Integrand size = 29, antiderivative size = 141 \[ \int \frac {\cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a \left (2 a^2+b^2\right ) x}{2 b^4}-\frac {2 a^2 \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{b^4 d}+\frac {\left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^3 d}-\frac {a \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}+\frac {\cosh (c+d x) \sinh ^2(c+d x)}{3 b d} \]
-1/2*a*(2*a^2+b^2)*x/b^4+1/3*(3*a^2+b^2)*cosh(d*x+c)/b^3/d-1/2*a*cosh(d*x+ c)*sinh(d*x+c)/b^2/d+1/3*cosh(d*x+c)*sinh(d*x+c)^2/b/d-2*a^2*arctanh((b-a* tanh(1/2*d*x+1/2*c))/(a^2+b^2)^(1/2))*(a^2+b^2)^(1/2)/b^4/d
Time = 0.54 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.87 \[ \int \frac {\cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {3 b \left (4 a^2+b^2\right ) \cosh (c+d x)+b^3 \cosh (3 (c+d x))-3 a \left (2 \left (2 a^2+b^2\right ) (c+d x)+8 a \sqrt {-a^2-b^2} \arctan \left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )+b^2 \sinh (2 (c+d x))\right )}{12 b^4 d} \]
(3*b*(4*a^2 + b^2)*Cosh[c + d*x] + b^3*Cosh[3*(c + d*x)] - 3*a*(2*(2*a^2 + b^2)*(c + d*x) + 8*a*Sqrt[-a^2 - b^2]*ArcTan[(b - a*Tanh[(c + d*x)/2])/Sq rt[-a^2 - b^2]] + b^2*Sinh[2*(c + d*x)]))/(12*b^4*d)
Result contains complex when optimal does not.
Time = 1.11 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.16, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.724, Rules used = {3042, 25, 3368, 25, 3042, 25, 3529, 26, 3042, 26, 3528, 25, 3042, 3502, 27, 3042, 3214, 3042, 3139, 1083, 217}
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.
| \(\displaystyle \int \frac {\sinh ^2(c+d x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sin (i c+i d x)^2 \cos (i c+i d x)^2}{a-i b \sin (i c+i d x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\cos (i c+i d x)^2 \sin (i c+i d x)^2}{a-i b \sin (i c+i d x)}dx\) |
| \(\Big \downarrow \) 3368 |
| \(\displaystyle -\int -\frac {\sinh ^2(c+d x) \left (\sinh ^2(c+d x)+1\right )}{a+b \sinh (c+d x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\sinh ^2(c+d x) \left (\sinh ^2(c+d x)+1\right )}{a+b \sinh (c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sin (i c+i d x)^2 \left (1-\sin (i c+i d x)^2\right )}{a-i b \sin (i c+i d x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\sin (i c+i d x)^2 \left (1-\sin (i c+i d x)^2\right )}{a-i b \sin (i c+i d x)}dx\) |
| \(\Big \downarrow \) 3529 |
| \(\displaystyle \frac {\sinh ^2(c+d x) \cosh (c+d x)}{3 b d}-\frac {i \int -\frac {i \sinh (c+d x) \left (3 a \sinh ^2(c+d x)-b \sinh (c+d x)+2 a\right )}{a+b \sinh (c+d x)}dx}{3 b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\sinh ^2(c+d x) \cosh (c+d x)}{3 b d}-\frac {\int \frac {\sinh (c+d x) \left (3 a \sinh ^2(c+d x)-b \sinh (c+d x)+2 a\right )}{a+b \sinh (c+d x)}dx}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh ^2(c+d x) \cosh (c+d x)}{3 b d}-\frac {\int -\frac {i \sin (i c+i d x) \left (-3 a \sin (i c+i d x)^2+i b \sin (i c+i d x)+2 a\right )}{a-i b \sin (i c+i d x)}dx}{3 b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\sinh ^2(c+d x) \cosh (c+d x)}{3 b d}+\frac {i \int \frac {\sin (i c+i d x) \left (-3 a \sin (i c+i d x)^2+i b \sin (i c+i d x)+2 a\right )}{a-i b \sin (i c+i d x)}dx}{3 b}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\sinh ^2(c+d x) \cosh (c+d x)}{3 b d}+\frac {i \left (\frac {i \int -\frac {3 a^2-b \sinh (c+d x) a+2 \left (3 a^2+b^2\right ) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{2 b}+\frac {3 i a \sinh (c+d x) \cosh (c+d x)}{2 b d}\right )}{3 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sinh ^2(c+d x) \cosh (c+d x)}{3 b d}+\frac {i \left (\frac {3 i a \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {i \int \frac {3 a^2-b \sinh (c+d x) a+2 \left (3 a^2+b^2\right ) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{2 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh ^2(c+d x) \cosh (c+d x)}{3 b d}+\frac {i \left (\frac {3 i a \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {i \int \frac {3 a^2+i b \sin (i c+i d x) a-2 \left (3 a^2+b^2\right ) \sin (i c+i d x)^2}{a-i b \sin (i c+i d x)}dx}{2 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\sinh ^2(c+d x) \cosh (c+d x)}{3 b d}+\frac {i \left (\frac {3 i a \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {i \left (\frac {2 \left (3 a^2+b^2\right ) \cosh (c+d x)}{b d}+\frac {i \int -\frac {3 i \left (a^2 b-a \left (2 a^2+b^2\right ) \sinh (c+d x)\right )}{a+b \sinh (c+d x)}dx}{b}\right )}{2 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sinh ^2(c+d x) \cosh (c+d x)}{3 b d}+\frac {i \left (\frac {3 i a \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {i \left (\frac {3 \int \frac {a^2 b-a \left (2 a^2+b^2\right ) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {2 \left (3 a^2+b^2\right ) \cosh (c+d x)}{b d}\right )}{2 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh ^2(c+d x) \cosh (c+d x)}{3 b d}+\frac {i \left (\frac {3 i a \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {i \left (\frac {2 \left (3 a^2+b^2\right ) \cosh (c+d x)}{b d}+\frac {3 \int \frac {b a^2+i \left (2 a^2+b^2\right ) \sin (i c+i d x) a}{a-i b \sin (i c+i d x)}dx}{b}\right )}{2 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\sinh ^2(c+d x) \cosh (c+d x)}{3 b d}+\frac {i \left (\frac {3 i a \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {i \left (\frac {3 \left (\frac {2 a^2 \left (a^2+b^2\right ) \int \frac {1}{a+b \sinh (c+d x)}dx}{b}-\frac {a x \left (2 a^2+b^2\right )}{b}\right )}{b}+\frac {2 \left (3 a^2+b^2\right ) \cosh (c+d x)}{b d}\right )}{2 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh ^2(c+d x) \cosh (c+d x)}{3 b d}+\frac {i \left (\frac {3 i a \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {i \left (\frac {2 \left (3 a^2+b^2\right ) \cosh (c+d x)}{b d}+\frac {3 \left (-\frac {a x \left (2 a^2+b^2\right )}{b}+\frac {2 a^2 \left (a^2+b^2\right ) \int \frac {1}{a-i b \sin (i c+i d x)}dx}{b}\right )}{b}\right )}{2 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\sinh ^2(c+d x) \cosh (c+d x)}{3 b d}+\frac {i \left (\frac {3 i a \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {i \left (\frac {2 \left (3 a^2+b^2\right ) \cosh (c+d x)}{b d}+\frac {3 \left (-\frac {a x \left (2 a^2+b^2\right )}{b}-\frac {4 i a^2 \left (a^2+b^2\right ) \int \frac {1}{-a \tanh ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tanh \left (\frac {1}{2} (c+d x)\right )+a}d\left (i \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{b d}\right )}{b}\right )}{2 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\sinh ^2(c+d x) \cosh (c+d x)}{3 b d}+\frac {i \left (\frac {3 i a \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {i \left (\frac {2 \left (3 a^2+b^2\right ) \cosh (c+d x)}{b d}+\frac {3 \left (-\frac {a x \left (2 a^2+b^2\right )}{b}+\frac {8 i a^2 \left (a^2+b^2\right ) \int \frac {1}{\tanh ^2\left (\frac {1}{2} (c+d x)\right )-4 \left (a^2+b^2\right )}d\left (2 i a \tanh \left (\frac {1}{2} (c+d x)\right )-2 i b\right )}{b d}\right )}{b}\right )}{2 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\sinh ^2(c+d x) \cosh (c+d x)}{3 b d}+\frac {i \left (\frac {3 i a \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {i \left (\frac {3 \left (\frac {4 a^2 \sqrt {a^2+b^2} \text {arctanh}\left (\frac {\tanh \left (\frac {1}{2} (c+d x)\right )}{2 \sqrt {a^2+b^2}}\right )}{b d}-\frac {a x \left (2 a^2+b^2\right )}{b}\right )}{b}+\frac {2 \left (3 a^2+b^2\right ) \cosh (c+d x)}{b d}\right )}{2 b}\right )}{3 b}\) |
(Cosh[c + d*x]*Sinh[c + d*x]^2)/(3*b*d) + ((I/3)*(((-1/2*I)*((3*(-((a*(2*a ^2 + b^2)*x)/b) + (4*a^2*Sqrt[a^2 + b^2]*ArcTanh[Tanh[(c + d*x)/2]/(2*Sqrt [a^2 + b^2])])/(b*d)))/b + (2*(3*a^2 + b^2)*Cosh[c + d*x])/(b*d)))/b + ((( 3*I)/2)*a*Cosh[c + d*x]*Sinh[c + d*x])/(b*d)))/b
3.4.70.3.1 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_)*(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, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[cos[(e_.) + (f_.)*(x_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, m, n }, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] : > Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x ])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*( n + 1)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + C* (a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f , A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Time = 9.18 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.73
| method | result | size |
| risch | \(-\frac {a^{3} x}{b^{4}}-\frac {a x}{2 b^{2}}+\frac {{\mathrm e}^{3 d x +3 c}}{24 b d}-\frac {a \,{\mathrm e}^{2 d x +2 c}}{8 b^{2} d}+\frac {{\mathrm e}^{d x +c} a^{2}}{2 b^{3} d}+\frac {{\mathrm e}^{d x +c}}{8 b d}+\frac {{\mathrm e}^{-d x -c} a^{2}}{2 b^{3} d}+\frac {{\mathrm e}^{-d x -c}}{8 b d}+\frac {a \,{\mathrm e}^{-2 d x -2 c}}{8 b^{2} d}+\frac {{\mathrm e}^{-3 d x -3 c}}{24 b d}+\frac {\sqrt {a^{2}+b^{2}}\, a^{2} \ln \left ({\mathrm e}^{d x +c}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}\right )}{d \,b^{4}}-\frac {\sqrt {a^{2}+b^{2}}\, a^{2} \ln \left ({\mathrm e}^{d x +c}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right )}{d \,b^{4}}\) | \(244\) |
| derivativedivides | \(\frac {\frac {2 a^{2} \sqrt {a^{2}+b^{2}}\, \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{4}}+\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {b -a}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-2 a^{2}+a b -b^{2}}{2 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {a \left (2 a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{4}}-\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {a +b}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 a^{2}+a b +b^{2}}{2 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a \left (2 a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{4}}}{d}\) | \(247\) |
| default | \(\frac {\frac {2 a^{2} \sqrt {a^{2}+b^{2}}\, \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{4}}+\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {b -a}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-2 a^{2}+a b -b^{2}}{2 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {a \left (2 a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{4}}-\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {a +b}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 a^{2}+a b +b^{2}}{2 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a \left (2 a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{4}}}{d}\) | \(247\) |
-a^3*x/b^4-1/2*a*x/b^2+1/24/b/d*exp(3*d*x+3*c)-1/8*a/b^2/d*exp(2*d*x+2*c)+ 1/2/b^3/d*exp(d*x+c)*a^2+1/8/b/d*exp(d*x+c)+1/2/b^3/d*exp(-d*x-c)*a^2+1/8/ b/d*exp(-d*x-c)+1/8*a/b^2/d*exp(-2*d*x-2*c)+1/24/b/d*exp(-3*d*x-3*c)+(a^2+ b^2)^(1/2)*a^2/d/b^4*ln(exp(d*x+c)-(-a+(a^2+b^2)^(1/2))/b)-(a^2+b^2)^(1/2) *a^2/d/b^4*ln(exp(d*x+c)+(a+(a^2+b^2)^(1/2))/b)
Leaf count of result is larger than twice the leaf count of optimal. 745 vs. \(2 (130) = 260\).
Time = 0.26 (sec) , antiderivative size = 745, normalized size of antiderivative = 5.28 \[ \int \frac {\cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b^{3} \cosh \left (d x + c\right )^{6} + b^{3} \sinh \left (d x + c\right )^{6} - 3 \, a b^{2} \cosh \left (d x + c\right )^{5} - 12 \, {\left (2 \, a^{3} + a b^{2}\right )} d x \cosh \left (d x + c\right )^{3} + 3 \, {\left (2 \, b^{3} \cosh \left (d x + c\right ) - a b^{2}\right )} \sinh \left (d x + c\right )^{5} + 3 \, {\left (4 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, b^{3} \cosh \left (d x + c\right )^{2} - 5 \, a b^{2} \cosh \left (d x + c\right ) + 4 \, a^{2} b + b^{3}\right )} \sinh \left (d x + c\right )^{4} + 3 \, a b^{2} \cosh \left (d x + c\right ) + 2 \, {\left (10 \, b^{3} \cosh \left (d x + c\right )^{3} - 15 \, a b^{2} \cosh \left (d x + c\right )^{2} - 6 \, {\left (2 \, a^{3} + a b^{2}\right )} d x + 6 \, {\left (4 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + b^{3} + 3 \, {\left (4 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, b^{3} \cosh \left (d x + c\right )^{4} - 10 \, a b^{2} \cosh \left (d x + c\right )^{3} - 12 \, {\left (2 \, a^{3} + a b^{2}\right )} d x \cosh \left (d x + c\right ) + 4 \, a^{2} b + b^{3} + 6 \, {\left (4 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 24 \, {\left (a^{2} \cosh \left (d x + c\right )^{3} + 3 \, a^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a^{2} \sinh \left (d x + c\right )^{3}\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) + a\right )}}{b \cosh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) + 2 \, {\left (b \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right ) - b}\right ) + 3 \, {\left (2 \, b^{3} \cosh \left (d x + c\right )^{5} - 5 \, a b^{2} \cosh \left (d x + c\right )^{4} - 12 \, {\left (2 \, a^{3} + a b^{2}\right )} d x \cosh \left (d x + c\right )^{2} + 4 \, {\left (4 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )^{3} + a b^{2} + 2 \, {\left (4 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{24 \, {\left (b^{4} d \cosh \left (d x + c\right )^{3} + 3 \, b^{4} d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, b^{4} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + b^{4} d \sinh \left (d x + c\right )^{3}\right )}} \]
1/24*(b^3*cosh(d*x + c)^6 + b^3*sinh(d*x + c)^6 - 3*a*b^2*cosh(d*x + c)^5 - 12*(2*a^3 + a*b^2)*d*x*cosh(d*x + c)^3 + 3*(2*b^3*cosh(d*x + c) - a*b^2) *sinh(d*x + c)^5 + 3*(4*a^2*b + b^3)*cosh(d*x + c)^4 + 3*(5*b^3*cosh(d*x + c)^2 - 5*a*b^2*cosh(d*x + c) + 4*a^2*b + b^3)*sinh(d*x + c)^4 + 3*a*b^2*c osh(d*x + c) + 2*(10*b^3*cosh(d*x + c)^3 - 15*a*b^2*cosh(d*x + c)^2 - 6*(2 *a^3 + a*b^2)*d*x + 6*(4*a^2*b + b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + b^3 + 3*(4*a^2*b + b^3)*cosh(d*x + c)^2 + 3*(5*b^3*cosh(d*x + c)^4 - 10*a*b^2 *cosh(d*x + c)^3 - 12*(2*a^3 + a*b^2)*d*x*cosh(d*x + c) + 4*a^2*b + b^3 + 6*(4*a^2*b + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 24*(a^2*cosh(d*x + c) ^3 + 3*a^2*cosh(d*x + c)^2*sinh(d*x + c) + 3*a^2*cosh(d*x + c)*sinh(d*x + c)^2 + a^2*sinh(d*x + c)^3)*sqrt(a^2 + b^2)*log((b^2*cosh(d*x + c)^2 + b^2 *sinh(d*x + c)^2 + 2*a*b*cosh(d*x + c) + 2*a^2 + b^2 + 2*(b^2*cosh(d*x + c ) + a*b)*sinh(d*x + c) - 2*sqrt(a^2 + b^2)*(b*cosh(d*x + c) + b*sinh(d*x + c) + a))/(b*cosh(d*x + c)^2 + b*sinh(d*x + c)^2 + 2*a*cosh(d*x + c) + 2*( b*cosh(d*x + c) + a)*sinh(d*x + c) - b)) + 3*(2*b^3*cosh(d*x + c)^5 - 5*a* b^2*cosh(d*x + c)^4 - 12*(2*a^3 + a*b^2)*d*x*cosh(d*x + c)^2 + 4*(4*a^2*b + b^3)*cosh(d*x + c)^3 + a*b^2 + 2*(4*a^2*b + b^3)*cosh(d*x + c))*sinh(d*x + c))/(b^4*d*cosh(d*x + c)^3 + 3*b^4*d*cosh(d*x + c)^2*sinh(d*x + c) + 3* b^4*d*cosh(d*x + c)*sinh(d*x + c)^2 + b^4*d*sinh(d*x + c)^3)
Timed out. \[ \int \frac {\cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.48 \[ \int \frac {\cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\sqrt {a^{2} + b^{2}} a^{2} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{b^{4} d} - \frac {{\left (3 \, a b e^{\left (-d x - c\right )} - b^{2} - 3 \, {\left (4 \, a^{2} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{24 \, b^{3} d} - \frac {{\left (2 \, a^{3} + a b^{2}\right )} {\left (d x + c\right )}}{2 \, b^{4} d} + \frac {3 \, a b e^{\left (-2 \, d x - 2 \, c\right )} + b^{2} e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, {\left (4 \, a^{2} + b^{2}\right )} e^{\left (-d x - c\right )}}{24 \, b^{3} d} \]
sqrt(a^2 + b^2)*a^2*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(b^4*d) - 1/24*(3*a*b*e^(-d*x - c) - b^2 - 3* (4*a^2 + b^2)*e^(-2*d*x - 2*c))*e^(3*d*x + 3*c)/(b^3*d) - 1/2*(2*a^3 + a*b ^2)*(d*x + c)/(b^4*d) + 1/24*(3*a*b*e^(-2*d*x - 2*c) + b^2*e^(-3*d*x - 3*c ) + 3*(4*a^2 + b^2)*e^(-d*x - c))/(b^3*d)
Time = 0.30 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.50 \[ \int \frac {\cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {12 \, {\left (2 \, a^{3} + a b^{2}\right )} {\left (d x + c\right )}}{b^{4}} - \frac {b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 3 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 12 \, a^{2} e^{\left (d x + c\right )} + 3 \, b^{2} e^{\left (d x + c\right )}}{b^{3}} - \frac {{\left (3 \, a b^{2} e^{\left (d x + c\right )} + b^{3} + 3 \, {\left (4 \, a^{2} b + b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{b^{4}} - \frac {24 \, {\left (a^{4} + a^{2} b^{2}\right )} \log \left (\frac {{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b^{4}}}{24 \, d} \]
-1/24*(12*(2*a^3 + a*b^2)*(d*x + c)/b^4 - (b^2*e^(3*d*x + 3*c) - 3*a*b*e^( 2*d*x + 2*c) + 12*a^2*e^(d*x + c) + 3*b^2*e^(d*x + c))/b^3 - (3*a*b^2*e^(d *x + c) + b^3 + 3*(4*a^2*b + b^3)*e^(2*d*x + 2*c))*e^(-3*d*x - 3*c)/b^4 - 24*(a^4 + a^2*b^2)*log(abs(2*b*e^(d*x + c) + 2*a - 2*sqrt(a^2 + b^2))/abs( 2*b*e^(d*x + c) + 2*a + 2*sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*b^4))/d
Time = 1.42 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.97 \[ \int \frac {\cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,b\,d}-\frac {x\,\left (2\,a^3+a\,b^2\right )}{2\,b^4}+\frac {{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,b\,d}+\frac {a\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,b^2\,d}-\frac {a\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,b^2\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (4\,a^2+b^2\right )}{8\,b^3\,d}+\frac {{\mathrm {e}}^{-c-d\,x}\,\left (4\,a^2+b^2\right )}{8\,b^3\,d}-\frac {a^2\,\ln \left (-\frac {2\,a^2\,\sqrt {a^2+b^2}\,\left (b-a\,{\mathrm {e}}^{c+d\,x}\right )}{b^5}-\frac {2\,a^2\,{\mathrm {e}}^{c+d\,x}\,\left (a^2+b^2\right )}{b^5}\right )\,\sqrt {a^2+b^2}}{b^4\,d}+\frac {a^2\,\ln \left (\frac {2\,a^2\,\sqrt {a^2+b^2}\,\left (b-a\,{\mathrm {e}}^{c+d\,x}\right )}{b^5}-\frac {2\,a^2\,{\mathrm {e}}^{c+d\,x}\,\left (a^2+b^2\right )}{b^5}\right )\,\sqrt {a^2+b^2}}{b^4\,d} \]
exp(- 3*c - 3*d*x)/(24*b*d) - (x*(a*b^2 + 2*a^3))/(2*b^4) + exp(3*c + 3*d* x)/(24*b*d) + (a*exp(- 2*c - 2*d*x))/(8*b^2*d) - (a*exp(2*c + 2*d*x))/(8*b ^2*d) + (exp(c + d*x)*(4*a^2 + b^2))/(8*b^3*d) + (exp(- c - d*x)*(4*a^2 + b^2))/(8*b^3*d) - (a^2*log(- (2*a^2*(a^2 + b^2)^(1/2)*(b - a*exp(c + d*x)) )/b^5 - (2*a^2*exp(c + d*x)*(a^2 + b^2))/b^5)*(a^2 + b^2)^(1/2))/(b^4*d) + (a^2*log((2*a^2*(a^2 + b^2)^(1/2)*(b - a*exp(c + d*x)))/b^5 - (2*a^2*exp( c + d*x)*(a^2 + b^2))/b^5)*(a^2 + b^2)^(1/2))/(b^4*d)