Integrand size = 29, antiderivative size = 247 \[ \int e^{2 (a+i b x)} \csc ^4(d+b x) \sec ^3(d+b x) \, dx=-\frac {32 i e^{2 (a-i d)+5 i (d+b x)}}{3 b \left (1-e^{2 i (d+b x)}\right )^3 \left (1+e^{2 i (d+b x)}\right )^2}+\frac {4 i e^{2 (a-i d)+i (d+b x)} \left (5+7 e^{2 i (d+b x)}\right )}{3 b \left (1-e^{4 i (d+b x)}\right )^2}-\frac {i e^{2 (a-i d)+i (d+b x)} \left (5+21 e^{2 i (d+b x)}\right )}{3 b \left (1-e^{4 i (d+b x)}\right )}+\frac {i e^{2 a-2 i d} \arctan \left (e^{i (d+b x)}\right )}{b}-\frac {6 i e^{2 a-2 i d} \text {arctanh}\left (e^{i (d+b x)}\right )}{b} \] Output:
-32/3*I*exp(2*a-2*I*d+5*I*(b*x+d))/b/(1-exp(2*I*(b*x+d)))^3/(1+exp(2*I*(b* x+d)))^2+4/3*I*exp(2*a-2*I*d+I*(b*x+d))*(5+7*exp(2*I*(b*x+d)))/b/(1-exp(4* I*(b*x+d)))^2-1/3*I*exp(2*a-2*I*d+I*(b*x+d))*(5+21*exp(2*I*(b*x+d)))/b/(1- exp(4*I*(b*x+d)))+I*exp(2*a-2*I*d)*arctan(exp(I*(b*x+d)))/b-6*I*exp(2*a-2* I*d)*arctanh(exp(I*(b*x+d)))/b
Time = 0.50 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.85 \[ \int e^{2 (a+i b x)} \csc ^4(d+b x) \sec ^3(d+b x) \, dx=\frac {i e^{2 a-2 i d} \left (\frac {8 e^{i (d+b x)}}{\left (-1+e^{2 i (d+b x)}\right )^3}+\frac {20 e^{i (d+b x)}}{\left (-1+e^{2 i (d+b x)}\right )^2}+\frac {6 e^{i (d+b x)}}{-1+e^{2 i (d+b x)}}-\frac {6 e^{i (d+b x)}}{\left (1+e^{2 i (d+b x)}\right )^2}+\frac {15 e^{i (d+b x)}}{1+e^{2 i (d+b x)}}+3 \arctan \left (e^{i (d+b x)}\right )+9 \log \left (1-e^{i (d+b x)}\right )-9 \log \left (1+e^{i (d+b x)}\right )\right )}{3 b} \] Input:
Integrate[E^(2*(a + I*b*x))*Csc[d + b*x]^4*Sec[d + b*x]^3,x]
Output:
((I/3)*E^(2*a - (2*I)*d)*((8*E^(I*(d + b*x)))/(-1 + E^((2*I)*(d + b*x)))^3 + (20*E^(I*(d + b*x)))/(-1 + E^((2*I)*(d + b*x)))^2 + (6*E^(I*(d + b*x))) /(-1 + E^((2*I)*(d + b*x))) - (6*E^(I*(d + b*x)))/(1 + E^((2*I)*(d + b*x)) )^2 + (15*E^(I*(d + b*x)))/(1 + E^((2*I)*(d + b*x))) + 3*ArcTan[E^(I*(d + b*x))] + 9*Log[1 - E^(I*(d + b*x))] - 9*Log[1 + E^(I*(d + b*x))]))/b
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(550\) vs. \(2(247)=494\).
Time = 0.98 (sec) , antiderivative size = 550, normalized size of antiderivative = 2.23, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, 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.
\(\displaystyle \int e^{2 (a+i b x)} \csc ^4(b x+d) \sec ^3(b x+d) \, dx\) |
\(\Big \downarrow \) 4974 |
\(\displaystyle \int \left (\frac {24 e^{2 a+9 i b x+7 i d}}{\left (-1+e^{2 i (b x+d)}\right )^2}+\frac {16 e^{2 a+9 i b x+7 i d}}{\left (1+e^{2 i (b x+d)}\right )^2}-\frac {24 e^{2 a+9 i b x+7 i d}}{\left (-1+e^{2 i (b x+d)}\right )^3}+\frac {8 e^{2 a+9 i b x+7 i d}}{\left (1+e^{2 i (b x+d)}\right )^3}+\frac {16 e^{2 a+9 i b x+7 i d}}{\left (-1+e^{2 i (b x+d)}\right )^4}-\frac {40 e^{2 a+9 i b x+7 i d}}{-1+e^{4 i (b x+d)}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {i e^{2 a-2 i d} \arctan \left (e^{i (b x+d)}\right )}{b}-\frac {6 i e^{2 a-2 i d} \text {arctanh}\left (e^{i (b x+d)}\right )}{b}+\frac {5 i e^{2 (a-i d)+i (b x+d)}}{b}+\frac {14 i e^{2 (a-i d)+3 i (b x+d)}}{b}-\frac {20 i e^{2 (a-i d)+5 i (b x+d)}}{b}-\frac {35 i e^{2 (a-i d)+3 i (b x+d)}}{3 b \left (1-e^{2 i (b x+d)}\right )}+\frac {21 i e^{2 (a-i d)+5 i (b x+d)}}{b \left (1-e^{2 i (b x+d)}\right )}-\frac {12 i e^{2 (a-i d)+7 i (b x+d)}}{b \left (1-e^{2 i (b x+d)}\right )}+\frac {7 i e^{2 (a-i d)+5 i (b x+d)}}{b \left (1+e^{2 i (b x+d)}\right )}+\frac {8 i e^{2 (a-i d)+7 i (b x+d)}}{b \left (1+e^{2 i (b x+d)}\right )}+\frac {14 i e^{2 (a-i d)+5 i (b x+d)}}{3 b \left (1-e^{2 i (b x+d)}\right )^2}-\frac {6 i e^{2 (a-i d)+7 i (b x+d)}}{b \left (1-e^{2 i (b x+d)}\right )^2}+\frac {2 i e^{2 (a-i d)+7 i (b x+d)}}{b \left (1+e^{2 i (b x+d)}\right )^2}-\frac {8 i e^{2 (a-i d)+7 i (b x+d)}}{3 b \left (1-e^{2 i (b x+d)}\right )^3}\) |
Input:
Int[E^(2*(a + I*b*x))*Csc[d + b*x]^4*Sec[d + b*x]^3,x]
Output:
((5*I)*E^(2*(a - I*d) + I*(d + b*x)))/b + ((14*I)*E^(2*(a - I*d) + (3*I)*( d + b*x)))/b - ((20*I)*E^(2*(a - I*d) + (5*I)*(d + b*x)))/b - (((8*I)/3)*E ^(2*(a - I*d) + (7*I)*(d + b*x)))/(b*(1 - E^((2*I)*(d + b*x)))^3) + (((14* I)/3)*E^(2*(a - I*d) + (5*I)*(d + b*x)))/(b*(1 - E^((2*I)*(d + b*x)))^2) - ((6*I)*E^(2*(a - I*d) + (7*I)*(d + b*x)))/(b*(1 - E^((2*I)*(d + b*x)))^2) - (((35*I)/3)*E^(2*(a - I*d) + (3*I)*(d + b*x)))/(b*(1 - E^((2*I)*(d + b* x)))) + ((21*I)*E^(2*(a - I*d) + (5*I)*(d + b*x)))/(b*(1 - E^((2*I)*(d + b *x)))) - ((12*I)*E^(2*(a - I*d) + (7*I)*(d + b*x)))/(b*(1 - E^((2*I)*(d + b*x)))) + ((2*I)*E^(2*(a - I*d) + (7*I)*(d + b*x)))/(b*(1 + E^((2*I)*(d + b*x)))^2) + ((7*I)*E^(2*(a - I*d) + (5*I)*(d + b*x)))/(b*(1 + E^((2*I)*(d + b*x)))) + ((8*I)*E^(2*(a - I*d) + (7*I)*(d + b*x)))/(b*(1 + E^((2*I)*(d + b*x)))) + (I*E^(2*a - (2*I)*d)*ArcTan[E^(I*(d + b*x))])/b - ((6*I)*E^(2* a - (2*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 = 26.70 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.77
method | result | size |
risch | \(-\frac {i \left (15 \,{\mathrm e}^{11 i b x} {\mathrm e}^{9 i d} {\mathrm e}^{2 a}-36 \,{\mathrm e}^{9 i b x} {\mathrm e}^{7 i d} {\mathrm e}^{2 a}-14 \,{\mathrm e}^{7 i b x} {\mathrm e}^{5 i d} {\mathrm e}^{2 a}-4 \,{\mathrm e}^{5 i b x} {\mathrm e}^{3 i d} {\mathrm e}^{2 a}+7 \,{\mathrm e}^{3 i b x} {\mathrm e}^{i d} {\mathrm e}^{2 a}\right )}{3 \left (-1+{\mathrm e}^{2 i \left (b x +d \right )}\right )^{3} \left (1+{\mathrm e}^{2 i \left (b x +d \right )}\right )^{2} b}-\frac {6 i {\mathrm e}^{2 a} {\mathrm e}^{-2 i d} \operatorname {arctanh}\left ({\mathrm e}^{i \left (b x +d \right )}\right )}{b}+\frac {i {\mathrm e}^{2 a} {\mathrm e}^{-2 i d} \arctan \left ({\mathrm e}^{i \left (b x +d \right )}\right )}{b}+\frac {5 i {\mathrm e}^{2 a} {\mathrm e}^{i b x} {\mathrm e}^{-i d}}{b}\) | \(191\) |
Input:
int(exp(2*a+2*I*b*x)*csc(b*x+d)^4*sec(b*x+d)^3,x,method=_RETURNVERBOSE)
Output:
-1/3*I/(-1+exp(2*I*(b*x+d)))^3/(1+exp(2*I*(b*x+d)))^2/b*(15*exp(11*I*b*x)* exp(9*I*d)*exp(2*a)-36*exp(9*I*b*x)*exp(7*I*d)*exp(2*a)-14*exp(7*I*b*x)*ex p(5*I*d)*exp(2*a)-4*exp(5*I*b*x)*exp(3*I*d)*exp(2*a)+7*exp(3*I*b*x)*exp(I* d)*exp(2*a))-6*I*exp(2*a)/b*exp(-2*I*d)*arctanh(exp(I*(b*x+d)))+I/b*exp(2* a)*exp(-2*I*d)*arctan(exp(I*(b*x+d)))+5*I/b*exp(2*a)*exp(I*b*x)*exp(-I*d)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (176) = 352\).
Time = 0.09 (sec) , antiderivative size = 497, normalized size of antiderivative = 2.01 \[ \int e^{2 (a+i b x)} \csc ^4(d+b x) \sec ^3(d+b x) \, dx =\text {Too large to display} \] Input:
integrate(exp(2*a+2*I*b*x)*csc(b*x+d)^4*sec(b*x+d)^3,x, algorithm="fricas" )
Output:
-1/6*(18*(I*e^(10*I*b*x + 2*a + 8*I*d) - I*e^(8*I*b*x + 2*a + 6*I*d) - 2*I *e^(6*I*b*x + 2*a + 4*I*d) + 2*I*e^(4*I*b*x + 2*a + 2*I*d) + I*e^(2*I*b*x + 2*a) - I*e^(2*a - 2*I*d))*log(e^(I*b*x + I*d) + 1) + 3*(e^(10*I*b*x + 2* a + 8*I*d) - e^(8*I*b*x + 2*a + 6*I*d) - 2*e^(6*I*b*x + 2*a + 4*I*d) + 2*e ^(4*I*b*x + 2*a + 2*I*d) + e^(2*I*b*x + 2*a) - e^(2*a - 2*I*d))*log(e^(I*b *x + I*d) + I) - 3*(e^(10*I*b*x + 2*a + 8*I*d) - e^(8*I*b*x + 2*a + 6*I*d) - 2*e^(6*I*b*x + 2*a + 4*I*d) + 2*e^(4*I*b*x + 2*a + 2*I*d) + e^(2*I*b*x + 2*a) - e^(2*a - 2*I*d))*log(e^(I*b*x + I*d) - I) + 18*(-I*e^(10*I*b*x + 2*a + 8*I*d) + I*e^(8*I*b*x + 2*a + 6*I*d) + 2*I*e^(6*I*b*x + 2*a + 4*I*d) - 2*I*e^(4*I*b*x + 2*a + 2*I*d) - I*e^(2*I*b*x + 2*a) + I*e^(2*a - 2*I*d) )*log(e^(I*b*x + I*d) - 1) - 42*I*e^(9*I*b*x + 2*a + 7*I*d) + 32*I*e^(7*I* b*x + 2*a + 5*I*d) - 68*I*e^(5*I*b*x + 2*a + 3*I*d) - 16*I*e^(3*I*b*x + 2* a + I*d) + 30*I*e^(I*b*x + 2*a - I*d))/(b*e^(10*I*b*x + 10*I*d) - b*e^(8*I *b*x + 8*I*d) - 2*b*e^(6*I*b*x + 6*I*d) + 2*b*e^(4*I*b*x + 4*I*d) + b*e^(2 *I*b*x + 2*I*d) - b)
Timed out. \[ \int e^{2 (a+i b x)} \csc ^4(d+b x) \sec ^3(d+b x) \, dx=\text {Timed out} \] Input:
integrate(exp(2*a+2*I*b*x)*csc(b*x+d)**4*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. 2798 vs. \(2 (176) = 352\).
Time = 0.31 (sec) , antiderivative size = 2798, normalized size of antiderivative = 11.33 \[ \int e^{2 (a+i b x)} \csc ^4(d+b x) \sec ^3(d+b x) \, dx=\text {Too large to display} \] Input:
integrate(exp(2*a+2*I*b*x)*csc(b*x+d)^4*sec(b*x+d)^3,x, algorithm="maxima" )
Output:
-(6*((cos(2*d)*e^(2*a) - I*e^(2*a)*sin(2*d))*cos(10*b*x + 11*d) - (cos(2*d )*e^(2*a) - I*e^(2*a)*sin(2*d))*cos(8*b*x + 9*d) - 2*(cos(2*d)*e^(2*a) - I *e^(2*a)*sin(2*d))*cos(6*b*x + 7*d) + 2*(cos(2*d)*e^(2*a) - I*e^(2*a)*sin( 2*d))*cos(4*b*x + 5*d) + (cos(2*d)*e^(2*a) - I*e^(2*a)*sin(2*d))*cos(2*b*x + 3*d) - (cos(d)*e^(2*a) + I*e^(2*a)*sin(d))*cos(2*d) - (-I*cos(2*d)*e^(2 *a) - e^(2*a)*sin(2*d))*sin(10*b*x + 11*d) - (I*cos(2*d)*e^(2*a) + e^(2*a) *sin(2*d))*sin(8*b*x + 9*d) - 2*(I*cos(2*d)*e^(2*a) + e^(2*a)*sin(2*d))*si n(6*b*x + 7*d) - 2*(-I*cos(2*d)*e^(2*a) - e^(2*a)*sin(2*d))*sin(4*b*x + 5* d) - (-I*cos(2*d)*e^(2*a) - e^(2*a)*sin(2*d))*sin(2*b*x + 3*d) - (-I*cos(d )*e^(2*a) + e^(2*a)*sin(d))*sin(2*d))*arctan2(2*(cos(b*x + 2*d)*cos(d) + s in(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)) - 36*((I*cos(2*d)*e^(2*a) + e^(2*a)*sin(2*d))*cos(10*b*x + 11* d) + (-I*cos(2*d)*e^(2*a) - e^(2*a)*sin(2*d))*cos(8*b*x + 9*d) + 2*(-I*cos (2*d)*e^(2*a) - e^(2*a)*sin(2*d))*cos(6*b*x + 7*d) + 2*(I*cos(2*d)*e^(2*a) + e^(2*a)*sin(2*d))*cos(4*b*x + 5*d) + (I*cos(2*d)*e^(2*a) + e^(2*a)*sin( 2*d))*cos(2*b*x + 3*d) + (-I*cos(d)*e^(2*a) + e^(2*a)*sin(d))*cos(2*d) - ( cos(2*d)*e^(2*a) - I*e^(2*a)*sin(2*d))*sin(10*b*x + 11*d) + (cos(2*d)*e...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 751 vs. \(2 (176) = 352\).
Time = 0.27 (sec) , antiderivative size = 751, normalized size of antiderivative = 3.04 \[ \int e^{2 (a+i b x)} \csc ^4(d+b x) \sec ^3(d+b x) \, dx=\text {Too large to display} \] Input:
integrate(exp(2*a+2*I*b*x)*csc(b*x+d)^4*sec(b*x+d)^3,x, algorithm="giac")
Output:
1/6*(-18*I*e^(10*I*b*x + 2*a + 8*I*d)*log(e^(I*b*x + I*d) + 1) + 18*I*e^(8 *I*b*x + 2*a + 6*I*d)*log(e^(I*b*x + I*d) + 1) + 36*I*e^(6*I*b*x + 2*a + 4 *I*d)*log(e^(I*b*x + I*d) + 1) - 36*I*e^(4*I*b*x + 2*a + 2*I*d)*log(e^(I*b *x + I*d) + 1) - 18*I*e^(2*I*b*x + 2*a)*log(e^(I*b*x + I*d) + 1) + 18*I*e^ (2*a - 2*I*d)*log(e^(I*b*x + I*d) + 1) + 18*I*e^(10*I*b*x + 2*a + 8*I*d)*l og(e^(I*b*x + I*d) - 1) - 18*I*e^(8*I*b*x + 2*a + 6*I*d)*log(e^(I*b*x + I* d) - 1) - 36*I*e^(6*I*b*x + 2*a + 4*I*d)*log(e^(I*b*x + I*d) - 1) + 36*I*e ^(4*I*b*x + 2*a + 2*I*d)*log(e^(I*b*x + I*d) - 1) + 18*I*e^(2*I*b*x + 2*a) *log(e^(I*b*x + I*d) - 1) - 18*I*e^(2*a - 2*I*d)*log(e^(I*b*x + I*d) - 1) + 3*e^(10*I*b*x + 2*a + 8*I*d)*log(I*e^(I*b*x + I*d) + 1) - 3*e^(8*I*b*x + 2*a + 6*I*d)*log(I*e^(I*b*x + I*d) + 1) - 6*e^(6*I*b*x + 2*a + 4*I*d)*log (I*e^(I*b*x + I*d) + 1) + 6*e^(4*I*b*x + 2*a + 2*I*d)*log(I*e^(I*b*x + I*d ) + 1) + 3*e^(2*I*b*x + 2*a)*log(I*e^(I*b*x + I*d) + 1) - 3*e^(2*a - 2*I*d )*log(I*e^(I*b*x + I*d) + 1) - 3*e^(10*I*b*x + 2*a + 8*I*d)*log(-I*e^(I*b* x + I*d) + 1) + 3*e^(8*I*b*x + 2*a + 6*I*d)*log(-I*e^(I*b*x + I*d) + 1) + 6*e^(6*I*b*x + 2*a + 4*I*d)*log(-I*e^(I*b*x + I*d) + 1) - 6*e^(4*I*b*x + 2 *a + 2*I*d)*log(-I*e^(I*b*x + I*d) + 1) - 3*e^(2*I*b*x + 2*a)*log(-I*e^(I* b*x + I*d) + 1) + 3*e^(2*a - 2*I*d)*log(-I*e^(I*b*x + I*d) + 1) + 42*I*e^( 9*I*b*x + 2*a + 7*I*d) - 32*I*e^(7*I*b*x + 2*a + 5*I*d) + 68*I*e^(5*I*b*x + 2*a + 3*I*d) + 16*I*e^(3*I*b*x + 2*a + I*d) - 30*I*e^(I*b*x + 2*a - I...
Timed out. \[ \int e^{2 (a+i b x)} \csc ^4(d+b x) \sec ^3(d+b x) \, dx=\int \frac {{\mathrm {e}}^{2\,a+b\,x\,2{}\mathrm {i}}}{{\cos \left (d+b\,x\right )}^3\,{\sin \left (d+b\,x\right )}^4} \,d x \] Input:
int(exp(2*a + b*x*2i)/(cos(d + b*x)^3*sin(d + b*x)^4),x)
Output:
int(exp(2*a + b*x*2i)/(cos(d + b*x)^3*sin(d + b*x)^4), x)
\[ \int e^{2 (a+i b x)} \csc ^4(d+b x) \sec ^3(d+b x) \, dx=e^{2 a} \left (\int e^{2 b i x} \csc \left (b x +d \right )^{4} \sec \left (b x +d \right )^{3}d x \right ) \] Input:
int(exp(2*a+2*I*b*x)*csc(b*x+d)^4*sec(b*x+d)^3,x)
Output:
e**(2*a)*int(e**(2*b*i*x)*csc(b*x + d)**4*sec(b*x + d)**3,x)