Integrand size = 31, antiderivative size = 76 \[ \int \frac {\cosh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\frac {\log (e+f x)}{a f}-\frac {i \text {Chi}\left (\frac {d e}{f}+d x\right ) \sinh \left (c-\frac {d e}{f}\right )}{a f}-\frac {i \cosh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{a f} \]
Time = 0.43 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.82 \[ \int \frac {\cosh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\frac {\log (e+f x)-i \text {Chi}\left (d \left (\frac {e}{f}+x\right )\right ) \sinh \left (c-\frac {d e}{f}\right )-i \cosh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (d \left (\frac {e}{f}+x\right )\right )}{a f} \]
(Log[e + f*x] - I*CoshIntegral[d*(e/f + x)]*Sinh[c - (d*e)/f] - I*Cosh[c - (d*e)/f]*SinhIntegral[d*(e/f + x)])/(a*f)
Time = 0.55 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {6097, 16, 3042, 26, 3784, 26, 3042, 26, 3779, 3782}
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.
\(\displaystyle \int \frac {\cosh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx\) |
\(\Big \downarrow \) 6097 |
\(\displaystyle \frac {\int \frac {1}{e+f x}dx}{a}-\frac {i \int \frac {\sinh (c+d x)}{e+f x}dx}{a}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {\log (e+f x)}{a f}-\frac {i \int \frac {\sinh (c+d x)}{e+f x}dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\log (e+f x)}{a f}-\frac {i \int -\frac {i \sin (i c+i d x)}{e+f x}dx}{a}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\log (e+f x)}{a f}-\frac {\int \frac {\sin (i c+i d x)}{e+f x}dx}{a}\) |
\(\Big \downarrow \) 3784 |
\(\displaystyle \frac {\log (e+f x)}{a f}-\frac {i \sinh \left (c-\frac {d e}{f}\right ) \int \frac {\cosh \left (\frac {d e}{f}+d x\right )}{e+f x}dx+\cosh \left (c-\frac {d e}{f}\right ) \int \frac {i \sinh \left (\frac {d e}{f}+d x\right )}{e+f x}dx}{a}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\log (e+f x)}{a f}-\frac {i \sinh \left (c-\frac {d e}{f}\right ) \int \frac {\cosh \left (\frac {d e}{f}+d x\right )}{e+f x}dx+i \cosh \left (c-\frac {d e}{f}\right ) \int \frac {\sinh \left (\frac {d e}{f}+d x\right )}{e+f x}dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\log (e+f x)}{a f}-\frac {i \sinh \left (c-\frac {d e}{f}\right ) \int \frac {\sin \left (\frac {i d e}{f}+i d x+\frac {\pi }{2}\right )}{e+f x}dx+i \cosh \left (c-\frac {d e}{f}\right ) \int -\frac {i \sin \left (\frac {i d e}{f}+i d x\right )}{e+f x}dx}{a}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\log (e+f x)}{a f}-\frac {i \sinh \left (c-\frac {d e}{f}\right ) \int \frac {\sin \left (\frac {i d e}{f}+i d x+\frac {\pi }{2}\right )}{e+f x}dx+\cosh \left (c-\frac {d e}{f}\right ) \int \frac {\sin \left (\frac {i d e}{f}+i d x\right )}{e+f x}dx}{a}\) |
\(\Big \downarrow \) 3779 |
\(\displaystyle \frac {\log (e+f x)}{a f}-\frac {i \sinh \left (c-\frac {d e}{f}\right ) \int \frac {\sin \left (\frac {i d e}{f}+i d x+\frac {\pi }{2}\right )}{e+f x}dx+\frac {i \cosh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{f}}{a}\) |
\(\Big \downarrow \) 3782 |
\(\displaystyle \frac {\log (e+f x)}{a f}-\frac {\frac {i \sinh \left (c-\frac {d e}{f}\right ) \text {Chi}\left (\frac {d e}{f}+d x\right )}{f}+\frac {i \cosh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{f}}{a}\) |
Log[e + f*x]/(a*f) - ((I*CoshIntegral[(d*e)/f + d*x]*Sinh[c - (d*e)/f])/f + (I*Cosh[c - (d*e)/f]*SinhIntegral[(d*e)/f + d*x])/f)/a
3.3.63.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f , fz}, x] && EqQ[d*e - c*f*fz*I, 0]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz }, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* f)/d] Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]
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]
Time = 12.95 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.36
method | result | size |
risch | \(\frac {\ln \left (f x +e \right )}{a f}+\frac {i {\mathrm e}^{\frac {c f -d e}{f}} \operatorname {Ei}_{1}\left (-d x -c -\frac {-c f +d e}{f}\right )}{2 a f}-\frac {i {\mathrm e}^{-\frac {c f -d e}{f}} \operatorname {Ei}_{1}\left (d x +c -\frac {c f -d e}{f}\right )}{2 a f}\) | \(103\) |
ln(f*x+e)/a/f+1/2*I/a/f*exp((c*f-d*e)/f)*Ei(1,-d*x-c-(-c*f+d*e)/f)-1/2*I/a /f*exp(-(c*f-d*e)/f)*Ei(1,d*x+c-(c*f-d*e)/f)
Time = 0.25 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.04 \[ \int \frac {\cosh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\frac {i \, {\rm Ei}\left (-\frac {d f x + d e}{f}\right ) e^{\left (\frac {d e - c f}{f}\right )} - i \, {\rm Ei}\left (\frac {d f x + d e}{f}\right ) e^{\left (-\frac {d e - c f}{f}\right )} + 2 \, \log \left (\frac {f x + e}{f}\right )}{2 \, a f} \]
1/2*(I*Ei(-(d*f*x + d*e)/f)*e^((d*e - c*f)/f) - I*Ei((d*f*x + d*e)/f)*e^(- (d*e - c*f)/f) + 2*log((f*x + e)/f))/(a*f)
\[ \int \frac {\cosh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=- \frac {i \int \frac {\cosh ^{2}{\left (c + d x \right )}}{e \sinh {\left (c + d x \right )} - i e + f x \sinh {\left (c + d x \right )} - i f x}\, dx}{a} \]
Time = 0.29 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=-\frac {i \, e^{\left (-c + \frac {d e}{f}\right )} E_{1}\left (\frac {{\left (f x + e\right )} d}{f}\right )}{2 \, a f} + \frac {i \, e^{\left (c - \frac {d e}{f}\right )} E_{1}\left (-\frac {{\left (f x + e\right )} d}{f}\right )}{2 \, a f} + \frac {\log \left (f x + e\right )}{a f} \]
-1/2*I*e^(-c + d*e/f)*exp_integral_e(1, (f*x + e)*d/f)/(a*f) + 1/2*I*e^(c - d*e/f)*exp_integral_e(1, -(f*x + e)*d/f)/(a*f) + log(f*x + e)/(a*f)
Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=-\frac {{\left (i \, {\rm Ei}\left (\frac {d f x + d e}{f}\right ) e^{\left (2 \, c - \frac {d e}{f}\right )} - i \, {\rm Ei}\left (-\frac {d f x + d e}{f}\right ) e^{\left (\frac {d e}{f}\right )} - 2 \, e^{c} \log \left (i \, f x + i \, e\right )\right )} e^{\left (-c\right )}}{2 \, a f} \]
-1/2*(I*Ei((d*f*x + d*e)/f)*e^(2*c - d*e/f) - I*Ei(-(d*f*x + d*e)/f)*e^(d* e/f) - 2*e^c*log(I*f*x + I*e))*e^(-c)/(a*f)
Timed out. \[ \int \frac {\cosh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2}{\left (e+f\,x\right )\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]