Integrand size = 23, antiderivative size = 241 \[ \int \frac {(c+d x)^2}{(a+i a \sinh (e+f x))^2} \, dx=\frac {(c+d x)^2}{3 a^2 f}-\frac {4 d (c+d x) \log \left (1+i e^{e+f x}\right )}{3 a^2 f^2}-\frac {4 d^2 \operatorname {PolyLog}\left (2,-i e^{e+f x}\right )}{3 a^2 f^3}+\frac {d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{3 a^2 f^2}-\frac {2 d^2 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{3 a^2 f^3}+\frac {(c+d x)^2 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{6 a^2 f} \]
1/3*(d*x+c)^2/a^2/f-4/3*d*(d*x+c)*ln(1+I*exp(f*x+e))/a^2/f^2-4/3*d^2*polyl og(2,-I*exp(f*x+e))/a^2/f^3+1/3*d*(d*x+c)*sech(1/2*e+1/4*I*Pi+1/2*f*x)^2/a ^2/f^2-2/3*d^2*tanh(1/2*e+1/4*I*Pi+1/2*f*x)/a^2/f^3+1/3*(d*x+c)^2*tanh(1/2 *e+1/4*I*Pi+1/2*f*x)/a^2/f+1/6*(d*x+c)^2*sech(1/2*e+1/4*I*Pi+1/2*f*x)^2*ta nh(1/2*e+1/4*I*Pi+1/2*f*x)/a^2/f
Time = 1.69 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.12 \[ \int \frac {(c+d x)^2}{(a+i a \sinh (e+f x))^2} \, dx=\frac {\frac {2 i f (c+d x) \left (f (c+d x)+2 d \left (1+i e^e\right ) \log \left (1-i e^{-e-f x}\right )\right )}{-i+e^e}+4 d^2 \operatorname {PolyLog}\left (2,i e^{-e-f x}\right )+\frac {2 d f (c+d x) \cosh \left (\frac {f x}{2}\right )+2 i d^2 \cosh \left (e+\frac {f x}{2}\right )+i \left (c^2 f^2+2 c d f^2 x+d^2 \left (-2+f^2 x^2\right )\right ) \cosh \left (e+\frac {3 f x}{2}\right )+\left (3 c^2 f^2+6 c d f^2 x+d^2 \left (-4+3 f^2 x^2\right )\right ) \sinh \left (\frac {f x}{2}\right )+2 i d f (c+d x) \sinh \left (e+\frac {f x}{2}\right )}{\left (\cosh \left (\frac {e}{2}\right )+i \sinh \left (\frac {e}{2}\right )\right ) \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )^3}}{3 a^2 f^3} \]
(((2*I)*f*(c + d*x)*(f*(c + d*x) + 2*d*(1 + I*E^e)*Log[1 - I*E^(-e - f*x)] ))/(-I + E^e) + 4*d^2*PolyLog[2, I*E^(-e - f*x)] + (2*d*f*(c + d*x)*Cosh[( f*x)/2] + (2*I)*d^2*Cosh[e + (f*x)/2] + I*(c^2*f^2 + 2*c*d*f^2*x + d^2*(-2 + f^2*x^2))*Cosh[e + (3*f*x)/2] + (3*c^2*f^2 + 6*c*d*f^2*x + d^2*(-4 + 3* f^2*x^2))*Sinh[(f*x)/2] + (2*I)*d*f*(c + d*x)*Sinh[e + (f*x)/2])/((Cosh[e/ 2] + I*Sinh[e/2])*(Cosh[(e + f*x)/2] + I*Sinh[(e + f*x)/2])^3))/(3*a^2*f^3 )
Time = 0.98 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.696, Rules used = {3042, 3799, 3042, 4674, 3042, 4254, 24, 4672, 26, 3042, 26, 4199, 26, 2620, 2715, 2838}
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.
| \(\displaystyle \int \frac {(c+d x)^2}{(a+i a \sinh (e+f x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d x)^2}{(a+a \sin (i e+i f x))^2}dx\) |
| \(\Big \downarrow \) 3799 |
| \(\displaystyle \frac {\int (c+d x)^2 \text {csch}^4\left (\frac {e}{2}+\frac {f x}{2}-\frac {i \pi }{4}\right )dx}{4 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (c+d x)^2 \csc \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^4dx}{4 a^2}\) |
| \(\Big \downarrow \) 4674 |
| \(\displaystyle \frac {\frac {2}{3} \int (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx-\frac {4 d^2 \int \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{3 f^2}+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}}{4 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2}{3} \int (c+d x)^2 \csc \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^2dx-\frac {4 d^2 \int \csc \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^2dx}{3 f^2}+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}}{4 a^2}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {2}{3} \int (c+d x)^2 \csc \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^2dx-\frac {8 i d^2 \int 1d\left (-i \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )\right )}{3 f^3}+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}}{4 a^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {2}{3} \int (c+d x)^2 \csc \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^2dx+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}-\frac {8 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^3}}{4 a^2}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i d \int -i (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}-\frac {8 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^3}}{4 a^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 d \int (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}-\frac {8 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^3}}{4 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 d \int -i (c+d x) \tan \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}\right )+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}-\frac {8 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^3}}{4 a^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {4 i d \int (c+d x) \tan \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}-\frac {8 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^3}}{4 a^2}\) |
| \(\Big \downarrow \) 4199 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {4 i d \left (2 i \int \frac {i e^{e+f x} (c+d x)}{1+i e^{e+f x}}dx-\frac {i (c+d x)^2}{2 d}\right )}{f}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}-\frac {8 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^3}}{4 a^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {4 i d \left (-2 \int \frac {e^{e+f x} (c+d x)}{1+i e^{e+f x}}dx-\frac {i (c+d x)^2}{2 d}\right )}{f}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}-\frac {8 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^3}}{4 a^2}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {4 i d \left (-2 \left (\frac {i d \int \log \left (1+i e^{e+f x}\right )dx}{f}-\frac {i (c+d x) \log \left (1+i e^{e+f x}\right )}{f}\right )-\frac {i (c+d x)^2}{2 d}\right )}{f}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}-\frac {8 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^3}}{4 a^2}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {4 i d \left (-2 \left (\frac {i d \int e^{-e-f x} \log \left (1+i e^{e+f x}\right )de^{e+f x}}{f^2}-\frac {i (c+d x) \log \left (1+i e^{e+f x}\right )}{f}\right )-\frac {i (c+d x)^2}{2 d}\right )}{f}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}-\frac {8 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^3}}{4 a^2}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {4 i d \left (-2 \left (-\frac {i (c+d x) \log \left (1+i e^{e+f x}\right )}{f}-\frac {i d \operatorname {PolyLog}\left (2,-i e^{e+f x}\right )}{f^2}\right )-\frac {i (c+d x)^2}{2 d}\right )}{f}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {4 d (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}-\frac {8 d^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^3}}{4 a^2}\) |
((4*d*(c + d*x)*Sech[e/2 + (I/4)*Pi + (f*x)/2]^2)/(3*f^2) - (8*d^2*Tanh[e/ 2 + (I/4)*Pi + (f*x)/2])/(3*f^3) + (2*(c + d*x)^2*Sech[e/2 + (I/4)*Pi + (f *x)/2]^2*Tanh[e/2 + (I/4)*Pi + (f*x)/2])/(3*f) + (2*(((4*I)*d*(((-1/2*I)*( c + d*x)^2)/d - 2*(((-I)*(c + d*x)*Log[1 + I*E^(e + f*x)])/f - (I*d*PolyLo g[2, (-I)*E^(e + f*x)])/f^2)))/f + (2*(c + d*x)^2*Tanh[e/2 + (I/4)*Pi + (f *x)/2])/f))/3)/(4*a^2)
3.2.14.3.1 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[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 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[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_ .)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp [2*I Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x ))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In tegerQ[4*k] && IGtQ[m, 0]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
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[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbo l] :> Simp[(-b^2)*(c + d*x)^m*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (-Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^ 2*(n - 1)*(n - 2))), x] + Simp[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))) Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Simp[b^2*((n - 2)/ (n - 1)) Int[(c + d*x)^m*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c , d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
Time = 1.41 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.55
| method | result | size |
| risch | \(\frac {-\frac {2 i c^{2} f^{2}}{3}-\frac {4 i f \,d^{2} x \,{\mathrm e}^{2 f x +2 e}}{3}-\frac {4 i f c d \,{\mathrm e}^{2 f x +2 e}}{3}-\frac {4 i d^{2} {\mathrm e}^{2 f x +2 e}}{3}-\frac {4 i f^{2} c d x}{3}+\frac {4 i d^{2}}{3}-\frac {4 f \,d^{2} x \,{\mathrm e}^{f x +e}}{3}-\frac {4 f c d \,{\mathrm e}^{f x +e}}{3}-\frac {2 i f^{2} d^{2} x^{2}}{3}-\frac {8 d^{2} {\mathrm e}^{f x +e}}{3}+2 f^{2} d^{2} x^{2} {\mathrm e}^{f x +e}+4 f^{2} c d x \,{\mathrm e}^{f x +e}+2 f^{2} c^{2} {\mathrm e}^{f x +e}}{\left ({\mathrm e}^{f x +e}-i\right )^{3} f^{3} a^{2}}-\frac {4 d \ln \left ({\mathrm e}^{f x +e}-i\right ) c}{3 a^{2} f^{2}}+\frac {4 d \ln \left ({\mathrm e}^{f x +e}\right ) c}{3 a^{2} f^{2}}+\frac {2 d^{2} x^{2}}{3 a^{2} f}+\frac {4 d^{2} e x}{3 a^{2} f^{2}}+\frac {2 d^{2} e^{2}}{3 a^{2} f^{3}}-\frac {4 d^{2} \ln \left (1+i {\mathrm e}^{f x +e}\right ) x}{3 a^{2} f^{2}}-\frac {4 d^{2} \ln \left (1+i {\mathrm e}^{f x +e}\right ) e}{3 a^{2} f^{3}}-\frac {4 d^{2} \operatorname {polylog}\left (2, -i {\mathrm e}^{f x +e}\right )}{3 a^{2} f^{3}}+\frac {4 d^{2} e \ln \left ({\mathrm e}^{f x +e}-i\right )}{3 a^{2} f^{3}}-\frac {4 d^{2} e \ln \left ({\mathrm e}^{f x +e}\right )}{3 a^{2} f^{3}}\) | \(374\) |
2/3*(-I*c^2*f^2-2*I*f*d^2*x*exp(2*f*x+2*e)-2*I*f*c*d*exp(2*f*x+2*e)-2*I*d^ 2*exp(2*f*x+2*e)-2*I*f^2*c*d*x+2*I*d^2-2*f*d^2*x*exp(f*x+e)-2*f*c*d*exp(f* x+e)-I*f^2*d^2*x^2-4*d^2*exp(f*x+e)+3*f^2*d^2*x^2*exp(f*x+e)+6*f^2*c*d*x*e xp(f*x+e)+3*f^2*c^2*exp(f*x+e))/(exp(f*x+e)-I)^3/f^3/a^2-4/3/a^2/f^2*d*ln( exp(f*x+e)-I)*c+4/3/a^2/f^2*d*ln(exp(f*x+e))*c+2/3/a^2/f*d^2*x^2+4/3/a^2/f ^2*d^2*e*x+2/3/a^2/f^3*d^2*e^2-4/3/a^2/f^2*d^2*ln(1+I*exp(f*x+e))*x-4/3/a^ 2/f^3*d^2*ln(1+I*exp(f*x+e))*e-4/3*d^2*polylog(2,-I*exp(f*x+e))/a^2/f^3+4/ 3/a^2/f^3*d^2*e*ln(exp(f*x+e)-I)-4/3/a^2/f^3*d^2*e*ln(exp(f*x+e))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 482 vs. \(2 (180) = 360\).
Time = 0.24 (sec) , antiderivative size = 482, normalized size of antiderivative = 2.00 \[ \int \frac {(c+d x)^2}{(a+i a \sinh (e+f x))^2} \, dx=-\frac {2 \, {\left (i \, d^{2} e^{2} - 2 i \, c d e f + i \, c^{2} f^{2} - 2 i \, d^{2} + 2 \, {\left (d^{2} e^{\left (3 \, f x + 3 \, e\right )} - 3 i \, d^{2} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, d^{2} e^{\left (f x + e\right )} + i \, d^{2}\right )} {\rm Li}_2\left (-i \, e^{\left (f x + e\right )}\right ) - {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x - d^{2} e^{2} + 2 \, c d e f\right )} e^{\left (3 \, f x + 3 \, e\right )} + {\left (3 i \, d^{2} f^{2} x^{2} - 3 i \, d^{2} e^{2} + 2 i \, d^{2} + 2 \, {\left (3 i \, c d e + i \, c d\right )} f + 2 \, {\left (3 i \, c d f^{2} + i \, d^{2} f\right )} x\right )} e^{\left (2 \, f x + 2 \, e\right )} - {\left (3 \, d^{2} e^{2} + 3 \, c^{2} f^{2} - 2 \, d^{2} f x - 4 \, d^{2} - 2 \, {\left (3 \, c d e + c d\right )} f\right )} e^{\left (f x + e\right )} + 2 \, {\left (-i \, d^{2} e + i \, c d f - {\left (d^{2} e - c d f\right )} e^{\left (3 \, f x + 3 \, e\right )} + 3 \, {\left (i \, d^{2} e - i \, c d f\right )} e^{\left (2 \, f x + 2 \, e\right )} + 3 \, {\left (d^{2} e - c d f\right )} e^{\left (f x + e\right )}\right )} \log \left (e^{\left (f x + e\right )} - i\right ) + 2 \, {\left (i \, d^{2} f x + i \, d^{2} e + {\left (d^{2} f x + d^{2} e\right )} e^{\left (3 \, f x + 3 \, e\right )} + 3 \, {\left (-i \, d^{2} f x - i \, d^{2} e\right )} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, {\left (d^{2} f x + d^{2} e\right )} e^{\left (f x + e\right )}\right )} \log \left (i \, e^{\left (f x + e\right )} + 1\right )\right )}}{3 \, {\left (a^{2} f^{3} e^{\left (3 \, f x + 3 \, e\right )} - 3 i \, a^{2} f^{3} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, a^{2} f^{3} e^{\left (f x + e\right )} + i \, a^{2} f^{3}\right )}} \]
-2/3*(I*d^2*e^2 - 2*I*c*d*e*f + I*c^2*f^2 - 2*I*d^2 + 2*(d^2*e^(3*f*x + 3* e) - 3*I*d^2*e^(2*f*x + 2*e) - 3*d^2*e^(f*x + e) + I*d^2)*dilog(-I*e^(f*x + e)) - (d^2*f^2*x^2 + 2*c*d*f^2*x - d^2*e^2 + 2*c*d*e*f)*e^(3*f*x + 3*e) + (3*I*d^2*f^2*x^2 - 3*I*d^2*e^2 + 2*I*d^2 + 2*(3*I*c*d*e + I*c*d)*f + 2*( 3*I*c*d*f^2 + I*d^2*f)*x)*e^(2*f*x + 2*e) - (3*d^2*e^2 + 3*c^2*f^2 - 2*d^2 *f*x - 4*d^2 - 2*(3*c*d*e + c*d)*f)*e^(f*x + e) + 2*(-I*d^2*e + I*c*d*f - (d^2*e - c*d*f)*e^(3*f*x + 3*e) + 3*(I*d^2*e - I*c*d*f)*e^(2*f*x + 2*e) + 3*(d^2*e - c*d*f)*e^(f*x + e))*log(e^(f*x + e) - I) + 2*(I*d^2*f*x + I*d^2 *e + (d^2*f*x + d^2*e)*e^(3*f*x + 3*e) + 3*(-I*d^2*f*x - I*d^2*e)*e^(2*f*x + 2*e) - 3*(d^2*f*x + d^2*e)*e^(f*x + e))*log(I*e^(f*x + e) + 1))/(a^2*f^ 3*e^(3*f*x + 3*e) - 3*I*a^2*f^3*e^(2*f*x + 2*e) - 3*a^2*f^3*e^(f*x + e) + I*a^2*f^3)
\[ \int \frac {(c+d x)^2}{(a+i a \sinh (e+f x))^2} \, dx=\frac {- 2 i c^{2} f^{2} - 4 i c d f^{2} x - 2 i d^{2} f^{2} x^{2} + 4 i d^{2} + \left (- 4 i c d f e^{2 e} - 4 i d^{2} f x e^{2 e} - 4 i d^{2} e^{2 e}\right ) e^{2 f x} + \left (6 c^{2} f^{2} e^{e} + 12 c d f^{2} x e^{e} - 4 c d f e^{e} + 6 d^{2} f^{2} x^{2} e^{e} - 4 d^{2} f x e^{e} - 8 d^{2} e^{e}\right ) e^{f x}}{3 a^{2} f^{3} e^{3 e} e^{3 f x} - 9 i a^{2} f^{3} e^{2 e} e^{2 f x} - 9 a^{2} f^{3} e^{e} e^{f x} + 3 i a^{2} f^{3}} - \frac {4 i d \left (\int \frac {c}{e^{e} e^{f x} - i}\, dx + \int \frac {d x}{e^{e} e^{f x} - i}\, dx\right )}{3 a^{2} f} \]
(-2*I*c**2*f**2 - 4*I*c*d*f**2*x - 2*I*d**2*f**2*x**2 + 4*I*d**2 + (-4*I*c *d*f*exp(2*e) - 4*I*d**2*f*x*exp(2*e) - 4*I*d**2*exp(2*e))*exp(2*f*x) + (6 *c**2*f**2*exp(e) + 12*c*d*f**2*x*exp(e) - 4*c*d*f*exp(e) + 6*d**2*f**2*x* *2*exp(e) - 4*d**2*f*x*exp(e) - 8*d**2*exp(e))*exp(f*x))/(3*a**2*f**3*exp( 3*e)*exp(3*f*x) - 9*I*a**2*f**3*exp(2*e)*exp(2*f*x) - 9*a**2*f**3*exp(e)*e xp(f*x) + 3*I*a**2*f**3) - 4*I*d*(Integral(c/(exp(e)*exp(f*x) - I), x) + I ntegral(d*x/(exp(e)*exp(f*x) - I), x))/(3*a**2*f)
\[ \int \frac {(c+d x)^2}{(a+i a \sinh (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
-2/3*d^2*((I*f^2*x^2 - 2*(-I*f*x*e^(2*e) - I*e^(2*e))*e^(2*f*x) - (3*f^2*x ^2*e^e - 2*f*x*e^e - 4*e^e)*e^(f*x) - 2*I)/(a^2*f^3*e^(3*f*x + 3*e) - 3*I* a^2*f^3*e^(2*f*x + 2*e) - 3*a^2*f^3*e^(f*x + e) + I*a^2*f^3) + 6*I*integra te(1/3*x/(a^2*f*e^(f*x + e) - I*a^2*f), x)) + 4/3*c*d*((f*x*e^(3*f*x + 3*e ) - (3*I*f*x*e^(2*e) + I*e^(2*e))*e^(2*f*x) - e^(f*x + e))/(a^2*f^2*e^(3*f *x + 3*e) - 3*I*a^2*f^2*e^(2*f*x + 2*e) - 3*a^2*f^2*e^(f*x + e) + I*a^2*f^ 2) - log(-I*(I*e^(f*x + e) + 1)*e^(-e))/(a^2*f^2)) + 2/3*c^2*(3*e^(-f*x - e)/((3*a^2*e^(-f*x - e) - 3*I*a^2*e^(-2*f*x - 2*e) - a^2*e^(-3*f*x - 3*e) + I*a^2)*f) + I/((3*a^2*e^(-f*x - e) - 3*I*a^2*e^(-2*f*x - 2*e) - a^2*e^(- 3*f*x - 3*e) + I*a^2)*f))
\[ \int \frac {(c+d x)^2}{(a+i a \sinh (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(c+d x)^2}{(a+i a \sinh (e+f x))^2} \, dx=\int \frac {{\left (c+d\,x\right )}^2}{{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]