Integrand size = 23, antiderivative size = 53 \[ \int e^{a+i b x} \csc (d+b x) \sec (d+b x) \, dx=\frac {2 e^{a-i d} \arctan \left (e^{i (d+b x)}\right )}{b}-\frac {2 e^{a-i d} \text {arctanh}\left (e^{i (d+b x)}\right )}{b} \] Output:
2*exp(a-I*d)*arctan(exp(I*(b*x+d)))/b-2*exp(a-I*d)*arctanh(exp(I*(b*x+d))) /b
Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.09 \[ \int e^{a+i b x} \csc (d+b x) \sec (d+b x) \, dx=\frac {2 e^a \left (\arctan \left (e^{i b x} (\cos (d)+i \sin (d))\right )-\text {arctanh}\left (e^{i b x} (\cos (d)+i \sin (d))\right )\right ) (\cos (d)-i \sin (d))}{b} \] Input:
Integrate[E^(a + I*b*x)*Csc[d + b*x]*Sec[d + b*x],x]
Output:
(2*E^a*(ArcTan[E^(I*b*x)*(Cos[d] + I*Sin[d])] - ArcTanh[E^(I*b*x)*(Cos[d] + I*Sin[d])])*(Cos[d] - I*Sin[d]))/b
Time = 0.41 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.75, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {4974, 2009}
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.
\(\displaystyle \int e^{a+i b x} \csc (b x+d) \sec (b x+d) \, dx\) |
\(\Big \downarrow \) 4974 |
\(\displaystyle \int \left (-\frac {i e^{a+3 i b x+2 i d}}{1+e^{i b x+i d}}-\frac {2 i e^{a+3 i b x+2 i d}}{1+e^{2 i b x+2 i d}}+\frac {i e^{a+3 i b x+2 i d}}{-1+e^{i b x+i d}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 e^{a-i d} \arctan \left (e^{i b x+i d}\right )}{b}+\frac {e^{a-i d} \log \left (1-e^{i b x+i d}\right )}{b}-\frac {e^{a-i d} \log \left (1+e^{i b x+i d}\right )}{b}\) |
Input:
Int[E^(a + I*b*x)*Csc[d + b*x]*Sec[d + b*x],x]
Output:
(2*E^(a - I*d)*ArcTan[E^(I*d + I*b*x)])/b + (E^(a - I*d)*Log[1 - E^(I*d + I*b*x)])/b - (E^(a - I*d)*Log[1 + E^(I*d + I*b*x)])/b
Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*(G_)[(d_.) + (e_.)*(x_)]^(m_.)*(H_)[( d_.) + (e_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigToExp[F^(c*(a + b*x)), G[d + e*x]^m*H[d + e*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e}, x] && IGtQ[ m, 0] && IGtQ[n, 0] && TrigQ[G] && TrigQ[H]
Time = 0.15 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.87
method | result | size |
risch | \(\frac {2 \,{\mathrm e}^{a} {\mathrm e}^{-i d} \arctan \left ({\mathrm e}^{i \left (b x +d \right )}\right )}{b}-\frac {2 \,{\mathrm e}^{a} {\mathrm e}^{-i d} \operatorname {arctanh}\left ({\mathrm e}^{i \left (b x +d \right )}\right )}{b}\) | \(46\) |
Input:
int(exp(a+I*b*x)*csc(b*x+d)*sec(b*x+d),x,method=_RETURNVERBOSE)
Output:
2*exp(a)/b*exp(-I*d)*arctan(exp(I*(b*x+d)))-2/b*exp(a)*exp(-I*d)*arctanh(e xp(I*(b*x+d)))
Time = 0.08 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.60 \[ \int e^{a+i b x} \csc (d+b x) \sec (d+b x) \, dx=-\frac {e^{\left (a - i \, d\right )} \log \left (e^{\left (i \, b x + i \, d\right )} + 1\right ) - i \, e^{\left (a - i \, d\right )} \log \left (e^{\left (i \, b x + i \, d\right )} + i\right ) + i \, e^{\left (a - i \, d\right )} \log \left (e^{\left (i \, b x + i \, d\right )} - i\right ) - e^{\left (a - i \, d\right )} \log \left (e^{\left (i \, b x + i \, d\right )} - 1\right )}{b} \] Input:
integrate(exp(a+I*b*x)*csc(b*x+d)*sec(b*x+d),x, algorithm="fricas")
Output:
-(e^(a - I*d)*log(e^(I*b*x + I*d) + 1) - I*e^(a - I*d)*log(e^(I*b*x + I*d) + I) + I*e^(a - I*d)*log(e^(I*b*x + I*d) - I) - e^(a - I*d)*log(e^(I*b*x + I*d) - 1))/b
\[ \int e^{a+i b x} \csc (d+b x) \sec (d+b x) \, dx=e^{a} \int e^{i b x} \csc {\left (b x + d \right )} \sec {\left (b x + d \right )}\, dx \] Input:
integrate(exp(a+I*b*x)*csc(b*x+d)*sec(b*x+d),x)
Output:
exp(a)*Integral(exp(I*b*x)*csc(b*x + d)*sec(b*x + d), x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 479 vs. \(2 (43) = 86\).
Time = 0.25 (sec) , antiderivative size = 479, normalized size of antiderivative = 9.04 \[ \int e^{a+i b x} \csc (d+b x) \sec (d+b x) \, dx =\text {Too large to display} \] Input:
integrate(exp(a+I*b*x)*csc(b*x+d)*sec(b*x+d),x, algorithm="maxima")
Output:
-1/2*(2*(cos(d)*e^a - I*e^a*sin(d))*arctan2(2*(cos(b*x + 2*d)*cos(d) + sin (b*x + 2*d)*sin(d))/(cos(b*x + 2*d)^2 + cos(d)^2 + 2*cos(d)*sin(b*x + 2*d) + sin(b*x + 2*d)^2 - 2*cos(b*x + 2*d)*sin(d) + sin(d)^2), (cos(b*x + 2*d) ^2 - cos(d)^2 + sin(b*x + 2*d)^2 - sin(d)^2)/(cos(b*x + 2*d)^2 + cos(d)^2 + 2*cos(d)*sin(b*x + 2*d) + sin(b*x + 2*d)^2 - 2*cos(b*x + 2*d)*sin(d) + s in(d)^2)) - 2*(I*cos(d)*e^a + e^a*sin(d))*arctan2(sin(b*x) + sin(d), cos(b *x) - cos(d)) - 2*(-I*cos(d)*e^a - e^a*sin(d))*arctan2(sin(b*x) - sin(d), cos(b*x) + cos(d)) + (cos(d)*e^a - I*e^a*sin(d))*log(cos(b*x)^2 + 2*cos(b* x)*cos(d) + cos(d)^2 + sin(b*x)^2 - 2*sin(b*x)*sin(d) + sin(d)^2) - (cos(d )*e^a - I*e^a*sin(d))*log(cos(b*x)^2 - 2*cos(b*x)*cos(d) + cos(d)^2 + sin( b*x)^2 + 2*sin(b*x)*sin(d) + sin(d)^2) + (I*cos(d)*e^a + e^a*sin(d))*log(( cos(b*x + 2*d)^2 + cos(d)^2 - 2*cos(d)*sin(b*x + 2*d) + sin(b*x + 2*d)^2 + 2*cos(b*x + 2*d)*sin(d) + sin(d)^2)/(cos(b*x + 2*d)^2 + cos(d)^2 + 2*cos( d)*sin(b*x + 2*d) + sin(b*x + 2*d)^2 - 2*cos(b*x + 2*d)*sin(d) + sin(d)^2) ))/b
Time = 0.15 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.43 \[ \int e^{a+i b x} \csc (d+b x) \sec (d+b x) \, dx=\frac {{\left (i \, e^{a} \log \left (e^{\left (i \, b x + i \, d\right )} + i\right ) - i \, e^{a} \log \left (e^{\left (i \, b x + i \, d\right )} - i\right ) - e^{a} \log \left (i \, e^{\left (i \, b x + i \, d\right )} + i\right ) + e^{a} \log \left (-i \, e^{\left (i \, b x + i \, d\right )} + i\right )\right )} e^{\left (-i \, d\right )}}{b} \] Input:
integrate(exp(a+I*b*x)*csc(b*x+d)*sec(b*x+d),x, algorithm="giac")
Output:
(I*e^a*log(e^(I*b*x + I*d) + I) - I*e^a*log(e^(I*b*x + I*d) - I) - e^a*log (I*e^(I*b*x + I*d) + I) + e^a*log(-I*e^(I*b*x + I*d) + I))*e^(-I*d)/b
Time = 18.06 (sec) , antiderivative size = 248, normalized size of antiderivative = 4.68 \[ \int e^{a+i b x} \csc (d+b x) \sec (d+b x) \, dx=-\frac {{\left ({\mathrm {e}}^{4\,a-d\,4{}\mathrm {i}}\right )}^{1/4}\,\ln \left (b\,{\mathrm {e}}^{12\,a}\,{\mathrm {e}}^{-d\,12{}\mathrm {i}}\,64{}\mathrm {i}-64\,b\,{\mathrm {e}}^{11\,a}\,{\mathrm {e}}^{-d\,10{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,{\left ({\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-d\,4{}\mathrm {i}}\right )}^{1/4}\right )\,1{}\mathrm {i}}{b}+\frac {{\left ({\mathrm {e}}^{4\,a-d\,4{}\mathrm {i}}\right )}^{1/4}\,\ln \left (b\,{\mathrm {e}}^{12\,a}\,{\mathrm {e}}^{-d\,12{}\mathrm {i}}\,64{}\mathrm {i}+64\,b\,{\mathrm {e}}^{11\,a}\,{\mathrm {e}}^{-d\,10{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,{\left ({\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-d\,4{}\mathrm {i}}\right )}^{1/4}\right )\,1{}\mathrm {i}}{b}+\frac {{\left ({\mathrm {e}}^{4\,a-d\,4{}\mathrm {i}}\right )}^{1/4}\,\ln \left (b\,{\mathrm {e}}^{12\,a}\,{\mathrm {e}}^{-d\,12{}\mathrm {i}}\,64{}\mathrm {i}-b\,{\mathrm {e}}^{11\,a}\,{\mathrm {e}}^{-d\,10{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,{\left ({\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-d\,4{}\mathrm {i}}\right )}^{1/4}\,64{}\mathrm {i}\right )}{b}-\frac {{\left ({\mathrm {e}}^{4\,a-d\,4{}\mathrm {i}}\right )}^{1/4}\,\ln \left (b\,{\mathrm {e}}^{12\,a}\,{\mathrm {e}}^{-d\,12{}\mathrm {i}}\,64{}\mathrm {i}+b\,{\mathrm {e}}^{11\,a}\,{\mathrm {e}}^{-d\,10{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,{\left ({\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-d\,4{}\mathrm {i}}\right )}^{1/4}\,64{}\mathrm {i}\right )}{b} \] Input:
int(exp(a + b*x*1i)/(cos(d + b*x)*sin(d + b*x)),x)
Output:
(exp(4*a - d*4i)^(1/4)*log(b*exp(12*a)*exp(-d*12i)*64i + 64*b*exp(11*a)*ex p(-d*10i)*exp(b*x*1i)*(exp(4*a)*exp(-d*4i))^(1/4))*1i)/b - (exp(4*a - d*4i )^(1/4)*log(b*exp(12*a)*exp(-d*12i)*64i - 64*b*exp(11*a)*exp(-d*10i)*exp(b *x*1i)*(exp(4*a)*exp(-d*4i))^(1/4))*1i)/b + (exp(4*a - d*4i)^(1/4)*log(b*e xp(12*a)*exp(-d*12i)*64i - b*exp(11*a)*exp(-d*10i)*exp(b*x*1i)*(exp(4*a)*e xp(-d*4i))^(1/4)*64i))/b - (exp(4*a - d*4i)^(1/4)*log(b*exp(12*a)*exp(-d*1 2i)*64i + b*exp(11*a)*exp(-d*10i)*exp(b*x*1i)*(exp(4*a)*exp(-d*4i))^(1/4)* 64i))/b
\[ \int e^{a+i b x} \csc (d+b x) \sec (d+b x) \, dx=e^{a} \left (\int e^{b i x} \csc \left (b x +d \right ) \sec \left (b x +d \right )d x \right ) \] Input:
int(exp(a+I*b*x)*csc(b*x+d)*sec(b*x+d),x)
Output:
e**a*int(e**(b*i*x)*csc(b*x + d)*sec(b*x + d),x)