Optimal. Leaf size=55 \[ \frac{i (a+i a \tan (c+d x))^6}{6 a^3 d}-\frac{2 i (a+i a \tan (c+d x))^5}{5 a^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0426855, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3487, 43} \[ \frac{i (a+i a \tan (c+d x))^6}{6 a^3 d}-\frac{2 i (a+i a \tan (c+d x))^5}{5 a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \sec ^4(c+d x) (a+i a \tan (c+d x))^3 \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x) (a+x)^4 \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (2 a (a+x)^4-(a+x)^5\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac{2 i (a+i a \tan (c+d x))^5}{5 a^2 d}+\frac{i (a+i a \tan (c+d x))^6}{6 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.888484, size = 97, normalized size = 1.76 \[ \frac{a^3 \sec (c) \sec ^6(c+d x) (15 \sin (c+2 d x)-15 \sin (3 c+2 d x)+12 \sin (3 c+4 d x)+2 \sin (5 c+6 d x)+15 i \cos (c+2 d x)+15 i \cos (3 c+2 d x)-20 \sin (c)+20 i \cos (c))}{60 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.06, size = 128, normalized size = 2.3 \begin{align*}{\frac{1}{d} \left ( -i{a}^{3} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{12\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}} \right ) -3\,{a}^{3} \left ( 1/5\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+2/15\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) +{\frac{{\frac{3\,i}{4}}{a}^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{a}^{3} \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \tan \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.09965, size = 111, normalized size = 2.02 \begin{align*} \frac{-10 i \, a^{3} \tan \left (d x + c\right )^{6} - 36 \, a^{3} \tan \left (d x + c\right )^{5} + 30 i \, a^{3} \tan \left (d x + c\right )^{4} - 40 \, a^{3} \tan \left (d x + c\right )^{3} + 90 i \, a^{3} \tan \left (d x + c\right )^{2} + 60 \, a^{3} \tan \left (d x + c\right )}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.12241, size = 419, normalized size = 7.62 \begin{align*} \frac{480 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} + 640 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} + 480 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 192 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 32 i \, a^{3}}{15 \,{\left (d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{3} \left (\int - 3 \tan ^{2}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int 3 i \tan{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int - i \tan ^{3}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \sec ^{4}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.25639, size = 111, normalized size = 2.02 \begin{align*} -\frac{5 i \, a^{3} \tan \left (d x + c\right )^{6} + 18 \, a^{3} \tan \left (d x + c\right )^{5} - 15 i \, a^{3} \tan \left (d x + c\right )^{4} + 20 \, a^{3} \tan \left (d x + c\right )^{3} - 45 i \, a^{3} \tan \left (d x + c\right )^{2} - 30 \, a^{3} \tan \left (d x + c\right )}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]