Integrand size = 29, antiderivative size = 170 \[ \int \frac {(e+f x) \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {3 i (e+f x)^2}{4 a f}+\frac {(e+f x) \cosh (c+d x)}{a d}+\frac {2 i f \log \left (\cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )\right )}{a d^2}-\frac {f \sinh (c+d x)}{a d^2}-\frac {i (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f \sinh ^2(c+d x)}{4 a d^2}-\frac {i (e+f x) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \] Output:
3/4*I*(f*x+e)^2/a/f+(f*x+e)*cosh(d*x+c)/a/d+2*I*f*ln(cosh(1/2*c+1/4*I*Pi+1 /2*d*x))/a/d^2-f*sinh(d*x+c)/a/d^2-1/2*I*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)/a /d+1/4*I*f*sinh(d*x+c)^2/a/d^2-I*(f*x+e)*tanh(1/2*c+1/4*I*Pi+1/2*d*x)/a/d
Time = 3.42 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.91 \[ \int \frac {(e+f x) \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {\left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right ) \left (-8 i d (e+f x) \cosh (c+d x)+f \cosh (2 (c+d x))+2 \left (6 c d e-4 i c f-3 c^2 f+6 d^2 e x-4 i d f x+3 d^2 f x^2+8 i f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+4 f \log (\cosh (c+d x))+4 i f \sinh (c+d x)-d (e+f x) \sinh (2 (c+d x))\right )\right )+\sinh \left (\frac {1}{2} (c+d x)\right ) \left (8 d (e+f x) \cosh (c+d x)+i \left (f \cosh (2 (c+d x))+2 \left (8 i d e+6 c d e-4 i c f-3 c^2 f+6 d^2 e x+4 i d f x+3 d^2 f x^2+8 i f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+4 f \log (\cosh (c+d x))+4 i f \sinh (c+d x)-d (e+f x) \sinh (2 (c+d x))\right )\right )\right )\right )}{8 a d^2 (-i+\sinh (c+d x))} \] Input:
Integrate[((e + f*x)*Sinh[c + d*x]^3)/(a + I*a*Sinh[c + d*x]),x]
Output:
((Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])*(Cosh[(c + d*x)/2]*((-8*I)*d*(e + f*x)*Cosh[c + d*x] + f*Cosh[2*(c + d*x)] + 2*(6*c*d*e - (4*I)*c*f - 3*c ^2*f + 6*d^2*e*x - (4*I)*d*f*x + 3*d^2*f*x^2 + (8*I)*f*ArcTan[Tanh[(c + d* x)/2]] + 4*f*Log[Cosh[c + d*x]] + (4*I)*f*Sinh[c + d*x] - d*(e + f*x)*Sinh [2*(c + d*x)])) + Sinh[(c + d*x)/2]*(8*d*(e + f*x)*Cosh[c + d*x] + I*(f*Co sh[2*(c + d*x)] + 2*((8*I)*d*e + 6*c*d*e - (4*I)*c*f - 3*c^2*f + 6*d^2*e*x + (4*I)*d*f*x + 3*d^2*f*x^2 + (8*I)*f*ArcTan[Tanh[(c + d*x)/2]] + 4*f*Log [Cosh[c + d*x]] + (4*I)*f*Sinh[c + d*x] - d*(e + f*x)*Sinh[2*(c + d*x)]))) ))/(8*a*d^2*(-I + Sinh[c + d*x]))
Time = 1.48 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.16, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.793, Rules used = {6091, 3042, 25, 3791, 17, 6091, 3042, 26, 3777, 3042, 3117, 6091, 17, 3042, 3799, 25, 25, 3042, 4672, 26, 3042, 26, 3956}
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.
| \(\displaystyle \int \frac {(e+f x) \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6091 |
| \(\displaystyle i \int \frac {(e+f x) \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx-\frac {i \int (e+f x) \sinh ^2(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \int \frac {(e+f x) \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx-\frac {i \int -\left ((e+f x) \sin (i c+i d x)^2\right )dx}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {i \int (e+f x) \sin (i c+i d x)^2dx}{a}+i \int \frac {(e+f x) \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {i \left (\frac {1}{2} \int (e+f x)dx+\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{a}+i \int \frac {(e+f x) \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle i \int \frac {(e+f x) \sinh ^2(c+d x)}{i \sinh (c+d x) a+a}dx+\frac {i \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 6091 |
| \(\displaystyle i \left (i \int \frac {(e+f x) \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {i \int (e+f x) \sinh (c+d x)dx}{a}\right )+\frac {i \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (i \int \frac {(e+f x) \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {i \int -i (e+f x) \sin (i c+i d x)dx}{a}\right )+\frac {i \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (i \int \frac {(e+f x) \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\int (e+f x) \sin (i c+i d x)dx}{a}\right )+\frac {i \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle i \left (i \int \frac {(e+f x) \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}}{a}\right )+\frac {i \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (i \int \frac {(e+f x) \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}}{a}\right )+\frac {i \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle i \left (i \int \frac {(e+f x) \sinh (c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}}{a}\right )+\frac {i \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 6091 |
| \(\displaystyle i \left (i \left (i \int \frac {e+f x}{i \sinh (c+d x) a+a}dx-\frac {i \int (e+f x)dx}{a}\right )-\frac {\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}}{a}\right )+\frac {i \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle i \left (i \left (i \int \frac {e+f x}{i \sinh (c+d x) a+a}dx-\frac {i (e+f x)^2}{2 a f}\right )-\frac {\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}}{a}\right )+\frac {i \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (i \left (i \int \frac {e+f x}{\sin (i c+i d x) a+a}dx-\frac {i (e+f x)^2}{2 a f}\right )-\frac {\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}}{a}\right )+\frac {i \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 3799 |
| \(\displaystyle i \left (i \left (\frac {i \int -\left ((e+f x) \text {csch}^2\left (\frac {c}{2}+\frac {d x}{2}-\frac {i \pi }{4}\right )\right )dx}{2 a}-\frac {i (e+f x)^2}{2 a f}\right )-\frac {\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}}{a}\right )+\frac {i \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (i \left (-\frac {i \int -\left ((e+f x) \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )dx}{2 a}-\frac {i (e+f x)^2}{2 a f}\right )-\frac {\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}}{a}\right )+\frac {i \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (i \left (\frac {i \int (e+f x) \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{2 a}-\frac {i (e+f x)^2}{2 a f}\right )-\frac {\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}}{a}\right )+\frac {i \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (i \left (\frac {i \int (e+f x) \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}-\frac {i (e+f x)^2}{2 a f}\right )-\frac {\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}}{a}\right )+\frac {i \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle i \left (i \left (\frac {i \left (\frac {2 (e+f x) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {2 i f \int -i \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )}{2 a}-\frac {i (e+f x)^2}{2 a f}\right )-\frac {\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}}{a}\right )+\frac {i \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (i \left (\frac {i \left (\frac {2 (e+f x) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {2 f \int \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )}{2 a}-\frac {i (e+f x)^2}{2 a f}\right )-\frac {\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}}{a}\right )+\frac {i \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (i \left (\frac {i \left (\frac {2 (e+f x) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {2 f \int -i \tan \left (\frac {i c}{2}+\frac {i d x}{2}-\frac {\pi }{4}\right )dx}{d}\right )}{2 a}-\frac {i (e+f x)^2}{2 a f}\right )-\frac {\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}}{a}\right )+\frac {i \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (i \left (\frac {i \left (\frac {2 i f \int \tan \left (\frac {i c}{2}+\frac {i d x}{2}-\frac {\pi }{4}\right )dx}{d}+\frac {2 (e+f x) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}-\frac {i (e+f x)^2}{2 a f}\right )-\frac {\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}}{a}\right )+\frac {i \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {i \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{a}+i \left (i \left (\frac {i \left (\frac {2 (e+f x) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 f \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{d^2}\right )}{2 a}-\frac {i (e+f x)^2}{2 a f}\right )-\frac {\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}}{a}\right )\) |
Input:
Int[((e + f*x)*Sinh[c + d*x]^3)/(a + I*a*Sinh[c + d*x]),x]
Output:
(I*((e + f*x)^2/(4*f) - ((e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*d) + (f *Sinh[c + d*x]^2)/(4*d^2)))/a + I*(-(((I*(e + f*x)*Cosh[c + d*x])/d - (I*f *Sinh[c + d*x])/d^2)/a) + I*(((-1/2*I)*(e + f*x)^2)/(a*f) + ((I/2)*((-4*f* Log[Cosh[c/2 + (I/4)*Pi + (d*x)/2]])/d^2 + (2*(e + f*x)*Tanh[c/2 + (I/4)*P i + (d*x)/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[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
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_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x ]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n) Int[(c + d* x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Simp[(2*a)^n Int[(c + d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^ 2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp [(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1) *Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_ .)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/b Int[(e + f*x)^m*Sinh[ c + d*x]^(n - 1), x], x] - Simp[a/b Int[(e + f*x)^m*(Sinh[c + d*x]^(n - 1 )/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
Time = 1.08 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.16
| method | result | size |
| risch | \(\frac {3 i f \,x^{2}}{4 a}+\frac {3 i e x}{2 a}-\frac {i \left (2 d x f +2 d e -f \right ) {\mathrm e}^{2 d x +2 c}}{16 a \,d^{2}}+\frac {\left (d x f +d e -f \right ) {\mathrm e}^{d x +c}}{2 a \,d^{2}}+\frac {\left (d x f +d e +f \right ) {\mathrm e}^{-d x -c}}{2 a \,d^{2}}+\frac {i \left (2 d x f +2 d e +f \right ) {\mathrm e}^{-2 d x -2 c}}{16 a \,d^{2}}-\frac {2 i f x}{a d}-\frac {2 i f c}{a \,d^{2}}+\frac {2 f x +2 e}{d a \left ({\mathrm e}^{d x +c}-i\right )}+\frac {2 i f \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{2}}\) | \(197\) |
| parallelrisch | \(\frac {32 f \left (i \cosh \left (\frac {d x}{2}+\frac {c}{2}\right )-\sinh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \ln \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-32 f \left (i \cosh \left (\frac {d x}{2}+\frac {c}{2}\right )-\sinh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (\left (6 i+12 i x^{2} d^{2}+\left (24-16 i\right ) x d \right ) f +24 i d^{2} e x +56 d e \right ) \cosh \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (\left (-6-12 x^{2} d^{2}+\left (16-24 i\right ) x d \right ) f +8 \left (-3 d x +i\right ) e d \right ) \sinh \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (\left (6 d x -7 i\right ) f +6 d e \right ) \cosh \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+\left (\left (2 d x +i\right ) f +2 d e \right ) \cosh \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+\left (6 i d f x +6 i e d -7 f \right ) \sinh \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-2 \left (\left (i d x +\frac {1}{2}\right ) f +i e d \right ) \sinh \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )}{16 a \,d^{2} \left (\cosh \left (\frac {d x}{2}+\frac {c}{2}\right )+i \sinh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(295\) |
Input:
int((f*x+e)*sinh(d*x+c)^3/(a+I*a*sinh(d*x+c)),x,method=_RETURNVERBOSE)
Output:
3/4*I/a*f*x^2+3/2*I/a*e*x-1/16*I*(2*d*f*x+2*d*e-f)/a/d^2*exp(2*d*x+2*c)+1/ 2*(d*f*x+d*e-f)/a/d^2*exp(d*x+c)+1/2*(d*f*x+d*e+f)/a/d^2*exp(-d*x-c)+1/16* I*(2*d*f*x+2*d*e+f)/a/d^2*exp(-2*d*x-2*c)-2*I*f/a/d*x-2*I*f/a/d^2*c+2*(f*x +e)/d/a/(exp(d*x+c)-I)+2*I*f/a/d^2*ln(exp(d*x+c)-I)
Time = 0.10 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.35 \[ \int \frac {(e+f x) \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {2 \, d f x + 2 \, d e + {\left (-2 i \, d f x - 2 i \, d e + i \, f\right )} e^{\left (5 \, d x + 5 \, c\right )} + {\left (6 \, d f x + 6 \, d e - 7 \, f\right )} e^{\left (4 \, d x + 4 \, c\right )} - 4 \, {\left (-3 i \, d^{2} f x^{2} + 2 i \, d e + 2 \, {\left (-3 i \, d^{2} e + 5 i \, d f\right )} x - 2 i \, f\right )} e^{\left (3 \, d x + 3 \, c\right )} + 4 \, {\left (3 \, d^{2} f x^{2} + 10 \, d e + 2 \, {\left (3 \, d^{2} e + d f\right )} x + 2 \, f\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (-6 i \, d f x - 6 i \, d e - 7 i \, f\right )} e^{\left (d x + c\right )} - 32 \, {\left (-i \, f e^{\left (3 \, d x + 3 \, c\right )} - f e^{\left (2 \, d x + 2 \, c\right )}\right )} \log \left (e^{\left (d x + c\right )} - i\right ) + f}{16 \, {\left (a d^{2} e^{\left (3 \, d x + 3 \, c\right )} - i \, a d^{2} e^{\left (2 \, d x + 2 \, c\right )}\right )}} \] Input:
integrate((f*x+e)*sinh(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")
Output:
1/16*(2*d*f*x + 2*d*e + (-2*I*d*f*x - 2*I*d*e + I*f)*e^(5*d*x + 5*c) + (6* d*f*x + 6*d*e - 7*f)*e^(4*d*x + 4*c) - 4*(-3*I*d^2*f*x^2 + 2*I*d*e + 2*(-3 *I*d^2*e + 5*I*d*f)*x - 2*I*f)*e^(3*d*x + 3*c) + 4*(3*d^2*f*x^2 + 10*d*e + 2*(3*d^2*e + d*f)*x + 2*f)*e^(2*d*x + 2*c) + (-6*I*d*f*x - 6*I*d*e - 7*I* f)*e^(d*x + c) - 32*(-I*f*e^(3*d*x + 3*c) - f*e^(2*d*x + 2*c))*log(e^(d*x + c) - I) + f)/(a*d^2*e^(3*d*x + 3*c) - I*a*d^2*e^(2*d*x + 2*c))
Time = 0.43 (sec) , antiderivative size = 396, normalized size of antiderivative = 2.33 \[ \int \frac {(e+f x) \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {2 e + 2 f x}{a d e^{c} e^{d x} - i a d} + \begin {cases} \frac {\left (\left (512 a^{3} d^{7} e e^{2 c} + 512 a^{3} d^{7} f x e^{2 c} + 512 a^{3} d^{6} f e^{2 c}\right ) e^{- d x} + \left (512 a^{3} d^{7} e e^{4 c} + 512 a^{3} d^{7} f x e^{4 c} - 512 a^{3} d^{6} f e^{4 c}\right ) e^{d x} + \left (128 i a^{3} d^{7} e e^{c} + 128 i a^{3} d^{7} f x e^{c} + 64 i a^{3} d^{6} f e^{c}\right ) e^{- 2 d x} + \left (- 128 i a^{3} d^{7} e e^{5 c} - 128 i a^{3} d^{7} f x e^{5 c} + 64 i a^{3} d^{6} f e^{5 c}\right ) e^{2 d x}\right ) e^{- 3 c}}{1024 a^{4} d^{8}} & \text {for}\: a^{4} d^{8} e^{3 c} \neq 0 \\\frac {x^{2} \left (- i f e^{4 c} + 2 f e^{3 c} - 2 f e^{c} - i f\right ) e^{- 2 c}}{8 a} + \frac {x \left (- i e e^{4 c} + 2 e e^{3 c} - 2 e e^{c} - i e\right ) e^{- 2 c}}{4 a} & \text {otherwise} \end {cases} + \frac {3 i f x^{2}}{4 a} + \frac {x \left (3 i d e - 4 i f\right )}{2 a d} + \frac {2 i f \log {\left (e^{d x} - i e^{- c} \right )}}{a d^{2}} \] Input:
integrate((f*x+e)*sinh(d*x+c)**3/(a+I*a*sinh(d*x+c)),x)
Output:
(2*e + 2*f*x)/(a*d*exp(c)*exp(d*x) - I*a*d) + Piecewise((((512*a**3*d**7*e *exp(2*c) + 512*a**3*d**7*f*x*exp(2*c) + 512*a**3*d**6*f*exp(2*c))*exp(-d* x) + (512*a**3*d**7*e*exp(4*c) + 512*a**3*d**7*f*x*exp(4*c) - 512*a**3*d** 6*f*exp(4*c))*exp(d*x) + (128*I*a**3*d**7*e*exp(c) + 128*I*a**3*d**7*f*x*e xp(c) + 64*I*a**3*d**6*f*exp(c))*exp(-2*d*x) + (-128*I*a**3*d**7*e*exp(5*c ) - 128*I*a**3*d**7*f*x*exp(5*c) + 64*I*a**3*d**6*f*exp(5*c))*exp(2*d*x))* exp(-3*c)/(1024*a**4*d**8), Ne(a**4*d**8*exp(3*c), 0)), (x**2*(-I*f*exp(4* c) + 2*f*exp(3*c) - 2*f*exp(c) - I*f)*exp(-2*c)/(8*a) + x*(-I*e*exp(4*c) + 2*e*exp(3*c) - 2*e*exp(c) - I*e)*exp(-2*c)/(4*a), True)) + 3*I*f*x**2/(4* a) + x*(3*I*d*e - 4*I*f)/(2*a*d) + 2*I*f*log(exp(d*x) - I*exp(-c))/(a*d**2 )
Exception generated. \[ \int \frac {(e+f x) \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((f*x+e)*sinh(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. 341 vs. \(2 (144) = 288\).
Time = 0.12 (sec) , antiderivative size = 341, normalized size of antiderivative = 2.01 \[ \int \frac {(e+f x) \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {12 i \, d^{2} f x^{2} e^{\left (3 \, d x + 3 \, c\right )} + 12 \, d^{2} f x^{2} e^{\left (2 \, d x + 2 \, c\right )} + 24 i \, d^{2} e x e^{\left (3 \, d x + 3 \, c\right )} + 24 \, d^{2} e x e^{\left (2 \, d x + 2 \, c\right )} - 2 i \, d f x e^{\left (5 \, d x + 5 \, c\right )} + 6 \, d f x e^{\left (4 \, d x + 4 \, c\right )} - 40 i \, d f x e^{\left (3 \, d x + 3 \, c\right )} + 8 \, d f x e^{\left (2 \, d x + 2 \, c\right )} - 6 i \, d f x e^{\left (d x + c\right )} + 2 \, d f x - 2 i \, d e e^{\left (5 \, d x + 5 \, c\right )} + 6 \, d e e^{\left (4 \, d x + 4 \, c\right )} - 8 i \, d e e^{\left (3 \, d x + 3 \, c\right )} + 40 \, d e e^{\left (2 \, d x + 2 \, c\right )} - 6 i \, d e e^{\left (d x + c\right )} + 32 i \, f e^{\left (3 \, d x + 3 \, c\right )} \log \left (e^{\left (d x + c\right )} - i\right ) + 32 \, f e^{\left (2 \, d x + 2 \, c\right )} \log \left (e^{\left (d x + c\right )} - i\right ) + 2 \, d e + i \, f e^{\left (5 \, d x + 5 \, c\right )} - 7 \, f e^{\left (4 \, d x + 4 \, c\right )} + 8 i \, f e^{\left (3 \, d x + 3 \, c\right )} + 8 \, f e^{\left (2 \, d x + 2 \, c\right )} - 7 i \, f e^{\left (d x + c\right )} + f}{16 \, {\left (a d^{2} e^{\left (3 \, d x + 3 \, c\right )} - i \, a d^{2} e^{\left (2 \, d x + 2 \, c\right )}\right )}} \] Input:
integrate((f*x+e)*sinh(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="giac")
Output:
1/16*(12*I*d^2*f*x^2*e^(3*d*x + 3*c) + 12*d^2*f*x^2*e^(2*d*x + 2*c) + 24*I *d^2*e*x*e^(3*d*x + 3*c) + 24*d^2*e*x*e^(2*d*x + 2*c) - 2*I*d*f*x*e^(5*d*x + 5*c) + 6*d*f*x*e^(4*d*x + 4*c) - 40*I*d*f*x*e^(3*d*x + 3*c) + 8*d*f*x*e ^(2*d*x + 2*c) - 6*I*d*f*x*e^(d*x + c) + 2*d*f*x - 2*I*d*e*e^(5*d*x + 5*c) + 6*d*e*e^(4*d*x + 4*c) - 8*I*d*e*e^(3*d*x + 3*c) + 40*d*e*e^(2*d*x + 2*c ) - 6*I*d*e*e^(d*x + c) + 32*I*f*e^(3*d*x + 3*c)*log(e^(d*x + c) - I) + 32 *f*e^(2*d*x + 2*c)*log(e^(d*x + c) - I) + 2*d*e + I*f*e^(5*d*x + 5*c) - 7* f*e^(4*d*x + 4*c) + 8*I*f*e^(3*d*x + 3*c) + 8*f*e^(2*d*x + 2*c) - 7*I*f*e^ (d*x + c) + f)/(a*d^2*e^(3*d*x + 3*c) - I*a*d^2*e^(2*d*x + 2*c))
Time = 1.56 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.26 \[ \int \frac {(e+f x) \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx={\mathrm {e}}^{-c-d\,x}\,\left (\frac {f+d\,e}{2\,a\,d^2}+\frac {f\,x}{2\,a\,d}\right )+{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (\frac {\left (f+2\,d\,e\right )\,1{}\mathrm {i}}{16\,a\,d^2}+\frac {f\,x\,1{}\mathrm {i}}{8\,a\,d}\right )+{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (\frac {\left (f-2\,d\,e\right )\,1{}\mathrm {i}}{16\,a\,d^2}-\frac {f\,x\,1{}\mathrm {i}}{8\,a\,d}\right )-{\mathrm {e}}^{c+d\,x}\,\left (\frac {f-d\,e}{2\,a\,d^2}-\frac {f\,x}{2\,a\,d}\right )+\frac {f\,x^2\,3{}\mathrm {i}}{4\,a}+\frac {2\,\left (e+f\,x\right )}{a\,d\,\left ({\mathrm {e}}^{c+d\,x}-\mathrm {i}\right )}-\frac {x\,\left (4\,f-3\,d\,e\right )\,1{}\mathrm {i}}{2\,a\,d}+\frac {f\,\ln \left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-\mathrm {i}\right )\,2{}\mathrm {i}}{a\,d^2} \] Input:
int((sinh(c + d*x)^3*(e + f*x))/(a + a*sinh(c + d*x)*1i),x)
Output:
exp(- c - d*x)*((f + d*e)/(2*a*d^2) + (f*x)/(2*a*d)) + exp(- 2*c - 2*d*x)* (((f + 2*d*e)*1i)/(16*a*d^2) + (f*x*1i)/(8*a*d)) + exp(2*c + 2*d*x)*(((f - 2*d*e)*1i)/(16*a*d^2) - (f*x*1i)/(8*a*d)) - exp(c + d*x)*((f - d*e)/(2*a* d^2) - (f*x)/(2*a*d)) + (f*x^2*3i)/(4*a) + (2*(e + f*x))/(a*d*(exp(c + d*x ) - 1i)) - (x*(4*f - 3*d*e)*1i)/(2*a*d) + (f*log(exp(d*x)*exp(c) - 1i)*2i) /(a*d^2)
\[ \int \frac {(e+f x) \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {\cosh \left (d x +c \right )^{2} d^{2} f i \,x^{2}+\cosh \left (d x +c \right )^{2} f i -2 \cosh \left (d x +c \right ) \sinh \left (d x +c \right ) d f i x +4 \cosh \left (d x +c \right ) d f x +4 \left (\int \frac {\sinh \left (d x +c \right )^{3}}{\sinh \left (d x +c \right ) i +1}d x \right ) d^{2} e +4 \left (\int -\frac {x}{\sinh \left (d x +c \right )-i}d x \right ) d^{2} f -\sinh \left (d x +c \right )^{2} d^{2} f i \,x^{2}-4 \sinh \left (d x +c \right ) f +2 d^{2} f i \,x^{2}}{4 a \,d^{2}} \] Input:
int((f*x+e)*sinh(d*x+c)^3/(a+I*a*sinh(d*x+c)),x)
Output:
(cosh(c + d*x)**2*d**2*f*i*x**2 + cosh(c + d*x)**2*f*i - 2*cosh(c + d*x)*s inh(c + d*x)*d*f*i*x + 4*cosh(c + d*x)*d*f*x + 4*int(sinh(c + d*x)**3/(sin h(c + d*x)*i + 1),x)*d**2*e + 4*int(( - x)/(sinh(c + d*x) - i),x)*d**2*f - sinh(c + d*x)**2*d**2*f*i*x**2 - 4*sinh(c + d*x)*f + 2*d**2*f*i*x**2)/(4* a*d**2)