Optimal. Leaf size=89 \[ 8 a^4 x-\frac {8 i a^4 \log (\cos (c+d x))}{d}-\frac {4 a^4 \tan (c+d x)}{d}+\frac {i a (a+i a \tan (c+d x))^3}{3 d}+\frac {i \left (a^2+i a^2 \tan (c+d x)\right )^2}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3559, 3558,
3556} \begin {gather*} -\frac {4 a^4 \tan (c+d x)}{d}-\frac {8 i a^4 \log (\cos (c+d x))}{d}+8 a^4 x+\frac {i \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}+\frac {i a (a+i a \tan (c+d x))^3}{3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3556
Rule 3558
Rule 3559
Rubi steps
\begin {align*} \int (a+i a \tan (c+d x))^4 \, dx &=\frac {i a (a+i a \tan (c+d x))^3}{3 d}+(2 a) \int (a+i a \tan (c+d x))^3 \, dx\\ &=\frac {i a (a+i a \tan (c+d x))^3}{3 d}+\frac {i \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}+\left (4 a^2\right ) \int (a+i a \tan (c+d x))^2 \, dx\\ &=8 a^4 x-\frac {4 a^4 \tan (c+d x)}{d}+\frac {i a (a+i a \tan (c+d x))^3}{3 d}+\frac {i \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}+\left (8 i a^4\right ) \int \tan (c+d x) \, dx\\ &=8 a^4 x-\frac {8 i a^4 \log (\cos (c+d x))}{d}-\frac {4 a^4 \tan (c+d x)}{d}+\frac {i a (a+i a \tan (c+d x))^3}{3 d}+\frac {i \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.91, size = 176, normalized size = 1.98 \begin {gather*} \frac {a^4 \sec (c) \sec ^3(c+d x) \left (6 d x \cos (2 c+3 d x)+6 d x \cos (4 c+3 d x)+3 \cos (d x) \left (-2 i+6 d x-3 i \log \left (\cos ^2(c+d x)\right )\right )+3 \cos (2 c+d x) \left (-2 i+6 d x-3 i \log \left (\cos ^2(c+d x)\right )\right )-3 i \cos (2 c+3 d x) \log \left (\cos ^2(c+d x)\right )-3 i \cos (4 c+3 d x) \log \left (\cos ^2(c+d x)\right )-21 \sin (d x)+12 \sin (2 c+d x)-11 \sin (2 c+3 d x)\right )}{6 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.04, size = 61, normalized size = 0.69
method | result | size |
derivativedivides | \(\frac {a^{4} \left (-7 \tan \left (d x +c \right )+\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-2 i \left (\tan ^{2}\left (d x +c \right )\right )+4 i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+8 \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(61\) |
default | \(\frac {a^{4} \left (-7 \tan \left (d x +c \right )+\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-2 i \left (\tan ^{2}\left (d x +c \right )\right )+4 i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+8 \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(61\) |
norman | \(8 a^{4} x -\frac {7 a^{4} \tan \left (d x +c \right )}{d}+\frac {a^{4} \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {2 i a^{4} \left (\tan ^{2}\left (d x +c \right )\right )}{d}+\frac {4 i a^{4} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\) | \(75\) |
risch | \(-\frac {16 a^{4} c}{d}-\frac {4 i a^{4} \left (18 \,{\mathrm e}^{4 i \left (d x +c \right )}+27 \,{\mathrm e}^{2 i \left (d x +c \right )}+11\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}-\frac {8 i a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.57, size = 108, normalized size = 1.21 \begin {gather*} a^{4} x + \frac {{\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{4}}{3 \, d} + \frac {6 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a^{4}}{d} + \frac {2 i \, a^{4} {\left (\frac {1}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )}}{d} + \frac {4 i \, a^{4} \log \left (\sec \left (d x + c\right )\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.43, size = 137, normalized size = 1.54 \begin {gather*} -\frac {4 \, {\left (18 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 27 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 11 i \, a^{4} + 6 \, {\left (i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )\right )}}{3 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, 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 = 0.21, size = 138, normalized size = 1.55 \begin {gather*} - \frac {8 i a^{4} \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac {- 72 i a^{4} e^{4 i c} e^{4 i d x} - 108 i a^{4} e^{2 i c} e^{2 i d x} - 44 i a^{4}}{3 d e^{6 i c} e^{6 i d x} + 9 d e^{4 i c} e^{4 i d x} + 9 d e^{2 i c} e^{2 i d x} + 3 d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 170 vs. \(2 (77) = 154\).
time = 0.51, size = 170, normalized size = 1.91 \begin {gather*} -\frac {4 \, {\left (6 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 18 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 18 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 18 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 27 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 6 i \, a^{4} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 11 i \, a^{4}\right )}}{3 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, 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]
________________________________________________________________________________________
Mupad [B]
time = 3.69, size = 59, normalized size = 0.66 \begin {gather*} \frac {\frac {a^4\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3}-7\,a^4\,\mathrm {tan}\left (c+d\,x\right )+a^4\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,8{}\mathrm {i}-a^4\,{\mathrm {tan}\left (c+d\,x\right )}^2\,2{}\mathrm {i}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________