Optimal. Leaf size=34 \[ -\frac {i \tan ^3(c+d x) (-\cot (c+d x)+i)^3}{3 a^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3088, 848, 37} \[ -\frac {i \tan ^3(c+d x) (-\cot (c+d x)+i)^3}{3 a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 848
Rule 3088
Rubi steps
\begin {align*} \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^4 (i a+a x)^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\left (-\frac {i}{a}+\frac {x}{a}\right )^2}{x^4} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac {i (i-\cot (c+d x))^3 \tan ^3(c+d x)}{3 a^2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.26, size = 68, normalized size = 2.00 \[ -\frac {\sec (c) \sec ^3(c+d x) (3 \sin (2 c+d x)-2 \sin (2 c+3 d x)+3 i \cos (2 c+d x)-3 \sin (d x)+3 i \cos (d x))}{6 a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.64, size = 54, normalized size = 1.59 \[ \frac {8 i}{3 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.63, size = 35, normalized size = 1.03 \[ -\frac {\tan \left (d x + c\right )^{3} + 3 i \, \tan \left (d x + c\right )^{2} - 3 \, \tan \left (d x + c\right )}{3 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.26, size = 36, normalized size = 1.06 \[ \frac {\tan \left (d x +c \right )-\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-i \left (\tan ^{2}\left (d x +c \right )\right )}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.51, size = 35, normalized size = 1.03 \[ -\frac {\tan \left (d x + c\right )^{3} + 3 i \, \tan \left (d x + c\right )^{2} - 3 \, \tan \left (d x + c\right )}{3 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.73, size = 49, normalized size = 1.44 \[ -\frac {\sin \left (c+d\,x\right )\,\left (-4\,{\cos \left (c+d\,x\right )}^2+3{}\mathrm {i}\,\sin \left (c+d\,x\right )\,\cos \left (c+d\,x\right )+1\right )}{3\,a^2\,d\,{\cos \left (c+d\,x\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sec ^{4}{\left (c + d x \right )}}{- \sin ^{2}{\left (c + d x \right )} + 2 i \sin {\left (c + d x \right )} \cos {\left (c + d x \right )} + \cos ^{2}{\left (c + d x \right )}}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________