Integrand size = 23, antiderivative size = 109 \[ \int \frac {\sinh ^7(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=-\frac {a^3 \arctan \left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{\sqrt {a-b} b^{7/2} d}+\frac {\left (a^2+a b+b^2\right ) \cosh (c+d x)}{b^3 d}-\frac {(a+2 b) \cosh ^3(c+d x)}{3 b^2 d}+\frac {\cosh ^5(c+d x)}{5 b d} \]
(a^2+a*b+b^2)*cosh(d*x+c)/b^3/d-1/3*(a+2*b)*cosh(d*x+c)^3/b^2/d+1/5*cosh(d *x+c)^5/b/d-a^3*arctan(cosh(d*x+c)*b^(1/2)/(a-b)^(1/2))/b^(7/2)/d/(a-b)^(1 /2)
Result contains complex when optimal does not.
Time = 1.43 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.51 \[ \int \frac {\sinh ^7(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {-\frac {240 a^3 \left (\arctan \left (\frac {\sqrt {b}-i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )+\arctan \left (\frac {\sqrt {b}+i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )\right )}{\sqrt {a-b}}+30 \sqrt {b} \left (8 a^2+6 a b+5 b^2\right ) \cosh (c+d x)-5 b^{3/2} (4 a+5 b) \cosh (3 (c+d x))+3 b^{5/2} \cosh (5 (c+d x))}{240 b^{7/2} d} \]
((-240*a^3*(ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]] + ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]]))/Sqrt[a - b] + 30*Sqrt[b]*(8*a^2 + 6*a*b + 5*b^2)*Cosh[c + d*x] - 5*b^(3/2)*(4*a + 5*b) *Cosh[3*(c + d*x)] + 3*b^(5/2)*Cosh[5*(c + d*x)])/(240*b^(7/2)*d)
Time = 0.32 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 26, 3665, 300, 2009}
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.
| \(\displaystyle \int \frac {\sinh ^7(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \sin (i c+i d x)^7}{a-b \sin (i c+i d x)^2}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\sin (i c+i d x)^7}{a-b \sin (i c+i d x)^2}dx\) |
| \(\Big \downarrow \) 3665 |
| \(\displaystyle -\frac {\int \frac {\left (1-\cosh ^2(c+d x)\right )^3}{b \cosh ^2(c+d x)+a-b}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 300 |
| \(\displaystyle -\frac {\int \left (-\frac {\cosh ^4(c+d x)}{b}+\frac {(a+2 b) \cosh ^2(c+d x)}{b^2}-\frac {a^2+b a+b^2}{b^3}+\frac {a^3}{b^3 \left (b \cosh ^2(c+d x)+a-b\right )}\right )d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {a^3 \arctan \left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{b^{7/2} \sqrt {a-b}}-\frac {\left (a^2+a b+b^2\right ) \cosh (c+d x)}{b^3}+\frac {(a+2 b) \cosh ^3(c+d x)}{3 b^2}-\frac {\cosh ^5(c+d x)}{5 b}}{d}\) |
-(((a^3*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(Sqrt[a - b]*b^(7/2)) - ((a^2 + a*b + b^2)*Cosh[c + d*x])/b^3 + ((a + 2*b)*Cosh[c + d*x]^3)/(3* b^2) - Cosh[c + d*x]^5/(5*b))/d)
3.1.28.3.1 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_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int [PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c , d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (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 - b*ff^2*x^2)^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. \(295\) vs. \(2(97)=194\).
Time = 1.20 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.72
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{5 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {-4 a -3 b}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-4 a +b}{12 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {8 a^{2}+4 a b +3 b^{2}}{8 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {1}{5 b \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {1}{2 b \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {-4 a -3 b}{8 b^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {4 a -b}{12 b^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {-8 a^{2}-4 a b -3 b^{2}}{8 b^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {a^{3} \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{b^{3} \sqrt {a b -b^{2}}}}{d}\) | \(296\) |
| default | \(\frac {-\frac {1}{5 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {-4 a -3 b}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-4 a +b}{12 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {8 a^{2}+4 a b +3 b^{2}}{8 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {1}{5 b \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {1}{2 b \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {-4 a -3 b}{8 b^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {4 a -b}{12 b^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {-8 a^{2}-4 a b -3 b^{2}}{8 b^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {a^{3} \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{b^{3} \sqrt {a b -b^{2}}}}{d}\) | \(296\) |
| risch | \(\frac {{\mathrm e}^{5 d x +5 c}}{160 b d}-\frac {5 \,{\mathrm e}^{3 d x +3 c}}{96 b d}-\frac {{\mathrm e}^{3 d x +3 c} a}{24 b^{2} d}+\frac {{\mathrm e}^{d x +c} a^{2}}{2 b^{3} d}+\frac {3 \,{\mathrm e}^{d x +c} a}{8 b^{2} d}+\frac {5 \,{\mathrm e}^{d x +c}}{16 b d}+\frac {{\mathrm e}^{-d x -c} a^{2}}{2 b^{3} d}+\frac {3 \,{\mathrm e}^{-d x -c} a}{8 b^{2} d}+\frac {5 \,{\mathrm e}^{-d x -c}}{16 b d}-\frac {5 \,{\mathrm e}^{-3 d x -3 c}}{96 b d}-\frac {{\mathrm e}^{-3 d x -3 c} a}{24 b^{2} d}+\frac {{\mathrm e}^{-5 d x -5 c}}{160 b d}-\frac {a^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a b +b^{2}}}+1\right )}{2 \sqrt {-a b +b^{2}}\, d \,b^{3}}+\frac {a^{3} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a -b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a b +b^{2}}}+1\right )}{2 \sqrt {-a b +b^{2}}\, d \,b^{3}}\) | \(319\) |
1/d*(-1/5/b/(tanh(1/2*d*x+1/2*c)-1)^5-1/2/b/(tanh(1/2*d*x+1/2*c)-1)^4-1/8* (-4*a-3*b)/b^2/(tanh(1/2*d*x+1/2*c)-1)^2-1/12*(-4*a+b)/b^2/(tanh(1/2*d*x+1 /2*c)-1)^3-1/8*(8*a^2+4*a*b+3*b^2)/b^3/(tanh(1/2*d*x+1/2*c)-1)+1/5/b/(1+ta nh(1/2*d*x+1/2*c))^5-1/2/b/(1+tanh(1/2*d*x+1/2*c))^4-1/8*(-4*a-3*b)/b^2/(1 +tanh(1/2*d*x+1/2*c))^2-1/12*(4*a-b)/b^2/(1+tanh(1/2*d*x+1/2*c))^3-1/8/b^3 *(-8*a^2-4*a*b-3*b^2)/(1+tanh(1/2*d*x+1/2*c))-a^3/b^3/(a*b-b^2)^(1/2)*arct an(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a-2*a+4*b)/(a*b-b^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 1588 vs. \(2 (97) = 194\).
Time = 0.33 (sec) , antiderivative size = 3242, normalized size of antiderivative = 29.74 \[ \int \frac {\sinh ^7(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\text {Too large to display} \]
[1/480*(3*(a*b^3 - b^4)*cosh(d*x + c)^10 + 30*(a*b^3 - b^4)*cosh(d*x + c)* sinh(d*x + c)^9 + 3*(a*b^3 - b^4)*sinh(d*x + c)^10 - 5*(4*a^2*b^2 + a*b^3 - 5*b^4)*cosh(d*x + c)^8 - 5*(4*a^2*b^2 + a*b^3 - 5*b^4 - 27*(a*b^3 - b^4) *cosh(d*x + c)^2)*sinh(d*x + c)^8 + 40*(9*(a*b^3 - b^4)*cosh(d*x + c)^3 - (4*a^2*b^2 + a*b^3 - 5*b^4)*cosh(d*x + c))*sinh(d*x + c)^7 + 30*(8*a^3*b - 2*a^2*b^2 - a*b^3 - 5*b^4)*cosh(d*x + c)^6 + 10*(63*(a*b^3 - b^4)*cosh(d* x + c)^4 + 24*a^3*b - 6*a^2*b^2 - 3*a*b^3 - 15*b^4 - 14*(4*a^2*b^2 + a*b^3 - 5*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(189*(a*b^3 - b^4)*cosh(d*x + c)^5 - 70*(4*a^2*b^2 + a*b^3 - 5*b^4)*cosh(d*x + c)^3 + 45*(8*a^3*b - 2 *a^2*b^2 - a*b^3 - 5*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 30*(8*a^3*b - 2 *a^2*b^2 - a*b^3 - 5*b^4)*cosh(d*x + c)^4 + 10*(63*(a*b^3 - b^4)*cosh(d*x + c)^6 - 35*(4*a^2*b^2 + a*b^3 - 5*b^4)*cosh(d*x + c)^4 + 24*a^3*b - 6*a^2 *b^2 - 3*a*b^3 - 15*b^4 + 45*(8*a^3*b - 2*a^2*b^2 - a*b^3 - 5*b^4)*cosh(d* x + c)^2)*sinh(d*x + c)^4 + 3*a*b^3 - 3*b^4 + 40*(9*(a*b^3 - b^4)*cosh(d*x + c)^7 - 7*(4*a^2*b^2 + a*b^3 - 5*b^4)*cosh(d*x + c)^5 + 15*(8*a^3*b - 2* a^2*b^2 - a*b^3 - 5*b^4)*cosh(d*x + c)^3 + 3*(8*a^3*b - 2*a^2*b^2 - a*b^3 - 5*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 - 5*(4*a^2*b^2 + a*b^3 - 5*b^4)*co sh(d*x + c)^2 + 5*(27*(a*b^3 - b^4)*cosh(d*x + c)^8 - 28*(4*a^2*b^2 + a*b^ 3 - 5*b^4)*cosh(d*x + c)^6 + 90*(8*a^3*b - 2*a^2*b^2 - a*b^3 - 5*b^4)*cosh (d*x + c)^4 - 4*a^2*b^2 - a*b^3 + 5*b^4 + 36*(8*a^3*b - 2*a^2*b^2 - a*b...
Timed out. \[ \int \frac {\sinh ^7(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {\sinh ^7(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\int { \frac {\sinh \left (d x + c\right )^{7}}{b \sinh \left (d x + c\right )^{2} + a} \,d x } \]
1/480*(3*b^2*e^(10*d*x + 10*c) + 3*b^2 - 5*(4*a*b*e^(8*c) + 5*b^2*e^(8*c)) *e^(8*d*x) + 30*(8*a^2*e^(6*c) + 6*a*b*e^(6*c) + 5*b^2*e^(6*c))*e^(6*d*x) + 30*(8*a^2*e^(4*c) + 6*a*b*e^(4*c) + 5*b^2*e^(4*c))*e^(4*d*x) - 5*(4*a*b* e^(2*c) + 5*b^2*e^(2*c))*e^(2*d*x))*e^(-5*d*x - 5*c)/(b^3*d) - 1/128*integ rate(256*(a^3*e^(3*d*x + 3*c) - a^3*e^(d*x + c))/(b^4*e^(4*d*x + 4*c) + b^ 4 + 2*(2*a*b^3*e^(2*c) - b^4*e^(2*c))*e^(2*d*x)), x)
\[ \int \frac {\sinh ^7(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\int { \frac {\sinh \left (d x + c\right )^{7}}{b \sinh \left (d x + c\right )^{2} + a} \,d x } \]
Time = 2.40 (sec) , antiderivative size = 415, normalized size of antiderivative = 3.81 \[ \int \frac {\sinh ^7(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {{\mathrm {e}}^{-5\,c-5\,d\,x}}{160\,b\,d}-\frac {\sqrt {a^6}\,\left (2\,\mathrm {atan}\left (\frac {a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {b^7\,d^2\,\left (a-b\right )}}{2\,b^3\,d\,\left (a-b\right )\,\sqrt {a^6}}\right )+2\,\mathrm {atan}\left (\frac {\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {2\,a^7}{b^{11}\,d\,{\left (a-b\right )}^2\,\sqrt {a^6}}-\frac {4\,\left (2\,a^4\,b^4\,d\,\sqrt {a^6}-2\,a^5\,b^3\,d\,\sqrt {a^6}\right )}{a^3\,b^8\,\left (a-b\right )\,\sqrt {a\,b^7\,d^2-b^8\,d^2}\,\sqrt {b^7\,d^2\,\left (a-b\right )}}\right )+\frac {2\,a^7\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}}{b^{11}\,d\,{\left (a-b\right )}^2\,\sqrt {a^6}}\right )\,\left (b^9\,\sqrt {a\,b^7\,d^2-b^8\,d^2}-a\,b^8\,\sqrt {a\,b^7\,d^2-b^8\,d^2}\right )}{4\,a^4}\right )\right )}{2\,\sqrt {a\,b^7\,d^2-b^8\,d^2}}+\frac {{\mathrm {e}}^{5\,c+5\,d\,x}}{160\,b\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (8\,a^2+6\,a\,b+5\,b^2\right )}{16\,b^3\,d}+\frac {{\mathrm {e}}^{-c-d\,x}\,\left (8\,a^2+6\,a\,b+5\,b^2\right )}{16\,b^3\,d}-\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}\,\left (4\,a+5\,b\right )}{96\,b^2\,d}-\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (4\,a+5\,b\right )}{96\,b^2\,d} \]
exp(- 5*c - 5*d*x)/(160*b*d) - ((a^6)^(1/2)*(2*atan((a^3*exp(d*x)*exp(c)*( b^7*d^2*(a - b))^(1/2))/(2*b^3*d*(a - b)*(a^6)^(1/2))) + 2*atan(((exp(d*x) *exp(c)*((2*a^7)/(b^11*d*(a - b)^2*(a^6)^(1/2)) - (4*(2*a^4*b^4*d*(a^6)^(1 /2) - 2*a^5*b^3*d*(a^6)^(1/2)))/(a^3*b^8*(a - b)*(a*b^7*d^2 - b^8*d^2)^(1/ 2)*(b^7*d^2*(a - b))^(1/2))) + (2*a^7*exp(3*c)*exp(3*d*x))/(b^11*d*(a - b) ^2*(a^6)^(1/2)))*(b^9*(a*b^7*d^2 - b^8*d^2)^(1/2) - a*b^8*(a*b^7*d^2 - b^8 *d^2)^(1/2)))/(4*a^4))))/(2*(a*b^7*d^2 - b^8*d^2)^(1/2)) + exp(5*c + 5*d*x )/(160*b*d) + (exp(c + d*x)*(6*a*b + 8*a^2 + 5*b^2))/(16*b^3*d) + (exp(- c - d*x)*(6*a*b + 8*a^2 + 5*b^2))/(16*b^3*d) - (exp(- 3*c - 3*d*x)*(4*a + 5 *b))/(96*b^2*d) - (exp(3*c + 3*d*x)*(4*a + 5*b))/(96*b^2*d)