Integrand size = 29, antiderivative size = 184 \[ \int \frac {\cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\left (8 a^4+4 a^2 b^2-b^4\right ) x}{8 b^5}+\frac {2 a^3 \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{b^5 d}-\frac {a \left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^4 d}+\frac {\left (4 a^2+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 b^3 d}-\frac {a \cosh (c+d x) \sinh ^2(c+d x)}{3 b^2 d}+\frac {\cosh (c+d x) \sinh ^3(c+d x)}{4 b d} \] Output:
1/8*(8*a^4+4*a^2*b^2-b^4)*x/b^5+2*a^3*(a^2+b^2)^(1/2)*arctanh((b-a*tanh(1/ 2*d*x+1/2*c))/(a^2+b^2)^(1/2))/b^5/d-1/3*a*(3*a^2+b^2)*cosh(d*x+c)/b^4/d+1 /8*(4*a^2+b^2)*cosh(d*x+c)*sinh(d*x+c)/b^3/d-1/3*a*cosh(d*x+c)*sinh(d*x+c) ^2/b^2/d+1/4*cosh(d*x+c)*sinh(d*x+c)^3/b/d
Time = 1.45 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.83 \[ \int \frac {\cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-24 a b \left (4 a^2+b^2\right ) \cosh (c+d x)-8 a b^3 \cosh (3 (c+d x))+3 \left (4 \left (8 a^4+4 a^2 b^2-b^4\right ) (c+d x)+64 a^3 \sqrt {-a^2-b^2} \arctan \left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )+8 a^2 b^2 \sinh (2 (c+d x))+b^4 \sinh (4 (c+d x))\right )}{96 b^5 d} \] Input:
Integrate[(Cosh[c + d*x]^2*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
Output:
(-24*a*b*(4*a^2 + b^2)*Cosh[c + d*x] - 8*a*b^3*Cosh[3*(c + d*x)] + 3*(4*(8 *a^4 + 4*a^2*b^2 - b^4)*(c + d*x) + 64*a^3*Sqrt[-a^2 - b^2]*ArcTan[(b - a* Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]] + 8*a^2*b^2*Sinh[2*(c + d*x)] + b^4*S inh[4*(c + d*x)]))/(96*b^5*d)
Result contains complex when optimal does not.
Time = 1.60 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.21, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.828, Rules used = {3042, 26, 3368, 26, 3042, 26, 3529, 3042, 25, 3528, 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.
| \(\displaystyle \int \frac {\sinh ^3(c+d x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \sin (i c+i d x)^3 \cos (i c+i d x)^2}{a-i b \sin (i c+i d x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\cos (i c+i d x)^2 \sin (i c+i d x)^3}{a-i b \sin (i c+i d x)}dx\) |
| \(\Big \downarrow \) 3368 |
| \(\displaystyle i \int -\frac {i \sinh ^3(c+d x) \left (\sinh ^2(c+d x)+1\right )}{a+b \sinh (c+d x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int \frac {\sinh ^3(c+d x) \left (\sinh ^2(c+d x)+1\right )}{a+b \sinh (c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \sin (i c+i d x)^3 \left (1-\sin (i c+i d x)^2\right )}{a-i b \sin (i c+i d x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\sin (i c+i d x)^3 \left (1-\sin (i c+i d x)^2\right )}{a-i b \sin (i c+i d x)}dx\) |
| \(\Big \downarrow \) 3529 |
| \(\displaystyle i \left (\frac {i \int \frac {\sinh ^2(c+d x) \left (4 a \sinh ^2(c+d x)-b \sinh (c+d x)+3 a\right )}{a+b \sinh (c+d x)}dx}{4 b}-\frac {i \sinh ^3(c+d x) \cosh (c+d x)}{4 b d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i \int -\frac {\sin (i c+i d x)^2 \left (-4 a \sin (i c+i d x)^2+i b \sin (i c+i d x)+3 a\right )}{a-i b \sin (i c+i d x)}dx}{4 b}-\frac {i \sinh ^3(c+d x) \cosh (c+d x)}{4 b d}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (-\frac {i \int \frac {\sin (i c+i d x)^2 \left (-4 a \sin (i c+i d x)^2+i b \sin (i c+i d x)+3 a\right )}{a-i b \sin (i c+i d x)}dx}{4 b}-\frac {i \sinh ^3(c+d x) \cosh (c+d x)}{4 b d}\right )\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {4 a \sinh ^2(c+d x) \cosh (c+d x)}{3 b d}+\frac {i \int -\frac {i \sinh (c+d x) \left (8 a^2-b \sinh (c+d x) a+3 \left (4 a^2+b^2\right ) \sinh ^2(c+d x)\right )}{a+b \sinh (c+d x)}dx}{3 b}\right )}{4 b}-\frac {i \sinh ^3(c+d x) \cosh (c+d x)}{4 b d}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\frac {i \left (\frac {\int \frac {\sinh (c+d x) \left (8 a^2-b \sinh (c+d x) a+3 \left (4 a^2+b^2\right ) \sinh ^2(c+d x)\right )}{a+b \sinh (c+d x)}dx}{3 b}-\frac {4 a \sinh ^2(c+d x) \cosh (c+d x)}{3 b d}\right )}{4 b}-\frac {i \sinh ^3(c+d x) \cosh (c+d x)}{4 b d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {4 a \sinh ^2(c+d x) \cosh (c+d x)}{3 b d}+\frac {\int -\frac {i \sin (i c+i d x) \left (8 a^2+i b \sin (i c+i d x) a-3 \left (4 a^2+b^2\right ) \sin (i c+i d x)^2\right )}{a-i b \sin (i c+i d x)}dx}{3 b}\right )}{4 b}-\frac {i \sinh ^3(c+d x) \cosh (c+d x)}{4 b d}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {4 a \sinh ^2(c+d x) \cosh (c+d x)}{3 b d}-\frac {i \int \frac {\sin (i c+i d x) \left (8 a^2+i b \sin (i c+i d x) a-3 \left (4 a^2+b^2\right ) \sin (i c+i d x)^2\right )}{a-i b \sin (i c+i d x)}dx}{3 b}\right )}{4 b}-\frac {i \sinh ^3(c+d x) \cosh (c+d x)}{4 b d}\right )\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {4 a \sinh ^2(c+d x) \cosh (c+d x)}{3 b d}-\frac {i \left (\frac {i \int -\frac {8 a \left (3 a^2+b^2\right ) \sinh ^2(c+d x)-b \left (4 a^2-3 b^2\right ) \sinh (c+d x)+3 a \left (4 a^2+b^2\right )}{a+b \sinh (c+d x)}dx}{2 b}+\frac {3 i \left (4 a^2+b^2\right ) \sinh (c+d x) \cosh (c+d x)}{2 b d}\right )}{3 b}\right )}{4 b}-\frac {i \sinh ^3(c+d x) \cosh (c+d x)}{4 b d}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {4 a \sinh ^2(c+d x) \cosh (c+d x)}{3 b d}-\frac {i \left (\frac {3 i \left (4 a^2+b^2\right ) \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {i \int \frac {8 a \left (3 a^2+b^2\right ) \sinh ^2(c+d x)-b \left (4 a^2-3 b^2\right ) \sinh (c+d x)+3 a \left (4 a^2+b^2\right )}{a+b \sinh (c+d x)}dx}{2 b}\right )}{3 b}\right )}{4 b}-\frac {i \sinh ^3(c+d x) \cosh (c+d x)}{4 b d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {4 a \sinh ^2(c+d x) \cosh (c+d x)}{3 b d}-\frac {i \left (\frac {3 i \left (4 a^2+b^2\right ) \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {i \int \frac {-8 a \left (3 a^2+b^2\right ) \sin (i c+i d x)^2+i b \left (4 a^2-3 b^2\right ) \sin (i c+i d x)+3 a \left (4 a^2+b^2\right )}{a-i b \sin (i c+i d x)}dx}{2 b}\right )}{3 b}\right )}{4 b}-\frac {i \sinh ^3(c+d x) \cosh (c+d x)}{4 b d}\right )\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {4 a \sinh ^2(c+d x) \cosh (c+d x)}{3 b d}-\frac {i \left (\frac {3 i \left (4 a^2+b^2\right ) \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {i \left (\frac {8 a \left (3 a^2+b^2\right ) \cosh (c+d x)}{b d}+\frac {i \int -\frac {3 i \left (a b \left (4 a^2+b^2\right )-\left (8 a^4+4 b^2 a^2-b^4\right ) \sinh (c+d x)\right )}{a+b \sinh (c+d x)}dx}{b}\right )}{2 b}\right )}{3 b}\right )}{4 b}-\frac {i \sinh ^3(c+d x) \cosh (c+d x)}{4 b d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {4 a \sinh ^2(c+d x) \cosh (c+d x)}{3 b d}-\frac {i \left (\frac {3 i \left (4 a^2+b^2\right ) \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {i \left (\frac {3 \int \frac {a b \left (4 a^2+b^2\right )-\left (8 a^4+4 b^2 a^2-b^4\right ) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {8 a \left (3 a^2+b^2\right ) \cosh (c+d x)}{b d}\right )}{2 b}\right )}{3 b}\right )}{4 b}-\frac {i \sinh ^3(c+d x) \cosh (c+d x)}{4 b d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {4 a \sinh ^2(c+d x) \cosh (c+d x)}{3 b d}-\frac {i \left (\frac {3 i \left (4 a^2+b^2\right ) \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {i \left (\frac {8 a \left (3 a^2+b^2\right ) \cosh (c+d x)}{b d}+\frac {3 \int \frac {a b \left (4 a^2+b^2\right )+i \left (8 a^4+4 b^2 a^2-b^4\right ) \sin (i c+i d x)}{a-i b \sin (i c+i d x)}dx}{b}\right )}{2 b}\right )}{3 b}\right )}{4 b}-\frac {i \sinh ^3(c+d x) \cosh (c+d x)}{4 b d}\right )\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {4 a \sinh ^2(c+d x) \cosh (c+d x)}{3 b d}-\frac {i \left (\frac {3 i \left (4 a^2+b^2\right ) \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {i \left (\frac {3 \left (\frac {8 a^3 \left (a^2+b^2\right ) \int \frac {1}{a+b \sinh (c+d x)}dx}{b}-\frac {x \left (8 a^4+4 a^2 b^2-b^4\right )}{b}\right )}{b}+\frac {8 a \left (3 a^2+b^2\right ) \cosh (c+d x)}{b d}\right )}{2 b}\right )}{3 b}\right )}{4 b}-\frac {i \sinh ^3(c+d x) \cosh (c+d x)}{4 b d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {4 a \sinh ^2(c+d x) \cosh (c+d x)}{3 b d}-\frac {i \left (\frac {3 i \left (4 a^2+b^2\right ) \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {i \left (\frac {8 a \left (3 a^2+b^2\right ) \cosh (c+d x)}{b d}+\frac {3 \left (-\frac {x \left (8 a^4+4 a^2 b^2-b^4\right )}{b}+\frac {8 a^3 \left (a^2+b^2\right ) \int \frac {1}{a-i b \sin (i c+i d x)}dx}{b}\right )}{b}\right )}{2 b}\right )}{3 b}\right )}{4 b}-\frac {i \sinh ^3(c+d x) \cosh (c+d x)}{4 b d}\right )\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {4 a \sinh ^2(c+d x) \cosh (c+d x)}{3 b d}-\frac {i \left (\frac {3 i \left (4 a^2+b^2\right ) \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {i \left (\frac {8 a \left (3 a^2+b^2\right ) \cosh (c+d x)}{b d}+\frac {3 \left (-\frac {x \left (8 a^4+4 a^2 b^2-b^4\right )}{b}-\frac {16 i a^3 \left (a^2+b^2\right ) \int \frac {1}{-a \tanh ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tanh \left (\frac {1}{2} (c+d x)\right )+a}d\left (i \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{b d}\right )}{b}\right )}{2 b}\right )}{3 b}\right )}{4 b}-\frac {i \sinh ^3(c+d x) \cosh (c+d x)}{4 b d}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {4 a \sinh ^2(c+d x) \cosh (c+d x)}{3 b d}-\frac {i \left (\frac {3 i \left (4 a^2+b^2\right ) \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {i \left (\frac {8 a \left (3 a^2+b^2\right ) \cosh (c+d x)}{b d}+\frac {3 \left (-\frac {x \left (8 a^4+4 a^2 b^2-b^4\right )}{b}+\frac {32 i a^3 \left (a^2+b^2\right ) \int \frac {1}{\tanh ^2\left (\frac {1}{2} (c+d x)\right )-4 \left (a^2+b^2\right )}d\left (2 i a \tanh \left (\frac {1}{2} (c+d x)\right )-2 i b\right )}{b d}\right )}{b}\right )}{2 b}\right )}{3 b}\right )}{4 b}-\frac {i \sinh ^3(c+d x) \cosh (c+d x)}{4 b d}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle i \left (-\frac {i \left (-\frac {4 a \sinh ^2(c+d x) \cosh (c+d x)}{3 b d}-\frac {i \left (\frac {3 i \left (4 a^2+b^2\right ) \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {i \left (\frac {8 a \left (3 a^2+b^2\right ) \cosh (c+d x)}{b d}+\frac {3 \left (\frac {16 a^3 \sqrt {a^2+b^2} \text {arctanh}\left (\frac {\tanh \left (\frac {1}{2} (c+d x)\right )}{2 \sqrt {a^2+b^2}}\right )}{b d}-\frac {x \left (8 a^4+4 a^2 b^2-b^4\right )}{b}\right )}{b}\right )}{2 b}\right )}{3 b}\right )}{4 b}-\frac {i \sinh ^3(c+d x) \cosh (c+d x)}{4 b d}\right )\) |
Input:
Int[(Cosh[c + d*x]^2*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
Output:
I*(((-1/4*I)*Cosh[c + d*x]*Sinh[c + d*x]^3)/(b*d) - ((I/4)*((-4*a*Cosh[c + d*x]*Sinh[c + d*x]^2)/(3*b*d) - ((I/3)*(((-1/2*I)*((3*(-(((8*a^4 + 4*a^2* b^2 - b^4)*x)/b) + (16*a^3*Sqrt[a^2 + b^2]*ArcTanh[Tanh[(c + d*x)/2]/(2*Sq rt[a^2 + b^2])])/(b*d)))/b + (8*a*(3*a^2 + b^2)*Cosh[c + d*x])/(b*d)))/b + (((3*I)/2)*(4*a^2 + b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(b*d)))/b))/b)
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 = 33.22 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.59
| method | result | size |
| risch | \(\frac {a^{4} x}{b^{5}}+\frac {x \,a^{2}}{2 b^{3}}-\frac {x}{8 b}+\frac {{\mathrm e}^{4 d x +4 c}}{64 b d}-\frac {a \,{\mathrm e}^{3 d x +3 c}}{24 b^{2} d}+\frac {{\mathrm e}^{2 d x +2 c} a^{2}}{8 b^{3} d}-\frac {a^{3} {\mathrm e}^{d x +c}}{2 b^{4} d}-\frac {a \,{\mathrm e}^{d x +c}}{8 b^{2} d}-\frac {a^{3} {\mathrm e}^{-d x -c}}{2 b^{4} d}-\frac {a \,{\mathrm e}^{-d x -c}}{8 b^{2} d}-\frac {{\mathrm e}^{-2 d x -2 c} a^{2}}{8 b^{3} d}-\frac {a \,{\mathrm e}^{-3 d x -3 c}}{24 b^{2} d}-\frac {{\mathrm e}^{-4 d x -4 c}}{64 b d}+\frac {\sqrt {a^{2}+b^{2}}\, a^{3} \ln \left ({\mathrm e}^{d x +c}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right )}{d \,b^{5}}-\frac {\sqrt {a^{2}+b^{2}}\, a^{3} \ln \left ({\mathrm e}^{d x +c}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}\right )}{d \,b^{5}}\) | \(293\) |
| derivativedivides | \(\frac {-\frac {1}{4 b \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {-3 b +2 a}{6 b^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {4 a^{2}-4 a b +3 b^{2}}{8 b^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (8 a^{4}+4 a^{2} b^{2}-b^{4}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 b^{5}}-\frac {8 a^{3}-4 a^{2} b +4 a \,b^{2}-b^{3}}{8 b^{4} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {2 a^{3} \sqrt {a^{2}+b^{2}}\, \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{5}}+\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {-3 b -2 a}{6 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-4 a^{2}-4 a b -3 b^{2}}{8 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (-8 a^{4}-4 a^{2} b^{2}+b^{4}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 b^{5}}-\frac {-8 a^{3}-4 a^{2} b -4 a \,b^{2}-b^{3}}{8 b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(355\) |
| default | \(\frac {-\frac {1}{4 b \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {-3 b +2 a}{6 b^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {4 a^{2}-4 a b +3 b^{2}}{8 b^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (8 a^{4}+4 a^{2} b^{2}-b^{4}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 b^{5}}-\frac {8 a^{3}-4 a^{2} b +4 a \,b^{2}-b^{3}}{8 b^{4} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {2 a^{3} \sqrt {a^{2}+b^{2}}\, \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{5}}+\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {-3 b -2 a}{6 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-4 a^{2}-4 a b -3 b^{2}}{8 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (-8 a^{4}-4 a^{2} b^{2}+b^{4}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 b^{5}}-\frac {-8 a^{3}-4 a^{2} b -4 a \,b^{2}-b^{3}}{8 b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(355\) |
Input:
int(cosh(d*x+c)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)
Output:
a^4*x/b^5+1/2*x/b^3*a^2-1/8*x/b+1/64/b/d*exp(4*d*x+4*c)-1/24*a/b^2/d*exp(3 *d*x+3*c)+1/8/b^3/d*exp(2*d*x+2*c)*a^2-1/2*a^3/b^4/d*exp(d*x+c)-1/8*a/b^2/ d*exp(d*x+c)-1/2*a^3/b^4/d*exp(-d*x-c)-1/8*a/b^2/d*exp(-d*x-c)-1/8/b^3/d*e xp(-2*d*x-2*c)*a^2-1/24*a/b^2/d*exp(-3*d*x-3*c)-1/64/b/d*exp(-4*d*x-4*c)+( a^2+b^2)^(1/2)*a^3/d/b^5*ln(exp(d*x+c)+(a+(a^2+b^2)^(1/2))/b)-(a^2+b^2)^(1 /2)*a^3/d/b^5*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. 1134 vs. \(2 (171) = 342\).
Time = 0.12 (sec) , antiderivative size = 1134, normalized size of antiderivative = 6.16 \[ \int \frac {\cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(cosh(d*x+c)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fric as")
Output:
1/192*(3*b^4*cosh(d*x + c)^8 + 3*b^4*sinh(d*x + c)^8 - 8*a*b^3*cosh(d*x + c)^7 + 24*a^2*b^2*cosh(d*x + c)^6 + 8*(3*b^4*cosh(d*x + c) - a*b^3)*sinh(d *x + c)^7 + 24*(8*a^4 + 4*a^2*b^2 - b^4)*d*x*cosh(d*x + c)^4 + 4*(21*b^4*c osh(d*x + c)^2 - 14*a*b^3*cosh(d*x + c) + 6*a^2*b^2)*sinh(d*x + c)^6 - 24* a^2*b^2*cosh(d*x + c)^2 - 24*(4*a^3*b + a*b^3)*cosh(d*x + c)^5 + 24*(7*b^4 *cosh(d*x + c)^3 - 7*a*b^3*cosh(d*x + c)^2 + 6*a^2*b^2*cosh(d*x + c) - 4*a ^3*b - a*b^3)*sinh(d*x + c)^5 - 8*a*b^3*cosh(d*x + c) + 2*(105*b^4*cosh(d* x + c)^4 - 140*a*b^3*cosh(d*x + c)^3 + 180*a^2*b^2*cosh(d*x + c)^2 + 12*(8 *a^4 + 4*a^2*b^2 - b^4)*d*x - 60*(4*a^3*b + a*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 - 3*b^4 - 24*(4*a^3*b + a*b^3)*cosh(d*x + c)^3 + 8*(21*b^4*cosh(d* x + c)^5 - 35*a*b^3*cosh(d*x + c)^4 + 60*a^2*b^2*cosh(d*x + c)^3 - 12*a^3* b - 3*a*b^3 + 12*(8*a^4 + 4*a^2*b^2 - b^4)*d*x*cosh(d*x + c) - 30*(4*a^3*b + a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 12*(7*b^4*cosh(d*x + c)^6 - 1 4*a*b^3*cosh(d*x + c)^5 + 30*a^2*b^2*cosh(d*x + c)^4 + 12*(8*a^4 + 4*a^2*b ^2 - b^4)*d*x*cosh(d*x + c)^2 - 2*a^2*b^2 - 20*(4*a^3*b + a*b^3)*cosh(d*x + c)^3 - 6*(4*a^3*b + a*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + 192*(a^3*cos h(d*x + c)^4 + 4*a^3*cosh(d*x + c)^3*sinh(d*x + c) + 6*a^3*cosh(d*x + c)^2 *sinh(d*x + c)^2 + 4*a^3*cosh(d*x + c)*sinh(d*x + c)^3 + a^3*sinh(d*x + c) ^4)*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...
Timed out. \[ \int \frac {\cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cosh(d*x+c)**2*sinh(d*x+c)**3/(a+b*sinh(d*x+c)),x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.40 \[ \int \frac {\cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\sqrt {a^{2} + b^{2}} a^{3} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{b^{5} d} - \frac {{\left (8 \, a b^{2} e^{\left (-d x - c\right )} - 24 \, a^{2} b e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, b^{3} + 24 \, {\left (4 \, a^{3} + a b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{192 \, b^{4} d} + \frac {{\left (8 \, a^{4} + 4 \, a^{2} b^{2} - b^{4}\right )} {\left (d x + c\right )}}{8 \, b^{5} d} - \frac {24 \, a^{2} b e^{\left (-2 \, d x - 2 \, c\right )} + 8 \, a b^{2} e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, b^{3} e^{\left (-4 \, d x - 4 \, c\right )} + 24 \, {\left (4 \, a^{3} + a b^{2}\right )} e^{\left (-d x - c\right )}}{192 \, b^{4} d} \] Input:
integrate(cosh(d*x+c)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxi ma")
Output:
-sqrt(a^2 + b^2)*a^3*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(b^5*d) - 1/192*(8*a*b^2*e^(-d*x - c) - 24*a ^2*b*e^(-2*d*x - 2*c) - 3*b^3 + 24*(4*a^3 + a*b^2)*e^(-3*d*x - 3*c))*e^(4* d*x + 4*c)/(b^4*d) + 1/8*(8*a^4 + 4*a^2*b^2 - b^4)*(d*x + c)/(b^5*d) - 1/1 92*(24*a^2*b*e^(-2*d*x - 2*c) + 8*a*b^2*e^(-3*d*x - 3*c) + 3*b^3*e^(-4*d*x - 4*c) + 24*(4*a^3 + a*b^2)*e^(-d*x - c))/(b^4*d)
Time = 0.17 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.40 \[ \int \frac {\cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {24 \, {\left (8 \, a^{4} + 4 \, a^{2} b^{2} - b^{4}\right )} {\left (d x + c\right )}}{b^{5}} + \frac {3 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 24 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 96 \, a^{3} e^{\left (d x + c\right )} - 24 \, a b^{2} e^{\left (d x + c\right )}}{b^{4}} - \frac {{\left (24 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a b^{3} e^{\left (d x + c\right )} + 3 \, b^{4} + 24 \, {\left (4 \, a^{3} b + a b^{3}\right )} e^{\left (3 \, d x + 3 \, c\right )}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{b^{5}} - \frac {192 \, {\left (a^{5} + a^{3} b^{2}\right )} \log \left (\frac {{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b^{5}}}{192 \, d} \] Input:
integrate(cosh(d*x+c)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac ")
Output:
1/192*(24*(8*a^4 + 4*a^2*b^2 - b^4)*(d*x + c)/b^5 + (3*b^3*e^(4*d*x + 4*c) - 8*a*b^2*e^(3*d*x + 3*c) + 24*a^2*b*e^(2*d*x + 2*c) - 96*a^3*e^(d*x + c) - 24*a*b^2*e^(d*x + c))/b^4 - (24*a^2*b^2*e^(2*d*x + 2*c) + 8*a*b^3*e^(d* x + c) + 3*b^4 + 24*(4*a^3*b + a*b^3)*e^(3*d*x + 3*c))*e^(-4*d*x - 4*c)/b^ 5 - 192*(a^5 + a^3*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^5))/d
Time = 1.72 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.79 \[ \int \frac {\cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {x\,\left (8\,a^4+4\,a^2\,b^2-b^4\right )}{8\,b^5}-\frac {{\mathrm {e}}^{-4\,c-4\,d\,x}}{64\,b\,d}+\frac {{\mathrm {e}}^{4\,c+4\,d\,x}}{64\,b\,d}-\frac {a\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,b^2\,d}-\frac {a\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,b^2\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (4\,a^3+a\,b^2\right )}{8\,b^4\,d}-\frac {a^2\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,b^3\,d}+\frac {a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,b^3\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (4\,a^3+a\,b^2\right )}{8\,b^4\,d}-\frac {a^3\,\ln \left (\frac {2\,a^3\,{\mathrm {e}}^{c+d\,x}\,\left (a^2+b^2\right )}{b^6}-\frac {2\,a^3\,\sqrt {a^2+b^2}\,\left (b-a\,{\mathrm {e}}^{c+d\,x}\right )}{b^6}\right )\,\sqrt {a^2+b^2}}{b^5\,d}+\frac {a^3\,\ln \left (\frac {2\,a^3\,\sqrt {a^2+b^2}\,\left (b-a\,{\mathrm {e}}^{c+d\,x}\right )}{b^6}+\frac {2\,a^3\,{\mathrm {e}}^{c+d\,x}\,\left (a^2+b^2\right )}{b^6}\right )\,\sqrt {a^2+b^2}}{b^5\,d} \] Input:
int((cosh(c + d*x)^2*sinh(c + d*x)^3)/(a + b*sinh(c + d*x)),x)
Output:
(x*(8*a^4 - b^4 + 4*a^2*b^2))/(8*b^5) - exp(- 4*c - 4*d*x)/(64*b*d) + exp( 4*c + 4*d*x)/(64*b*d) - (a*exp(- 3*c - 3*d*x))/(24*b^2*d) - (a*exp(3*c + 3 *d*x))/(24*b^2*d) - (exp(c + d*x)*(a*b^2 + 4*a^3))/(8*b^4*d) - (a^2*exp(- 2*c - 2*d*x))/(8*b^3*d) + (a^2*exp(2*c + 2*d*x))/(8*b^3*d) - (exp(- c - d* x)*(a*b^2 + 4*a^3))/(8*b^4*d) - (a^3*log((2*a^3*exp(c + d*x)*(a^2 + b^2))/ b^6 - (2*a^3*(a^2 + b^2)^(1/2)*(b - a*exp(c + d*x)))/b^6)*(a^2 + b^2)^(1/2 ))/(b^5*d) + (a^3*log((2*a^3*(a^2 + b^2)^(1/2)*(b - a*exp(c + d*x)))/b^6 + (2*a^3*exp(c + d*x)*(a^2 + b^2))/b^6)*(a^2 + b^2)^(1/2))/(b^5*d)
Time = 0.28 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.49 \[ \int \frac {\cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-384 e^{4 d x +4 c} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{d x +c} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{3} i +3 e^{8 d x +8 c} b^{4}-8 e^{7 d x +7 c} a \,b^{3}+24 e^{6 d x +6 c} a^{2} b^{2}-96 e^{5 d x +5 c} a^{3} b -24 e^{5 d x +5 c} a \,b^{3}+192 e^{4 d x +4 c} a^{4} d x +96 e^{4 d x +4 c} a^{2} b^{2} d x -24 e^{4 d x +4 c} b^{4} d x -96 e^{3 d x +3 c} a^{3} b -24 e^{3 d x +3 c} a \,b^{3}-24 e^{2 d x +2 c} a^{2} b^{2}-8 e^{d x +c} a \,b^{3}-3 b^{4}}{192 e^{4 d x +4 c} b^{5} d} \] Input:
int(cosh(d*x+c)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Output:
( - 384*e**(4*c + 4*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/s qrt(a**2 + b**2))*a**3*i + 3*e**(8*c + 8*d*x)*b**4 - 8*e**(7*c + 7*d*x)*a* b**3 + 24*e**(6*c + 6*d*x)*a**2*b**2 - 96*e**(5*c + 5*d*x)*a**3*b - 24*e** (5*c + 5*d*x)*a*b**3 + 192*e**(4*c + 4*d*x)*a**4*d*x + 96*e**(4*c + 4*d*x) *a**2*b**2*d*x - 24*e**(4*c + 4*d*x)*b**4*d*x - 96*e**(3*c + 3*d*x)*a**3*b - 24*e**(3*c + 3*d*x)*a*b**3 - 24*e**(2*c + 2*d*x)*a**2*b**2 - 8*e**(c + d*x)*a*b**3 - 3*b**4)/(192*e**(4*c + 4*d*x)*b**5*d)