cosh[(c + d*x )/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 6*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Si nh[(c + d*x)/2]*#1] - 3*a*c*#1^2 + 17*b*c*#1^2 - 3*a*d*x*#1^2 + 17*b*d*x*# 1^2 - 6*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 + 34*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 + 3*a*c*#1^4 - 17*b*c*#1^4 + 3*a*d*x*#1^4 - 17*b*d*x*#1^4 + 6*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 - 34 *b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sin h[(c + d*x)/2]*#1]*#1^4 - a*c*#1^6 + 3*b*c*#1^6 - a*d*x*#1^6 + 3*b*d*x*#1^ 6 - 2*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 + 6*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) & ])/(256*(a - b)^2*b*d)
Time = 0.64 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.05, 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.
| \(\displaystyle \int \frac {\sinh ^7(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \sin (i c+i d x)^7}{\left (a-b \sin (i c+i d x)^4\right )^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\sin (i c+i d x)^7}{\left (a-b \sin (i c+i d x)^4\right )^3}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle -\frac {\int \frac {\left (1-\cosh ^2(c+d x)\right )^3}{\left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )^3}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1517 |
| \(\displaystyle -\frac {\frac {a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 b (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}-\frac {\int \frac {2 a \left (2 (a-4 b)-(3 a-8 b) \cosh ^2(c+d x)\right )}{\left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )^2}d\cosh (c+d x)}{16 a b (a-b)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 b (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}-\frac {\int \frac {2 (a-4 b)-(3 a-8 b) \cosh ^2(c+d x)}{\left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )^2}d\cosh (c+d x)}{8 b (a-b)}}{d}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle -\frac {\frac {a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 b (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}-\frac {\frac {\cosh (c+d x) \left (-3 (a-3 b) \cosh ^2(c+d x)+5 a-17 b\right )}{4 (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac {\int -\frac {6 a b \left (-\left ((a-3 b) \cosh ^2(c+d x)\right )+a-5 b\right )}{-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b}d\cosh (c+d x)}{8 a b (a-b)}}{8 b (a-b)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 b (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}-\frac {\frac {3 \int \frac {-\left ((a-3 b) \cosh ^2(c+d x)\right )+a-5 b}{-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b}d\cosh (c+d x)}{4 (a-b)}+\frac {\cosh (c+d x) \left (-3 (a-3 b) \cosh ^2(c+d x)+5 a-17 b\right )}{4 (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}}{8 b (a-b)}}{d}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {\frac {a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 b (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}-\frac {\frac {3 \left (-\frac {1}{2} \left (-\frac {2 b^{3/2}}{\sqrt {a}}+a-3 b\right ) \int \frac {1}{-b \cosh ^2(c+d x)-\left (\sqrt {a}-\sqrt {b}\right ) \sqrt {b}}d\cosh (c+d x)-\frac {1}{2} \left (\frac {2 b^{3/2}}{\sqrt {a}}+a-3 b\right ) \int \frac {1}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}-b \cosh ^2(c+d x)}d\cosh (c+d x)\right )}{4 (a-b)}+\frac {\cosh (c+d x) \left (-3 (a-3 b) \cosh ^2(c+d x)+5 a-17 b\right )}{4 (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}}{8 b (a-b)}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\frac {a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 b (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}-\frac {\frac {3 \left (\frac {\left (-\frac {2 b^{3/2}}{\sqrt {a}}+a-3 b\right ) \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{3/4} \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {1}{2} \left (\frac {2 b^{3/2}}{\sqrt {a}}+a-3 b\right ) \int \frac {1}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}-b \cosh ^2(c+d x)}d\cosh (c+d x)\right )}{4 (a-b)}+\frac {\cosh (c+d x) \left (-3 (a-3 b) \cosh ^2(c+d x)+5 a-17 b\right )}{4 (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}}{8 b (a-b)}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 b (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}-\frac {\frac {3 \left (\frac {\left (-\frac {2 b^{3/2}}{\sqrt {a}}+a-3 b\right ) \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{3/4} \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\left (\frac {2 b^{3/2}}{\sqrt {a}}+a-3 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 b^{3/4} \sqrt {\sqrt {a}+\sqrt {b}}}\right )}{4 (a-b)}+\frac {\cosh (c+d x) \left (-3 (a-3 b) \cosh ^2(c+d x)+5 a-17 b\right )}{4 (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}}{8 b (a-b)}}{d}\) |
Input:
Int[Sinh[c + d*x]^7/(a - b*Sinh[c + d*x]^4)^3,x]
Output:
-(((a*Cosh[c + d*x]*(2 - Cosh[c + d*x]^2))/(8*(a - b)*b*(a - b + 2*b*Cosh[ c + d*x]^2 - b*Cosh[c + d*x]^4)^2) - ((3*(((a - 3*b - (2*b^(3/2))/Sqrt[a]) *ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(3/4)) - ((a - 3*b + (2*b^(3/2))/Sqrt[a])*ArcTanh[(b^(1/4)*Co sh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(3/4)) ))/(4*(a - b)) + (Cosh[c + d*x]*(5*a - 17*b - 3*(a - 3*b)*Cosh[c + d*x]^2) )/(4*(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. \(584\) vs. \(2(238)=476\).
Time = 21.82 (sec) , antiderivative size = 585, normalized size of antiderivative = 2.02
| method | result | size |
| derivativedivides | \(\frac {\frac {-\frac {3 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{8 \left (a^{2}-2 a b +b^{2}\right )}-\frac {3 \left (a -10 b \right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{8 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (16 a^{2}-111 a b +80 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (35 a^{3}-26 a^{2} b -64 b^{2} a +256 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8 a b \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (40 a^{2}+95 a b -336 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{8 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (25 a^{2}+54 a b -64 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (8 a +19 b \right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {a \left (a +2 b \right )}{8 b \left (a^{2}-2 a b +b^{2}\right )}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a -4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a +6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -16 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a \right )^{2}}+\frac {3 a \left (-\frac {\left (-\sqrt {a b}\, a +3 \sqrt {a b}\, b -2 b^{2}\right ) \arctan \left (\frac {-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}+2 a}{4 \sqrt {-a b -\sqrt {a b}\, a}}\right )}{8 a b \sqrt {-a b -\sqrt {a b}\, a}}+\frac {\left (\sqrt {a b}\, a -3 \sqrt {a b}\, b -2 b^{2}\right ) \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}-2 a}{4 \sqrt {-a b +\sqrt {a b}\, a}}\right )}{8 a b \sqrt {-a b +\sqrt {a b}\, a}}\right )}{8 b \left (a^{2}-2 a b +b^{2}\right )}}{d}\) | \(585\) |
| default | \(\frac {\frac {-\frac {3 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{8 \left (a^{2}-2 a b +b^{2}\right )}-\frac {3 \left (a -10 b \right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{8 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (16 a^{2}-111 a b +80 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (35 a^{3}-26 a^{2} b -64 b^{2} a +256 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8 a b \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (40 a^{2}+95 a b -336 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{8 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (25 a^{2}+54 a b -64 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (8 a +19 b \right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {a \left (a +2 b \right )}{8 b \left (a^{2}-2 a b +b^{2}\right )}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a -4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a +6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -16 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a \right )^{2}}+\frac {3 a \left (-\frac {\left (-\sqrt {a b}\, a +3 \sqrt {a b}\, b -2 b^{2}\right ) \arctan \left (\frac {-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}+2 a}{4 \sqrt {-a b -\sqrt {a b}\, a}}\right )}{8 a b \sqrt {-a b -\sqrt {a b}\, a}}+\frac {\left (\sqrt {a b}\, a -3 \sqrt {a b}\, b -2 b^{2}\right ) \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}-2 a}{4 \sqrt {-a b +\sqrt {a b}\, a}}\right )}{8 a b \sqrt {-a b +\sqrt {a b}\, a}}\right )}{8 b \left (a^{2}-2 a b +b^{2}\right )}}{d}\) | \(585\) |
| risch | \(\text {Expression too large to display}\) | \(1108\) |
Input:
int(sinh(d*x+c)^7/(a-b*sinh(d*x+c)^4)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(128*(-3/1024*a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^14-3/1024/b*(a-10* b)*a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^12+1/1024/b*(16*a^2-111*a*b+80*b^ 2)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^10-1/1024*(35*a^3-26*a^2*b-64*a*b^2 +256*b^3)/a/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^8+1/1024*(40*a^2+95*a*b- 336*b^2)/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^6-1/1024*(25*a^2+54*a*b-64* b^2)/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4+1/1024*(8*a+19*b)*a/b/(a^2-2* a*b+b^2)*tanh(1/2*d*x+1/2*c)^2-1/1024*a*(a+2*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+3/8/b/(a^2-2*a*b+b^2 )*a*(-1/8*(-(a*b)^(1/2)*a+3*(a*b)^(1/2)*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*((a*b)^(1/2)*a-3*(a*b)^(1/2)*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. 20362 vs. \(2 (234) = 468\).
Time = 0.46 (sec) , antiderivative size = 20362, normalized size of antiderivative = 70.21 \[ \int \frac {\sinh ^7(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(sinh(d*x+c)^7/(a-b*sinh(d*x+c)^4)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\sinh ^7(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\text {Timed out} \] Input:
integrate(sinh(d*x+c)**7/(a-b*sinh(d*x+c)**4)**3,x)
Output:
Timed out
\[ \int \frac {\sinh ^7(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\int { -\frac {\sinh \left (d x + c\right )^{7}}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \] Input:
integrate(sinh(d*x+c)^7/(a-b*sinh(d*x+c)^4)^3,x, algorithm="maxima")
Output:
1/16*(3*(a*b*e^(15*c) - 3*b^2*e^(15*c))*e^(15*d*x) - (23*a*b*e^(13*c) - 77 *b^2*e^(13*c))*e^(13*d*x) + (16*a^2*e^(11*c) + 131*a*b*e^(11*c) - 177*b^2* e^(11*c))*e^(11*d*x) - (144*a^2*e^(9*c) + 367*a*b*e^(9*c) - 109*b^2*e^(9*c ))*e^(9*d*x) - (144*a^2*e^(7*c) + 367*a*b*e^(7*c) - 109*b^2*e^(7*c))*e^(7* d*x) + (16*a^2*e^(5*c) + 131*a*b*e^(5*c) - 177*b^2*e^(5*c))*e^(5*d*x) - (2 3*a*b*e^(3*c) - 77*b^2*e^(3*c))*e^(3*d*x) + 3*(a*b*e^c - 3*b^2*e^c)*e^(d*x ))/(a^2*b^3*d - 2*a*b^4*d + b^5*d + (a^2*b^3*d*e^(16*c) - 2*a*b^4*d*e^(16* c) + b^5*d*e^(16*c))*e^(16*d*x) - 8*(a^2*b^3*d*e^(14*c) - 2*a*b^4*d*e^(14* c) + b^5*d*e^(14*c))*e^(14*d*x) - 4*(8*a^3*b^2*d*e^(12*c) - 23*a^2*b^3*d*e ^(12*c) + 22*a*b^4*d*e^(12*c) - 7*b^5*d*e^(12*c))*e^(12*d*x) + 8*(16*a^3*b ^2*d*e^(10*c) - 39*a^2*b^3*d*e^(10*c) + 30*a*b^4*d*e^(10*c) - 7*b^5*d*e^(1 0*c))*e^(10*d*x) + 2*(128*a^4*b*d*e^(8*c) - 352*a^3*b^2*d*e^(8*c) + 355*a^ 2*b^3*d*e^(8*c) - 166*a*b^4*d*e^(8*c) + 35*b^5*d*e^(8*c))*e^(8*d*x) + 8*(1 6*a^3*b^2*d*e^(6*c) - 39*a^2*b^3*d*e^(6*c) + 30*a*b^4*d*e^(6*c) - 7*b^5*d* e^(6*c))*e^(6*d*x) - 4*(8*a^3*b^2*d*e^(4*c) - 23*a^2*b^3*d*e^(4*c) + 22*a* b^4*d*e^(4*c) - 7*b^5*d*e^(4*c))*e^(4*d*x) - 8*(a^2*b^3*d*e^(2*c) - 2*a*b^ 4*d*e^(2*c) + b^5*d*e^(2*c))*e^(2*d*x)) + 1/128*integrate(24*((a*e^(7*c) - 3*b*e^(7*c))*e^(7*d*x) - (3*a*e^(5*c) - 17*b*e^(5*c))*e^(5*d*x) + (3*a*e^ (3*c) - 17*b*e^(3*c))*e^(3*d*x) - (a*e^c - 3*b*e^c)*e^(d*x))/(a^2*b^2 - 2* a*b^3 + b^4 + (a^2*b^2*e^(8*c) - 2*a*b^3*e^(8*c) + b^4*e^(8*c))*e^(8*d*...
\[ \int \frac {\sinh ^7(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\int { -\frac {\sinh \left (d x + c\right )^{7}}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \] Input:
integrate(sinh(d*x+c)^7/(a-b*sinh(d*x+c)^4)^3,x, algorithm="giac")
Output:
sage0*x
Timed out. \[ \int \frac {\sinh ^7(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^7}{{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^3} \,d x \] Input:
int(sinh(c + d*x)^7/(a - b*sinh(c + d*x)^4)^3,x)
Output:
int(sinh(c + d*x)^7/(a - b*sinh(c + d*x)^4)^3, x)
\[ \int \frac {\sinh ^7(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\text {too large to display} \] Input:
int(sinh(d*x+c)^7/(a-b*sinh(d*x+c)^4)^3,x)
Output:
(32*e**c*(1576960*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 + 4 95*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**3* b**2*d + 1664768*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**(...