Integrand size = 31, antiderivative size = 60 \[ \int \frac {\sec ^4(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx=\frac {\text {arctanh}(\sin (c+d x))}{2 a d}-\frac {i \sec ^3(c+d x)}{3 a d}+\frac {\sec (c+d x) \tan (c+d x)}{2 a d} \]
Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.90 \[ \int \frac {\sec ^4(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx=-\frac {i \left (12 i \text {arctanh}\left (\sin (c)+\cos (c) \tan \left (\frac {d x}{2}\right )\right )+\sec ^3(c+d x) (4+3 i \sin (2 (c+d x)))\right )}{12 a d} \]
((-1/12*I)*((12*I)*ArcTanh[Sin[c] + Cos[c]*Tan[(d*x)/2]] + Sec[c + d*x]^3* (4 + (3*I)*Sin[2*(c + d*x)])))/(a*d)
Time = 0.34 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3042, 3571, 3042, 3569, 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.
\(\displaystyle \int \frac {\sec ^4(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\cos (c+d x)^4 (a \cos (c+d x)+i a \sin (c+d x))}dx\) |
\(\Big \downarrow \) 3571 |
\(\displaystyle -\frac {i \int \sec ^4(c+d x) (i a \cos (c+d x)+a \sin (c+d x))dx}{a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {i \int \frac {i a \cos (c+d x)+a \sin (c+d x)}{\cos (c+d x)^4}dx}{a^2}\) |
\(\Big \downarrow \) 3569 |
\(\displaystyle -\frac {i \int \left (i a \sec ^3(c+d x)+a \tan (c+d x) \sec ^3(c+d x)\right )dx}{a^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {i \left (\frac {i a \text {arctanh}(\sin (c+d x))}{2 d}+\frac {a \sec ^3(c+d x)}{3 d}+\frac {i a \tan (c+d x) \sec (c+d x)}{2 d}\right )}{a^2}\) |
((-I)*(((I/2)*a*ArcTanh[Sin[c + d*x]])/d + (a*Sec[c + d*x]^3)/(3*d) + ((I/ 2)*a*Sec[c + d*x]*Tan[c + d*x])/d))/a^2
3.2.59.3.1 Defintions of rubi rules used
Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*si n[(c_.) + (d_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandTrig[cos[c + d*x]^m*(a *cos[c + d*x] + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] && Inte gerQ[m] && IGtQ[n, 0]
Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*si n[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[a^n*b^n Int[Cos[c + d*x]^m /(b*Cos[c + d*x] + a*Sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, m}, x] & & EqQ[a^2 + b^2, 0] && ILtQ[n, 0]
Time = 0.66 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.67
method | result | size |
risch | \(-\frac {i \left (3 \,{\mathrm e}^{5 i \left (d x +c \right )}+8 \,{\mathrm e}^{3 i \left (d x +c \right )}-3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {\ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{2 a d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 a d}\) | \(100\) |
norman | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {2 i}{3 a d}+\frac {2 i \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{a d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{3}}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a d}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a d}\) | \(123\) |
derivativedivides | \(\frac {\frac {i}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {2 \left (\frac {1}{4}+\frac {i}{4}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {2 \left (\frac {1}{4}+\frac {i}{4}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2}-\frac {i}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {2 \left (\frac {1}{4}-\frac {i}{4}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {2 \left (-\frac {1}{4}+\frac {i}{4}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2}}{a d}\) | \(138\) |
default | \(\frac {\frac {i}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {2 \left (\frac {1}{4}+\frac {i}{4}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {2 \left (\frac {1}{4}+\frac {i}{4}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2}-\frac {i}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {2 \left (\frac {1}{4}-\frac {i}{4}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {2 \left (-\frac {1}{4}+\frac {i}{4}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2}}{a d}\) | \(138\) |
-1/3*I/d/a/(exp(2*I*(d*x+c))+1)^3*(3*exp(5*I*(d*x+c))+8*exp(3*I*(d*x+c))-3 *exp(I*(d*x+c)))+1/2/a/d*ln(I+exp(I*(d*x+c)))-1/2/a/d*ln(exp(I*(d*x+c))-I)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (52) = 104\).
Time = 0.25 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.90 \[ \int \frac {\sec ^4(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx=\frac {3 \, {\left (e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 3 \, {\left (e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right ) - 6 i \, e^{\left (5 i \, d x + 5 i \, c\right )} - 16 i \, e^{\left (3 i \, d x + 3 i \, c\right )} + 6 i \, e^{\left (i \, d x + i \, c\right )}}{6 \, {\left (a d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )}} \]
1/6*(3*(e^(6*I*d*x + 6*I*c) + 3*e^(4*I*d*x + 4*I*c) + 3*e^(2*I*d*x + 2*I*c ) + 1)*log(e^(I*d*x + I*c) + I) - 3*(e^(6*I*d*x + 6*I*c) + 3*e^(4*I*d*x + 4*I*c) + 3*e^(2*I*d*x + 2*I*c) + 1)*log(e^(I*d*x + I*c) - I) - 6*I*e^(5*I* d*x + 5*I*c) - 16*I*e^(3*I*d*x + 3*I*c) + 6*I*e^(I*d*x + I*c))/(a*d*e^(6*I *d*x + 6*I*c) + 3*a*d*e^(4*I*d*x + 4*I*c) + 3*a*d*e^(2*I*d*x + 2*I*c) + a* d)
\[ \int \frac {\sec ^4(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx=\frac {\int \frac {\sec ^{4}{\left (c + d x \right )}}{i \sin {\left (c + d x \right )} + \cos {\left (c + d x \right )}}\, dx}{a} \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (52) = 104\).
Time = 0.20 (sec) , antiderivative size = 186, normalized size of antiderivative = 3.10 \[ \int \frac {\sec ^4(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx=\frac {\frac {4 \, {\left (\frac {3 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {6 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {3 i \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + 2\right )}}{6 i \, a - \frac {18 i \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {18 i \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {6 i \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} - \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a}}{2 \, d} \]
1/2*(4*(3*I*sin(d*x + c)/(cos(d*x + c) + 1) + 6*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 3*I*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 2)/(6*I*a - 18*I*a*s in(d*x + c)^2/(cos(d*x + c) + 1)^2 + 18*I*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 6*I*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6) + log(sin(d*x + c)/(cos (d*x + c) + 1) + 1)/a - log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a)/d
Time = 0.30 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.65 \[ \int \frac {\sec ^4(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx=\frac {\frac {3 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a} - \frac {3 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}{a} + \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 i\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a}}{6 \, d} \]
1/6*(3*log(tan(1/2*d*x + 1/2*c) + 1)/a - 3*log(tan(1/2*d*x + 1/2*c) - 1)/a + 2*(3*tan(1/2*d*x + 1/2*c)^5 + 6*I*tan(1/2*d*x + 1/2*c)^4 - 3*tan(1/2*d* x + 1/2*c) + 2*I)/((tan(1/2*d*x + 1/2*c)^2 - 1)^3*a))/d
Time = 24.86 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.93 \[ \int \frac {\sec ^4(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx=\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}+\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{a}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,2{}\mathrm {i}}{a}+\frac {2{}\mathrm {i}}{3\,a}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]