Integrand size = 31, antiderivative size = 231 \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {3 i f^3 x}{8 a d^3}-\frac {i (e+f x)^3}{4 a d}-\frac {6 f^3 \cosh (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{a d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{a d}+\frac {3 i f^3 \cosh (c+d x) \sinh (c+d x)}{8 a d^4}+\frac {3 i f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^2}-\frac {3 i f^2 (e+f x) \sinh ^2(c+d x)}{4 a d^3}-\frac {i (e+f x)^3 \sinh ^2(c+d x)}{2 a d} \] Output:
-3/8*I*f^3*x/a/d^3-1/4*I*(f*x+e)^3/a/d-6*f^3*cosh(d*x+c)/a/d^4-3*f*(f*x+e) ^2*cosh(d*x+c)/a/d^2+6*f^2*(f*x+e)*sinh(d*x+c)/a/d^3+(f*x+e)^3*sinh(d*x+c) /a/d+3/8*I*f^3*cosh(d*x+c)*sinh(d*x+c)/a/d^4+3/4*I*f*(f*x+e)^2*cosh(d*x+c) *sinh(d*x+c)/a/d^2-3/4*I*f^2*(f*x+e)*sinh(d*x+c)^2/a/d^3-1/2*I*(f*x+e)^3*s inh(d*x+c)^2/a/d
Time = 1.67 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.58 \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {-96 f \left (2 f^2+d^2 (e+f x)^2\right ) \cosh (c+d x)-4 i d (e+f x) \left (3 f^2+2 d^2 (e+f x)^2\right ) \cosh (2 (c+d x))+4 \left (8 d (e+f x) \left (6 f^2+d^2 (e+f x)^2\right )+3 i f \left (f^2+2 d^2 (e+f x)^2\right ) \cosh (c+d x)\right ) \sinh (c+d x)}{32 a d^4} \] Input:
Integrate[((e + f*x)^3*Cosh[c + d*x]^3)/(a + I*a*Sinh[c + d*x]),x]
Output:
(-96*f*(2*f^2 + d^2*(e + f*x)^2)*Cosh[c + d*x] - (4*I)*d*(e + f*x)*(3*f^2 + 2*d^2*(e + f*x)^2)*Cosh[2*(c + d*x)] + 4*(8*d*(e + f*x)*(6*f^2 + d^2*(e + f*x)^2) + (3*I)*f*(f^2 + 2*d^2*(e + f*x)^2)*Cosh[c + d*x])*Sinh[c + d*x] )/(32*a*d^4)
Time = 1.21 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.97, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.742, Rules used = {6097, 3042, 3777, 26, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3118, 5969, 3042, 25, 3792, 17, 25, 3042, 25, 3115, 24}
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.
| \(\displaystyle \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6097 |
| \(\displaystyle \frac {\int (e+f x)^3 \cosh (c+d x)dx}{a}-\frac {i \int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (e+f x)^3 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{a}-\frac {i \int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {3 i f \int -i (e+f x)^2 \sinh (c+d x)dx}{d}}{a}-\frac {i \int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {3 f \int (e+f x)^2 \sinh (c+d x)dx}{d}}{a}-\frac {i \int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {3 f \int -i (e+f x)^2 \sin (i c+i d x)dx}{d}}{a}-\frac {i \int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \int (e+f x)^2 \sin (i c+i d x)dx}{d}}{a}-\frac {i \int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \cosh (c+d x)dx}{d}\right )}{d}}{a}-\frac {i \int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}}{a}-\frac {i \int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )}{d}\right )}{d}}{a}-\frac {i \int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )}{d}\right )}{d}}{a}-\frac {i \int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )}{d}\right )}{d}}{a}-\frac {i \int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )}{d}\right )}{d}}{a}-\frac {i \int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}-\frac {i \int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 5969 |
| \(\displaystyle \frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}-\frac {i \left (\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}-\frac {3 f \int (e+f x)^2 \sinh ^2(c+d x)dx}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}-\frac {i \left (\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}-\frac {3 f \int -(e+f x)^2 \sin (i c+i d x)^2dx}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}-\frac {i \left (\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}+\frac {3 f \int (e+f x)^2 \sin (i c+i d x)^2dx}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}-\frac {i \left (\frac {3 f \left (\frac {f^2 \int -\sinh ^2(c+d x)dx}{2 d^2}+\frac {1}{2} \int (e+f x)^2dx+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}-\frac {i \left (\frac {3 f \left (\frac {f^2 \int -\sinh ^2(c+d x)dx}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}-\frac {i \left (\frac {3 f \left (-\frac {f^2 \int \sinh ^2(c+d x)dx}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}-\frac {i \left (\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}+\frac {3 f \left (-\frac {f^2 \int -\sin (i c+i d x)^2dx}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}-\frac {i \left (\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}+\frac {3 f \left (\frac {f^2 \int \sin (i c+i d x)^2dx}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}-\frac {i \left (\frac {3 f \left (\frac {f^2 \left (\frac {\int 1dx}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{a}-\frac {i \left (\frac {3 f \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}\right )}{a}\) |
Input:
Int[((e + f*x)^3*Cosh[c + d*x]^3)/(a + I*a*Sinh[c + d*x]),x]
Output:
(((e + f*x)^3*Sinh[c + d*x])/d + ((3*I)*f*((I*(e + f*x)^2*Cosh[c + d*x])/d - ((2*I)*f*(-((f*Cosh[c + d*x])/d^2) + ((e + f*x)*Sinh[c + d*x])/d))/d))/ d)/a - (I*(((e + f*x)^3*Sinh[c + d*x]^2)/(2*d) + (3*f*((e + f*x)^3/(6*f) - ((e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(2*d) + (f*(e + f*x)*Sinh[c + d *x]^2)/(2*d^2) + (f^2*(x/2 - (Cosh[c + d*x]*Sinh[c + d*x])/(2*d)))/(2*d^2) ))/(2*d)))/a
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
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[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
Int[Cosh[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)* (x_)]^(n_.), x_Symbol] :> Simp[(c + d*x)^m*(Sinh[a + b*x]^(n + 1)/(b*(n + 1 ))), x] - Simp[d*(m/(b*(n + 1))) Int[(c + d*x)^(m - 1)*Sinh[a + b*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_. )*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/a Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x], x] + Simp[1/b Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2)* Sinh[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 1] && E qQ[a^2 + b^2, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 428 vs. \(2 (213 ) = 426\).
Time = 31.39 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.86
| method | result | size |
| risch | \(-\frac {i \left (4 d^{3} x^{3} f^{3}+12 d^{3} e \,f^{2} x^{2}+12 d^{3} e^{2} f x -6 d^{2} f^{3} x^{2}+4 d^{3} e^{3}-12 d^{2} e \,f^{2} x -6 d^{2} e^{2} f +6 d \,f^{3} x +6 d e \,f^{2}-3 f^{3}\right ) {\mathrm e}^{2 d x +2 c}}{32 d^{4} a}+\frac {\left (d^{3} x^{3} f^{3}+3 d^{3} e \,f^{2} x^{2}+3 d^{3} e^{2} f x -3 d^{2} f^{3} x^{2}+d^{3} e^{3}-6 d^{2} e \,f^{2} x -3 d^{2} e^{2} f +6 d \,f^{3} x +6 d e \,f^{2}-6 f^{3}\right ) {\mathrm e}^{d x +c}}{2 d^{4} a}-\frac {\left (d^{3} x^{3} f^{3}+3 d^{3} e \,f^{2} x^{2}+3 d^{3} e^{2} f x +3 d^{2} f^{3} x^{2}+d^{3} e^{3}+6 d^{2} e \,f^{2} x +3 d^{2} e^{2} f +6 d \,f^{3} x +6 d e \,f^{2}+6 f^{3}\right ) {\mathrm e}^{-d x -c}}{2 d^{4} a}-\frac {i \left (4 d^{3} x^{3} f^{3}+12 d^{3} e \,f^{2} x^{2}+12 d^{3} e^{2} f x +6 d^{2} f^{3} x^{2}+4 d^{3} e^{3}+12 d^{2} e \,f^{2} x +6 d^{2} e^{2} f +6 d \,f^{3} x +6 d e \,f^{2}+3 f^{3}\right ) {\mathrm e}^{-2 d x -2 c}}{32 d^{4} a}\) | \(429\) |
| derivativedivides | \(-\frac {3 i d^{2} e^{2} f \left (\frac {\left (d x +c \right ) \cosh \left (d x +c \right )^{2}}{2}-\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{4}-\frac {d x}{4}-\frac {c}{4}\right )-6 i c d e \,f^{2} \left (\frac {\left (d x +c \right ) \cosh \left (d x +c \right )^{2}}{2}-\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{4}-\frac {d x}{4}-\frac {c}{4}\right )+3 i c^{2} f^{3} \left (\frac {\left (d x +c \right ) \cosh \left (d x +c \right )^{2}}{2}-\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{4}-\frac {d x}{4}-\frac {c}{4}\right )-3 i c \,f^{3} \left (\frac {\left (d x +c \right )^{2} \cosh \left (d x +c \right )^{2}}{2}-\frac {\left (d x +c \right ) \cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {\left (d x +c \right )^{2}}{4}+\frac {\cosh \left (d x +c \right )^{2}}{4}\right )+i f^{3} \left (\frac {\left (d x +c \right )^{3} \cosh \left (d x +c \right )^{2}}{2}-\frac {3 \left (d x +c \right )^{2} \cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{4}-\frac {\left (d x +c \right )^{3}}{4}+\frac {3 \left (d x +c \right ) \cosh \left (d x +c \right )^{2}}{4}-\frac {3 \sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{8}-\frac {3 d x}{8}-\frac {3 c}{8}\right )+\frac {3 i c^{2} d e \,f^{2} \cosh \left (d x +c \right )^{2}}{2}-\frac {3 i c \,d^{2} e^{2} f \cosh \left (d x +c \right )^{2}}{2}-\frac {i c^{3} f^{3} \cosh \left (d x +c \right )^{2}}{2}+\frac {i d^{3} e^{3} \cosh \left (d x +c \right )^{2}}{2}+3 i d e \,f^{2} \left (\frac {\left (d x +c \right )^{2} \cosh \left (d x +c \right )^{2}}{2}-\frac {\left (d x +c \right ) \cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {\left (d x +c \right )^{2}}{4}+\frac {\cosh \left (d x +c \right )^{2}}{4}\right )+\sinh \left (d x +c \right ) c^{3} f^{3}-3 \sinh \left (d x +c \right ) c^{2} d e \,f^{2}-3 c^{2} f^{3} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )+3 \sinh \left (d x +c \right ) c \,d^{2} e^{2} f +6 c d e \,f^{2} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )+3 c \,f^{3} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )-\sinh \left (d x +c \right ) d^{3} e^{3}-3 d^{2} e^{2} f \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )-3 d e \,f^{2} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )-f^{3} \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{4} a}\) | \(726\) |
| default | \(-\frac {3 i d^{2} e^{2} f \left (\frac {\left (d x +c \right ) \cosh \left (d x +c \right )^{2}}{2}-\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{4}-\frac {d x}{4}-\frac {c}{4}\right )-6 i c d e \,f^{2} \left (\frac {\left (d x +c \right ) \cosh \left (d x +c \right )^{2}}{2}-\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{4}-\frac {d x}{4}-\frac {c}{4}\right )+3 i c^{2} f^{3} \left (\frac {\left (d x +c \right ) \cosh \left (d x +c \right )^{2}}{2}-\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{4}-\frac {d x}{4}-\frac {c}{4}\right )-3 i c \,f^{3} \left (\frac {\left (d x +c \right )^{2} \cosh \left (d x +c \right )^{2}}{2}-\frac {\left (d x +c \right ) \cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {\left (d x +c \right )^{2}}{4}+\frac {\cosh \left (d x +c \right )^{2}}{4}\right )+i f^{3} \left (\frac {\left (d x +c \right )^{3} \cosh \left (d x +c \right )^{2}}{2}-\frac {3 \left (d x +c \right )^{2} \cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{4}-\frac {\left (d x +c \right )^{3}}{4}+\frac {3 \left (d x +c \right ) \cosh \left (d x +c \right )^{2}}{4}-\frac {3 \sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{8}-\frac {3 d x}{8}-\frac {3 c}{8}\right )+\frac {3 i c^{2} d e \,f^{2} \cosh \left (d x +c \right )^{2}}{2}-\frac {3 i c \,d^{2} e^{2} f \cosh \left (d x +c \right )^{2}}{2}-\frac {i c^{3} f^{3} \cosh \left (d x +c \right )^{2}}{2}+\frac {i d^{3} e^{3} \cosh \left (d x +c \right )^{2}}{2}+3 i d e \,f^{2} \left (\frac {\left (d x +c \right )^{2} \cosh \left (d x +c \right )^{2}}{2}-\frac {\left (d x +c \right ) \cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {\left (d x +c \right )^{2}}{4}+\frac {\cosh \left (d x +c \right )^{2}}{4}\right )+\sinh \left (d x +c \right ) c^{3} f^{3}-3 \sinh \left (d x +c \right ) c^{2} d e \,f^{2}-3 c^{2} f^{3} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )+3 \sinh \left (d x +c \right ) c \,d^{2} e^{2} f +6 c d e \,f^{2} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )+3 c \,f^{3} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )-\sinh \left (d x +c \right ) d^{3} e^{3}-3 d^{2} e^{2} f \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )-3 d e \,f^{2} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )-f^{3} \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{4} a}\) | \(726\) |
Input:
int((f*x+e)^3*cosh(d*x+c)^3/(a+I*a*sinh(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-1/32*I*(4*d^3*f^3*x^3+12*d^3*e*f^2*x^2+12*d^3*e^2*f*x-6*d^2*f^3*x^2+4*d^3 *e^3-12*d^2*e*f^2*x-6*d^2*e^2*f+6*d*f^3*x+6*d*e*f^2-3*f^3)/d^4/a*exp(2*d*x +2*c)+1/2*(d^3*f^3*x^3+3*d^3*e*f^2*x^2+3*d^3*e^2*f*x-3*d^2*f^3*x^2+d^3*e^3 -6*d^2*e*f^2*x-3*d^2*e^2*f+6*d*f^3*x+6*d*e*f^2-6*f^3)/d^4/a*exp(d*x+c)-1/2 *(d^3*f^3*x^3+3*d^3*e*f^2*x^2+3*d^3*e^2*f*x+3*d^2*f^3*x^2+d^3*e^3+6*d^2*e* f^2*x+3*d^2*e^2*f+6*d*f^3*x+6*d*e*f^2+6*f^3)/d^4/a*exp(-d*x-c)-1/32*I*(4*d ^3*f^3*x^3+12*d^3*e*f^2*x^2+12*d^3*e^2*f*x+6*d^2*f^3*x^2+4*d^3*e^3+12*d^2* e*f^2*x+6*d^2*e^2*f+6*d*f^3*x+6*d*e*f^2+3*f^3)/a/d^4*exp(-2*d*x-2*c)
Time = 0.10 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.75 \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {{\left (-4 i \, d^{3} f^{3} x^{3} - 4 i \, d^{3} e^{3} - 6 i \, d^{2} e^{2} f - 6 i \, d e f^{2} - 3 i \, f^{3} - 6 \, {\left (2 i \, d^{3} e f^{2} + i \, d^{2} f^{3}\right )} x^{2} - 6 \, {\left (2 i \, d^{3} e^{2} f + 2 i \, d^{2} e f^{2} + i \, d f^{3}\right )} x + {\left (-4 i \, d^{3} f^{3} x^{3} - 4 i \, d^{3} e^{3} + 6 i \, d^{2} e^{2} f - 6 i \, d e f^{2} + 3 i \, f^{3} - 6 \, {\left (2 i \, d^{3} e f^{2} - i \, d^{2} f^{3}\right )} x^{2} - 6 \, {\left (2 i \, d^{3} e^{2} f - 2 i \, d^{2} e f^{2} + i \, d f^{3}\right )} x\right )} e^{\left (4 \, d x + 4 \, c\right )} + 16 \, {\left (d^{3} f^{3} x^{3} + d^{3} e^{3} - 3 \, d^{2} e^{2} f + 6 \, d e f^{2} - 6 \, f^{3} + 3 \, {\left (d^{3} e f^{2} - d^{2} f^{3}\right )} x^{2} + 3 \, {\left (d^{3} e^{2} f - 2 \, d^{2} e f^{2} + 2 \, d f^{3}\right )} x\right )} e^{\left (3 \, d x + 3 \, c\right )} - 16 \, {\left (d^{3} f^{3} x^{3} + d^{3} e^{3} + 3 \, d^{2} e^{2} f + 6 \, d e f^{2} + 6 \, f^{3} + 3 \, {\left (d^{3} e f^{2} + d^{2} f^{3}\right )} x^{2} + 3 \, {\left (d^{3} e^{2} f + 2 \, d^{2} e f^{2} + 2 \, d f^{3}\right )} x\right )} e^{\left (d x + c\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{32 \, a d^{4}} \] Input:
integrate((f*x+e)^3*cosh(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="fricas ")
Output:
1/32*(-4*I*d^3*f^3*x^3 - 4*I*d^3*e^3 - 6*I*d^2*e^2*f - 6*I*d*e*f^2 - 3*I*f ^3 - 6*(2*I*d^3*e*f^2 + I*d^2*f^3)*x^2 - 6*(2*I*d^3*e^2*f + 2*I*d^2*e*f^2 + I*d*f^3)*x + (-4*I*d^3*f^3*x^3 - 4*I*d^3*e^3 + 6*I*d^2*e^2*f - 6*I*d*e*f ^2 + 3*I*f^3 - 6*(2*I*d^3*e*f^2 - I*d^2*f^3)*x^2 - 6*(2*I*d^3*e^2*f - 2*I* d^2*e*f^2 + I*d*f^3)*x)*e^(4*d*x + 4*c) + 16*(d^3*f^3*x^3 + d^3*e^3 - 3*d^ 2*e^2*f + 6*d*e*f^2 - 6*f^3 + 3*(d^3*e*f^2 - d^2*f^3)*x^2 + 3*(d^3*e^2*f - 2*d^2*e*f^2 + 2*d*f^3)*x)*e^(3*d*x + 3*c) - 16*(d^3*f^3*x^3 + d^3*e^3 + 3 *d^2*e^2*f + 6*d*e*f^2 + 6*f^3 + 3*(d^3*e*f^2 + d^2*f^3)*x^2 + 3*(d^3*e^2* f + 2*d^2*e*f^2 + 2*d*f^3)*x)*e^(d*x + c))*e^(-2*d*x - 2*c)/(a*d^4)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1040 vs. \(2 (214) = 428\).
Time = 0.56 (sec) , antiderivative size = 1040, normalized size of antiderivative = 4.50 \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:
integrate((f*x+e)**3*cosh(d*x+c)**3/(a+I*a*sinh(d*x+c)),x)
Output:
Piecewise((((-2048*a**3*d**15*e**3*exp(2*c) - 6144*a**3*d**15*e**2*f*x*exp (2*c) - 6144*a**3*d**15*e*f**2*x**2*exp(2*c) - 2048*a**3*d**15*f**3*x**3*e xp(2*c) - 6144*a**3*d**14*e**2*f*exp(2*c) - 12288*a**3*d**14*e*f**2*x*exp( 2*c) - 6144*a**3*d**14*f**3*x**2*exp(2*c) - 12288*a**3*d**13*e*f**2*exp(2* c) - 12288*a**3*d**13*f**3*x*exp(2*c) - 12288*a**3*d**12*f**3*exp(2*c))*ex p(-d*x) + (2048*a**3*d**15*e**3*exp(4*c) + 6144*a**3*d**15*e**2*f*x*exp(4* c) + 6144*a**3*d**15*e*f**2*x**2*exp(4*c) + 2048*a**3*d**15*f**3*x**3*exp( 4*c) - 6144*a**3*d**14*e**2*f*exp(4*c) - 12288*a**3*d**14*e*f**2*x*exp(4*c ) - 6144*a**3*d**14*f**3*x**2*exp(4*c) + 12288*a**3*d**13*e*f**2*exp(4*c) + 12288*a**3*d**13*f**3*x*exp(4*c) - 12288*a**3*d**12*f**3*exp(4*c))*exp(d *x) + (-512*I*a**3*d**15*e**3*exp(c) - 1536*I*a**3*d**15*e**2*f*x*exp(c) - 1536*I*a**3*d**15*e*f**2*x**2*exp(c) - 512*I*a**3*d**15*f**3*x**3*exp(c) - 768*I*a**3*d**14*e**2*f*exp(c) - 1536*I*a**3*d**14*e*f**2*x*exp(c) - 768 *I*a**3*d**14*f**3*x**2*exp(c) - 768*I*a**3*d**13*e*f**2*exp(c) - 768*I*a* *3*d**13*f**3*x*exp(c) - 384*I*a**3*d**12*f**3*exp(c))*exp(-2*d*x) + (-512 *I*a**3*d**15*e**3*exp(5*c) - 1536*I*a**3*d**15*e**2*f*x*exp(5*c) - 1536*I *a**3*d**15*e*f**2*x**2*exp(5*c) - 512*I*a**3*d**15*f**3*x**3*exp(5*c) + 7 68*I*a**3*d**14*e**2*f*exp(5*c) + 1536*I*a**3*d**14*e*f**2*x*exp(5*c) + 76 8*I*a**3*d**14*f**3*x**2*exp(5*c) - 768*I*a**3*d**13*e*f**2*exp(5*c) - 768 *I*a**3*d**13*f**3*x*exp(5*c) + 384*I*a**3*d**12*f**3*exp(5*c))*exp(2*d...
Exception generated. \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((f*x+e)^3*cosh(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="maxima ")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 618 vs. \(2 (207) = 414\).
Time = 0.14 (sec) , antiderivative size = 618, normalized size of antiderivative = 2.68 \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {{\left (4 i \, d^{3} f^{3} x^{3} e^{\left (4 \, d x + 4 \, c\right )} - 16 \, d^{3} f^{3} x^{3} e^{\left (3 \, d x + 3 \, c\right )} + 16 \, d^{3} f^{3} x^{3} e^{\left (d x + c\right )} + 4 i \, d^{3} f^{3} x^{3} + 12 i \, d^{3} e f^{2} x^{2} e^{\left (4 \, d x + 4 \, c\right )} - 48 \, d^{3} e f^{2} x^{2} e^{\left (3 \, d x + 3 \, c\right )} + 48 \, d^{3} e f^{2} x^{2} e^{\left (d x + c\right )} + 12 i \, d^{3} e f^{2} x^{2} + 12 i \, d^{3} e^{2} f x e^{\left (4 \, d x + 4 \, c\right )} - 6 i \, d^{2} f^{3} x^{2} e^{\left (4 \, d x + 4 \, c\right )} - 48 \, d^{3} e^{2} f x e^{\left (3 \, d x + 3 \, c\right )} + 48 \, d^{2} f^{3} x^{2} e^{\left (3 \, d x + 3 \, c\right )} + 48 \, d^{3} e^{2} f x e^{\left (d x + c\right )} + 48 \, d^{2} f^{3} x^{2} e^{\left (d x + c\right )} + 12 i \, d^{3} e^{2} f x + 6 i \, d^{2} f^{3} x^{2} + 4 i \, d^{3} e^{3} e^{\left (4 \, d x + 4 \, c\right )} - 12 i \, d^{2} e f^{2} x e^{\left (4 \, d x + 4 \, c\right )} - 16 \, d^{3} e^{3} e^{\left (3 \, d x + 3 \, c\right )} + 96 \, d^{2} e f^{2} x e^{\left (3 \, d x + 3 \, c\right )} + 16 \, d^{3} e^{3} e^{\left (d x + c\right )} + 96 \, d^{2} e f^{2} x e^{\left (d x + c\right )} + 4 i \, d^{3} e^{3} + 12 i \, d^{2} e f^{2} x - 6 i \, d^{2} e^{2} f e^{\left (4 \, d x + 4 \, c\right )} + 6 i \, d f^{3} x e^{\left (4 \, d x + 4 \, c\right )} + 48 \, d^{2} e^{2} f e^{\left (3 \, d x + 3 \, c\right )} - 96 \, d f^{3} x e^{\left (3 \, d x + 3 \, c\right )} + 48 \, d^{2} e^{2} f e^{\left (d x + c\right )} + 96 \, d f^{3} x e^{\left (d x + c\right )} + 6 i \, d^{2} e^{2} f + 6 i \, d f^{3} x + 6 i \, d e f^{2} e^{\left (4 \, d x + 4 \, c\right )} - 96 \, d e f^{2} e^{\left (3 \, d x + 3 \, c\right )} + 96 \, d e f^{2} e^{\left (d x + c\right )} + 6 i \, d e f^{2} - 3 i \, f^{3} e^{\left (4 \, d x + 4 \, c\right )} + 96 \, f^{3} e^{\left (3 \, d x + 3 \, c\right )} + 96 \, f^{3} e^{\left (d x + c\right )} + 3 i \, f^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{32 \, a d^{4}} \] Input:
integrate((f*x+e)^3*cosh(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="giac")
Output:
-1/32*(4*I*d^3*f^3*x^3*e^(4*d*x + 4*c) - 16*d^3*f^3*x^3*e^(3*d*x + 3*c) + 16*d^3*f^3*x^3*e^(d*x + c) + 4*I*d^3*f^3*x^3 + 12*I*d^3*e*f^2*x^2*e^(4*d*x + 4*c) - 48*d^3*e*f^2*x^2*e^(3*d*x + 3*c) + 48*d^3*e*f^2*x^2*e^(d*x + c) + 12*I*d^3*e*f^2*x^2 + 12*I*d^3*e^2*f*x*e^(4*d*x + 4*c) - 6*I*d^2*f^3*x^2* e^(4*d*x + 4*c) - 48*d^3*e^2*f*x*e^(3*d*x + 3*c) + 48*d^2*f^3*x^2*e^(3*d*x + 3*c) + 48*d^3*e^2*f*x*e^(d*x + c) + 48*d^2*f^3*x^2*e^(d*x + c) + 12*I*d ^3*e^2*f*x + 6*I*d^2*f^3*x^2 + 4*I*d^3*e^3*e^(4*d*x + 4*c) - 12*I*d^2*e*f^ 2*x*e^(4*d*x + 4*c) - 16*d^3*e^3*e^(3*d*x + 3*c) + 96*d^2*e*f^2*x*e^(3*d*x + 3*c) + 16*d^3*e^3*e^(d*x + c) + 96*d^2*e*f^2*x*e^(d*x + c) + 4*I*d^3*e^ 3 + 12*I*d^2*e*f^2*x - 6*I*d^2*e^2*f*e^(4*d*x + 4*c) + 6*I*d*f^3*x*e^(4*d* x + 4*c) + 48*d^2*e^2*f*e^(3*d*x + 3*c) - 96*d*f^3*x*e^(3*d*x + 3*c) + 48* d^2*e^2*f*e^(d*x + c) + 96*d*f^3*x*e^(d*x + c) + 6*I*d^2*e^2*f + 6*I*d*f^3 *x + 6*I*d*e*f^2*e^(4*d*x + 4*c) - 96*d*e*f^2*e^(3*d*x + 3*c) + 96*d*e*f^2 *e^(d*x + c) + 6*I*d*e*f^2 - 3*I*f^3*e^(4*d*x + 4*c) + 96*f^3*e^(3*d*x + 3 *c) + 96*f^3*e^(d*x + c) + 3*I*f^3)*e^(-2*d*x - 2*c)/(a*d^4)
Time = 1.96 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.94 \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-{\mathrm {e}}^{c+d\,x}\,\left (\frac {-d^3\,e^3+3\,d^2\,e^2\,f-6\,d\,e\,f^2+6\,f^3}{2\,a\,d^4}-\frac {f^3\,x^3}{2\,a\,d}+\frac {3\,f^2\,x^2\,\left (f-d\,e\right )}{2\,a\,d^2}-\frac {3\,f\,x\,\left (d^2\,e^2-2\,d\,e\,f+2\,f^2\right )}{2\,a\,d^3}\right )-{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (\frac {\left (4\,d^3\,e^3+6\,d^2\,e^2\,f+6\,d\,e\,f^2+3\,f^3\right )\,1{}\mathrm {i}}{32\,a\,d^4}+\frac {f^3\,x^3\,1{}\mathrm {i}}{8\,a\,d}+\frac {f\,x\,\left (2\,d^2\,e^2+2\,d\,e\,f+f^2\right )\,3{}\mathrm {i}}{16\,a\,d^3}+\frac {f^2\,x^2\,\left (f+2\,d\,e\right )\,3{}\mathrm {i}}{16\,a\,d^2}\right )+{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (\frac {\left (-4\,d^3\,e^3+6\,d^2\,e^2\,f-6\,d\,e\,f^2+3\,f^3\right )\,1{}\mathrm {i}}{32\,a\,d^4}-\frac {f^3\,x^3\,1{}\mathrm {i}}{8\,a\,d}-\frac {f\,x\,\left (2\,d^2\,e^2-2\,d\,e\,f+f^2\right )\,3{}\mathrm {i}}{16\,a\,d^3}+\frac {f^2\,x^2\,\left (f-2\,d\,e\right )\,3{}\mathrm {i}}{16\,a\,d^2}\right )-{\mathrm {e}}^{-c-d\,x}\,\left (\frac {d^3\,e^3+3\,d^2\,e^2\,f+6\,d\,e\,f^2+6\,f^3}{2\,a\,d^4}+\frac {f^3\,x^3}{2\,a\,d}+\frac {3\,f^2\,x^2\,\left (f+d\,e\right )}{2\,a\,d^2}+\frac {3\,f\,x\,\left (d^2\,e^2+2\,d\,e\,f+2\,f^2\right )}{2\,a\,d^3}\right ) \] Input:
int((cosh(c + d*x)^3*(e + f*x)^3)/(a + a*sinh(c + d*x)*1i),x)
Output:
exp(2*c + 2*d*x)*(((3*f^3 - 4*d^3*e^3 + 6*d^2*e^2*f - 6*d*e*f^2)*1i)/(32*a *d^4) - (f^3*x^3*1i)/(8*a*d) - (f*x*(f^2 + 2*d^2*e^2 - 2*d*e*f)*3i)/(16*a* d^3) + (f^2*x^2*(f - 2*d*e)*3i)/(16*a*d^2)) - exp(- 2*c - 2*d*x)*(((3*f^3 + 4*d^3*e^3 + 6*d^2*e^2*f + 6*d*e*f^2)*1i)/(32*a*d^4) + (f^3*x^3*1i)/(8*a* d) + (f*x*(f^2 + 2*d^2*e^2 + 2*d*e*f)*3i)/(16*a*d^3) + (f^2*x^2*(f + 2*d*e )*3i)/(16*a*d^2)) - exp(c + d*x)*((6*f^3 - d^3*e^3 + 3*d^2*e^2*f - 6*d*e*f ^2)/(2*a*d^4) - (f^3*x^3)/(2*a*d) + (3*f^2*x^2*(f - d*e))/(2*a*d^2) - (3*f *x*(2*f^2 + d^2*e^2 - 2*d*e*f))/(2*a*d^3)) - exp(- c - d*x)*((6*f^3 + d^3* e^3 + 3*d^2*e^2*f + 6*d*e*f^2)/(2*a*d^4) + (f^3*x^3)/(2*a*d) + (3*f^2*x^2* (f + d*e))/(2*a*d^2) + (3*f*x*(2*f^2 + d^2*e^2 + 2*d*e*f))/(2*a*d^3))
\[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {\left (\int \frac {\cosh \left (d x +c \right )^{3}}{\sinh \left (d x +c \right ) i +1}d x \right ) e^{3}+\left (\int \frac {\cosh \left (d x +c \right )^{3} x^{3}}{\sinh \left (d x +c \right ) i +1}d x \right ) f^{3}+3 \left (\int \frac {\cosh \left (d x +c \right )^{3} x^{2}}{\sinh \left (d x +c \right ) i +1}d x \right ) e \,f^{2}+3 \left (\int \frac {\cosh \left (d x +c \right )^{3} x}{\sinh \left (d x +c \right ) i +1}d x \right ) e^{2} f}{a} \] Input:
int((f*x+e)^3*cosh(d*x+c)^3/(a+I*a*sinh(d*x+c)),x)
Output:
(int(cosh(c + d*x)**3/(sinh(c + d*x)*i + 1),x)*e**3 + int((cosh(c + d*x)** 3*x**3)/(sinh(c + d*x)*i + 1),x)*f**3 + 3*int((cosh(c + d*x)**3*x**2)/(sin h(c + d*x)*i + 1),x)*e*f**2 + 3*int((cosh(c + d*x)**3*x)/(sinh(c + d*x)*i + 1),x)*e**2*f)/a