Optimal. Leaf size=46 \[ \frac {i a \sec ^4(c+d x)}{4 d}+\frac {a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3567, 3852}
\begin {gather*} \frac {a \tan ^3(c+d x)}{3 d}+\frac {a \tan (c+d x)}{d}+\frac {i a \sec ^4(c+d x)}{4 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3567
Rule 3852
Rubi steps
\begin {align*} \int \sec ^4(c+d x) (a+i a \tan (c+d x)) \, dx &=\frac {i a \sec ^4(c+d x)}{4 d}+a \int \sec ^4(c+d x) \, dx\\ &=\frac {i a \sec ^4(c+d x)}{4 d}-\frac {a \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac {i a \sec ^4(c+d x)}{4 d}+\frac {a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 43, normalized size = 0.93 \begin {gather*} \frac {i a \sec ^4(c+d x)}{4 d}+\frac {a \left (\tan (c+d x)+\frac {1}{3} \tan ^3(c+d x)\right )}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.29, size = 39, normalized size = 0.85
method | result | size |
derivativedivides | \(\frac {\frac {i a}{4 \cos \left (d x +c \right )^{4}}-a \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(39\) |
default | \(\frac {\frac {i a}{4 \cos \left (d x +c \right )^{4}}-a \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(39\) |
risch | \(\frac {4 i a \left (6 \,{\mathrm e}^{4 i \left (d x +c \right )}+4 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 48, normalized size = 1.04 \begin {gather*} \frac {3 i \, a \tan \left (d x + c\right )^{4} + 4 \, a \tan \left (d x + c\right )^{3} + 6 i \, a \tan \left (d x + c\right )^{2} + 12 \, a \tan \left (d x + c\right )}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 81 vs. \(2 (40) = 80\).
time = 0.35, size = 81, normalized size = 1.76 \begin {gather*} -\frac {4 \, {\left (-6 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} - 4 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a\right )}}{3 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 1.75, size = 48, normalized size = 1.04 \begin {gather*} \begin {cases} \frac {a \left (\frac {\tan ^{3}{\left (c + d x \right )}}{3} + \tan {\left (c + d x \right )}\right ) + \frac {i a \sec ^{4}{\left (c + d x \right )}}{4}}{d} & \text {for}\: d \neq 0 \\x \left (i a \tan {\left (c \right )} + a\right ) \sec ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.58, size = 48, normalized size = 1.04 \begin {gather*} -\frac {-3 i \, a \tan \left (d x + c\right )^{4} - 4 \, a \tan \left (d x + c\right )^{3} - 6 i \, a \tan \left (d x + c\right )^{2} - 12 \, a \tan \left (d x + c\right )}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.24, size = 48, normalized size = 1.04 \begin {gather*} \frac {\frac {1{}\mathrm {i}\,a\,{\mathrm {tan}\left (c+d\,x\right )}^4}{4}+\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3}+\frac {1{}\mathrm {i}\,a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+a\,\mathrm {tan}\left (c+d\,x\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________