Integrand size = 25, antiderivative size = 128 \[ \int e^{a+i b x} \csc (d+b x) \sec ^3(d+b x) \, dx=-\frac {2 e^{a-i d+i (d+b x)}}{b \left (1+e^{2 i (d+b x)}\right )^2}+\frac {3 e^{a-i d+i (d+b x)}}{b \left (1+e^{2 i (d+b x)}\right )}+\frac {e^{a-i d} \arctan \left (e^{i (d+b x)}\right )}{b}-\frac {2 e^{a-i d} \text {arctanh}\left (e^{i (d+b x)}\right )}{b} \] Output:
-2*exp(a-I*d+I*(b*x+d))/b/(1+exp(2*I*(b*x+d)))^2+3*exp(a-I*d+I*(b*x+d))/b/ (1+exp(2*I*(b*x+d)))+exp(a-I*d)*arctan(exp(I*(b*x+d)))/b-2*exp(a-I*d)*arct anh(exp(I*(b*x+d)))/b
Time = 0.37 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.98 \[ \int e^{a+i b x} \csc (d+b x) \sec ^3(d+b x) \, dx=\frac {e^a \left (-\frac {2 e^{i b x}}{\left (1+e^{2 i (d+b x)}\right )^2}+\frac {3 e^{i b x}}{1+e^{2 i (d+b x)}}+e^{-i d} \arctan \left (e^{i (d+b x)}\right )+e^{-i d} \log \left (1-e^{i (d+b x)}\right )-e^{-i d} \log \left (1+e^{i (d+b x)}\right )\right )}{b} \] Input:
Integrate[E^(a + I*b*x)*Csc[d + b*x]*Sec[d + b*x]^3,x]
Output:
(E^a*((-2*E^(I*b*x))/(1 + E^((2*I)*(d + b*x)))^2 + (3*E^(I*b*x))/(1 + E^(( 2*I)*(d + b*x))) + ArcTan[E^(I*(d + b*x))]/E^(I*d) + Log[1 - E^(I*(d + b*x ))]/E^(I*d) - Log[1 + E^(I*(d + b*x))]/E^(I*d)))/b
Time = 0.49 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.48, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.
\(\displaystyle \int e^{a+i b x} \csc (b x+d) \sec ^3(b x+d) \, dx\) |
\(\Big \downarrow \) 4974 |
\(\displaystyle \int \left (-\frac {4 i e^{a+5 i b x+4 i d}}{\left (1+e^{2 i (b x+d)}\right )^2}-\frac {8 i e^{a+5 i b x+4 i d}}{\left (1+e^{2 i (b x+d)}\right )^3}+\frac {4 i e^{a+5 i b x+4 i d}}{-1+e^{4 i (b x+d)}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e^{a-i d} \arctan \left (e^{i (b x+d)}\right )}{b}-\frac {2 e^{a-i d} \text {arctanh}\left (e^{i (b x+d)}\right )}{b}-\frac {2 e^{a+i (b x+d)-i d}}{b}+\frac {3 e^{a+i (b x+d)-i d}}{b \left (1+e^{2 i (b x+d)}\right )}+\frac {2 e^{a+3 i (b x+d)-i d}}{b \left (1+e^{2 i (b x+d)}\right )}+\frac {2 e^{a+3 i (b x+d)-i d}}{b \left (1+e^{2 i (b x+d)}\right )^2}\) |
Input:
Int[E^(a + I*b*x)*Csc[d + b*x]*Sec[d + b*x]^3,x]
Output:
(-2*E^(a - I*d + I*(d + b*x)))/b + (2*E^(a - I*d + (3*I)*(d + b*x)))/(b*(1 + E^((2*I)*(d + b*x)))^2) + (3*E^(a - I*d + I*(d + b*x)))/(b*(1 + E^((2*I )*(d + b*x)))) + (2*E^(a - I*d + (3*I)*(d + b*x)))/(b*(1 + E^((2*I)*(d + b *x)))) + (E^(a - I*d)*ArcTan[E^(I*(d + b*x))])/b - (2*E^(a - I*d)*ArcTanh[ E^(I*(d + 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 = 0.59 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.68
method | result | size |
risch | \(\frac {3 \,{\mathrm e}^{a} {\mathrm e}^{3 i b x} {\mathrm e}^{2 i d}+{\mathrm e}^{a} {\mathrm e}^{i b x}}{b \left (1+{\mathrm e}^{2 i \left (b x +d \right )}\right )^{2}}-\frac {2 \,{\mathrm e}^{a} {\mathrm e}^{-i d} \operatorname {arctanh}\left ({\mathrm e}^{i \left (b x +d \right )}\right )}{b}+\frac {{\mathrm e}^{a} {\mathrm e}^{-i d} \arctan \left ({\mathrm e}^{i \left (b x +d \right )}\right )}{b}\) | \(87\) |
Input:
int(exp(a+I*b*x)*csc(b*x+d)*sec(b*x+d)^3,x,method=_RETURNVERBOSE)
Output:
1/b/(1+exp(2*I*(b*x+d)))^2*(3*exp(a)*exp(3*I*b*x)*exp(2*I*d)+exp(a)*exp(I* b*x))-2/b*exp(a)*exp(-I*d)*arctanh(exp(I*(b*x+d)))+exp(a)/b*exp(-I*d)*arct an(exp(I*(b*x+d)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (92) = 184\).
Time = 0.08 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.80 \[ \int e^{a+i b x} \csc (d+b x) \sec ^3(d+b x) \, dx=-\frac {2 \, {\left (e^{\left (4 i \, b x + a + 3 i \, d\right )} + 2 \, e^{\left (2 i \, b x + a + i \, d\right )} + e^{\left (a - i \, d\right )}\right )} \log \left (e^{\left (i \, b x + i \, d\right )} + 1\right ) - {\left (i \, e^{\left (4 i \, b x + a + 3 i \, d\right )} + 2 i \, e^{\left (2 i \, b x + a + i \, d\right )} + i \, e^{\left (a - i \, d\right )}\right )} \log \left (e^{\left (i \, b x + i \, d\right )} + i\right ) - {\left (-i \, e^{\left (4 i \, b x + a + 3 i \, d\right )} - 2 i \, e^{\left (2 i \, b x + a + i \, d\right )} - i \, e^{\left (a - i \, d\right )}\right )} \log \left (e^{\left (i \, b x + i \, d\right )} - i\right ) - 2 \, {\left (e^{\left (4 i \, b x + a + 3 i \, d\right )} + 2 \, e^{\left (2 i \, b x + a + i \, d\right )} + e^{\left (a - i \, d\right )}\right )} \log \left (e^{\left (i \, b x + i \, d\right )} - 1\right ) - 6 \, e^{\left (3 i \, b x + a + 2 i \, d\right )} - 2 \, e^{\left (i \, b x + a\right )}}{2 \, {\left (b e^{\left (4 i \, b x + 4 i \, d\right )} + 2 \, b e^{\left (2 i \, b x + 2 i \, d\right )} + b\right )}} \] Input:
integrate(exp(a+I*b*x)*csc(b*x+d)*sec(b*x+d)^3,x, algorithm="fricas")
Output:
-1/2*(2*(e^(4*I*b*x + a + 3*I*d) + 2*e^(2*I*b*x + a + I*d) + e^(a - I*d))* log(e^(I*b*x + I*d) + 1) - (I*e^(4*I*b*x + a + 3*I*d) + 2*I*e^(2*I*b*x + a + I*d) + I*e^(a - I*d))*log(e^(I*b*x + I*d) + I) - (-I*e^(4*I*b*x + a + 3 *I*d) - 2*I*e^(2*I*b*x + a + I*d) - I*e^(a - I*d))*log(e^(I*b*x + I*d) - I ) - 2*(e^(4*I*b*x + a + 3*I*d) + 2*e^(2*I*b*x + a + I*d) + e^(a - I*d))*lo g(e^(I*b*x + I*d) - 1) - 6*e^(3*I*b*x + a + 2*I*d) - 2*e^(I*b*x + a))/(b*e ^(4*I*b*x + 4*I*d) + 2*b*e^(2*I*b*x + 2*I*d) + b)
Timed out. \[ \int e^{a+i b x} \csc (d+b x) \sec ^3(d+b x) \, dx=\text {Timed out} \] Input:
integrate(exp(a+I*b*x)*csc(b*x+d)*sec(b*x+d)**3,x)
Output:
Timed out
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1121 vs. \(2 (92) = 184\).
Time = 0.19 (sec) , antiderivative size = 1121, normalized size of antiderivative = 8.76 \[ \int e^{a+i b x} \csc (d+b x) \sec ^3(d+b x) \, dx=\text {Too large to display} \] Input:
integrate(exp(a+I*b*x)*csc(b*x+d)*sec(b*x+d)^3,x, algorithm="maxima")
Output:
(2*((I*cos(d)*e^a + e^a*sin(d))*cos(4*b*x + 4*d) + 2*(I*cos(d)*e^a + e^a*s in(d))*cos(2*b*x + 2*d) + I*cos(d)*e^a - (cos(d)*e^a - I*e^a*sin(d))*sin(4 *b*x + 4*d) - 2*(cos(d)*e^a - I*e^a*sin(d))*sin(2*b*x + 2*d) + 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) + sin(d)^2)) + 4*((cos(d)*e^a - I*e^a* sin(d))*cos(4*b*x + 4*d) + 2*(cos(d)*e^a - I*e^a*sin(d))*cos(2*b*x + 2*d) + cos(d)*e^a + (I*cos(d)*e^a + e^a*sin(d))*sin(4*b*x + 4*d) + 2*(I*cos(d)* e^a + e^a*sin(d))*sin(2*b*x + 2*d) - I*e^a*sin(d))*arctan2(sin(b*x) + sin( d), cos(b*x) - cos(d)) - 4*((cos(d)*e^a - I*e^a*sin(d))*cos(4*b*x + 4*d) + 2*(cos(d)*e^a - I*e^a*sin(d))*cos(2*b*x + 2*d) + cos(d)*e^a - (-I*cos(d)* e^a - e^a*sin(d))*sin(4*b*x + 4*d) - 2*(-I*cos(d)*e^a - e^a*sin(d))*sin(2* b*x + 2*d) - I*e^a*sin(d))*arctan2(sin(b*x) - sin(d), cos(b*x) + cos(d)) - 12*I*cos(3*b*x + 2*d)*e^a - 4*I*cos(b*x)*e^a + 2*((I*cos(d)*e^a + e^a*sin (d))*cos(4*b*x + 4*d) + 2*(I*cos(d)*e^a + e^a*sin(d))*cos(2*b*x + 2*d) + I *cos(d)*e^a - (cos(d)*e^a - I*e^a*sin(d))*sin(4*b*x + 4*d) - 2*(cos(d)*e^a - I*e^a*sin(d))*sin(2*b*x + 2*d) + 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) + 2*((...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (92) = 184\).
Time = 0.16 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.54 \[ \int e^{a+i b x} \csc (d+b x) \sec ^3(d+b x) \, dx=\frac {i \, e^{\left (4 i \, b x + a + 4 i \, d\right )} \log \left (e^{\left (i \, b x + i \, d\right )} + i\right ) + 2 i \, e^{\left (2 i \, b x + a + 2 i \, d\right )} \log \left (e^{\left (i \, b x + i \, d\right )} + i\right ) + i \, e^{a} \log \left (e^{\left (i \, b x + i \, d\right )} + i\right ) - i \, e^{\left (4 i \, b x + a + 4 i \, d\right )} \log \left (e^{\left (i \, b x + i \, d\right )} - i\right ) - 2 i \, e^{\left (2 i \, b x + a + 2 i \, d\right )} \log \left (e^{\left (i \, b x + i \, d\right )} - i\right ) - i \, e^{a} \log \left (e^{\left (i \, b x + i \, d\right )} - i\right ) - 2 \, e^{\left (4 i \, b x + a + 4 i \, d\right )} \log \left (i \, e^{\left (i \, b x + i \, d\right )} + i\right ) - 4 \, e^{\left (2 i \, b x + a + 2 i \, d\right )} \log \left (i \, e^{\left (i \, b x + i \, d\right )} + i\right ) - 2 \, e^{a} \log \left (i \, e^{\left (i \, b x + i \, d\right )} + i\right ) + 2 \, e^{\left (4 i \, b x + a + 4 i \, d\right )} \log \left (-i \, e^{\left (i \, b x + i \, d\right )} + i\right ) + 4 \, e^{\left (2 i \, b x + a + 2 i \, d\right )} \log \left (-i \, e^{\left (i \, b x + i \, d\right )} + i\right ) + 2 \, e^{a} \log \left (-i \, e^{\left (i \, b x + i \, d\right )} + i\right ) + 6 \, e^{\left (3 i \, b x + a + 3 i \, d\right )} + 2 \, e^{\left (i \, b x + a + i \, d\right )}}{2 \, b {\left (e^{\left (4 i \, b x + 5 i \, d\right )} + 2 \, e^{\left (2 i \, b x + 3 i \, d\right )} + e^{\left (i \, d\right )}\right )}} \] Input:
integrate(exp(a+I*b*x)*csc(b*x+d)*sec(b*x+d)^3,x, algorithm="giac")
Output:
1/2*(I*e^(4*I*b*x + a + 4*I*d)*log(e^(I*b*x + I*d) + I) + 2*I*e^(2*I*b*x + a + 2*I*d)*log(e^(I*b*x + I*d) + I) + I*e^a*log(e^(I*b*x + I*d) + I) - I* e^(4*I*b*x + a + 4*I*d)*log(e^(I*b*x + I*d) - I) - 2*I*e^(2*I*b*x + a + 2* I*d)*log(e^(I*b*x + I*d) - I) - I*e^a*log(e^(I*b*x + I*d) - I) - 2*e^(4*I* b*x + a + 4*I*d)*log(I*e^(I*b*x + I*d) + I) - 4*e^(2*I*b*x + a + 2*I*d)*lo g(I*e^(I*b*x + I*d) + I) - 2*e^a*log(I*e^(I*b*x + I*d) + I) + 2*e^(4*I*b*x + a + 4*I*d)*log(-I*e^(I*b*x + I*d) + I) + 4*e^(2*I*b*x + a + 2*I*d)*log( -I*e^(I*b*x + I*d) + I) + 2*e^a*log(-I*e^(I*b*x + I*d) + I) + 6*e^(3*I*b*x + a + 3*I*d) + 2*e^(I*b*x + a + I*d))/(b*(e^(4*I*b*x + 5*I*d) + 2*e^(2*I* b*x + 3*I*d) + e^(I*d)))
Timed out. \[ \int e^{a+i b x} \csc (d+b x) \sec ^3(d+b x) \, dx=\int \frac {{\mathrm {e}}^{a+b\,x\,1{}\mathrm {i}}}{{\cos \left (d+b\,x\right )}^3\,\sin \left (d+b\,x\right )} \,d x \] Input:
int(exp(a + b*x*1i)/(cos(d + b*x)^3*sin(d + b*x)),x)
Output:
int(exp(a + b*x*1i)/(cos(d + b*x)^3*sin(d + b*x)), x)
\[ \int e^{a+i b x} \csc (d+b x) \sec ^3(d+b x) \, dx=e^{a} \left (\int e^{b i x} \csc \left (b x +d \right ) \sec \left (b x +d \right )^{3}d x \right ) \] Input:
int(exp(a+I*b*x)*csc(b*x+d)*sec(b*x+d)^3,x)
Output:
e**a*int(e**(b*i*x)*csc(b*x + d)*sec(b*x + d)**3,x)