Optimal. Leaf size=82 \[ \frac{i (a-i a \tan (c+d x))^8}{8 a^{11} d}-\frac{4 i (a-i a \tan (c+d x))^7}{7 a^{10} d}+\frac{2 i (a-i a \tan (c+d x))^6}{3 a^9 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0613849, antiderivative size = 82, 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} \[ \frac{i (a-i a \tan (c+d x))^8}{8 a^{11} d}-\frac{4 i (a-i a \tan (c+d x))^7}{7 a^{10} d}+\frac{2 i (a-i a \tan (c+d x))^6}{3 a^9 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \frac{\sec ^{12}(c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^5 (a+x)^2 \, dx,x,i a \tan (c+d x)\right )}{a^{11} d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (4 a^2 (a-x)^5-4 a (a-x)^6+(a-x)^7\right ) \, dx,x,i a \tan (c+d x)\right )}{a^{11} d}\\ &=\frac{2 i (a-i a \tan (c+d x))^6}{3 a^9 d}-\frac{4 i (a-i a \tan (c+d x))^7}{7 a^{10} d}+\frac{i (a-i a \tan (c+d x))^8}{8 a^{11} d}\\ \end{align*}
Mathematica [A] time = 0.468781, size = 106, normalized size = 1.29 \[ \frac{\sec (c) \sec ^8(c+d x) (28 \sin (c+2 d x)-28 \sin (3 c+2 d x)+28 \sin (3 c+4 d x)+8 \sin (5 c+6 d x)+\sin (7 c+8 d x)-28 i \cos (c+2 d x)-28 i \cos (3 c+2 d x)-35 \sin (c)-35 i \cos (c))}{168 a^3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.079, size = 89, normalized size = 1.1 \begin{align*}{\frac{1}{d{a}^{3}} \left ( \tan \left ( dx+c \right ) +{\frac{i}{8}} \left ( \tan \left ( dx+c \right ) \right ) ^{8}-{\frac{3\, \left ( \tan \left ( dx+c \right ) \right ) ^{7}}{7}}-{\frac{i}{6}} \left ( \tan \left ( dx+c \right ) \right ) ^{6}- \left ( \tan \left ( dx+c \right ) \right ) ^{5}-{\frac{5\,i}{4}} \left ( \tan \left ( dx+c \right ) \right ) ^{4}-{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3}}-{\frac{3\,i}{2}} \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.986612, size = 117, normalized size = 1.43 \begin{align*} -\frac{-21 i \, \tan \left (d x + c\right )^{8} + 72 \, \tan \left (d x + c\right )^{7} + 28 i \, \tan \left (d x + c\right )^{6} + 168 \, \tan \left (d x + c\right )^{5} + 210 i \, \tan \left (d x + c\right )^{4} + 56 \, \tan \left (d x + c\right )^{3} + 252 i \, \tan \left (d x + c\right )^{2} - 168 \, \tan \left (d x + c\right )}{168 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.34707, size = 443, normalized size = 5.4 \begin{align*} \frac{896 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 256 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 32 i}{21 \,{\left (a^{3} d e^{\left (16 i \, d x + 16 i \, c\right )} + 8 \, a^{3} d e^{\left (14 i \, d x + 14 i \, c\right )} + 28 \, a^{3} d e^{\left (12 i \, d x + 12 i \, c\right )} + 56 \, a^{3} d e^{\left (10 i \, d x + 10 i \, c\right )} + 70 \, a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + 56 \, a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )} + 28 \, a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 8 \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.20249, size = 117, normalized size = 1.43 \begin{align*} -\frac{-21 i \, \tan \left (d x + c\right )^{8} + 72 \, \tan \left (d x + c\right )^{7} + 28 i \, \tan \left (d x + c\right )^{6} + 168 \, \tan \left (d x + c\right )^{5} + 210 i \, \tan \left (d x + c\right )^{4} + 56 \, \tan \left (d x + c\right )^{3} + 252 i \, \tan \left (d x + c\right )^{2} - 168 \, \tan \left (d x + c\right )}{168 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]