Optimal. Leaf size=55 \[ -\frac {i (a+i a \tan (c+d x))^4}{2 a^2 d}+\frac {i (a+i a \tan (c+d x))^5}{5 a^3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, 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 = {3568, 45}
\begin {gather*} \frac {i (a+i a \tan (c+d x))^5}{5 a^3 d}-\frac {i (a+i a \tan (c+d x))^4}{2 a^2 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 3568
Rubi steps
\begin {align*} \int \sec ^4(c+d x) (a+i a \tan (c+d x))^2 \, dx &=-\frac {i \text {Subst}\left (\int (a-x) (a+x)^3 \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac {i \text {Subst}\left (\int \left (2 a (a+x)^3-(a+x)^4\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac {i (a+i a \tan (c+d x))^4}{2 a^2 d}+\frac {i (a+i a \tan (c+d x))^5}{5 a^3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.36, size = 77, normalized size = 1.40 \begin {gather*} \frac {a^2 \sec (c) \sec ^5(c+d x) (5 i \cos (d x)+5 i \cos (2 c+d x)+5 \sin (d x)-5 \sin (2 c+d x)+5 \sin (2 c+3 d x)+\sin (4 c+5 d x))}{20 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.24, size = 85, normalized size = 1.55
method | result | size |
risch | \(\frac {8 i a^{2} \left (10 \,{\mathrm e}^{6 i \left (d x +c \right )}+10 \,{\mathrm e}^{4 i \left (d x +c \right )}+5 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{5 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}\) | \(58\) |
derivativedivides | \(\frac {-a^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )+\frac {i a^{2}}{2 \cos \left (d x +c \right )^{4}}-a^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(85\) |
default | \(\frac {-a^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )+\frac {i a^{2}}{2 \cos \left (d x +c \right )^{4}}-a^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 56, normalized size = 1.02 \begin {gather*} -\frac {2 \, a^{2} \tan \left (d x + c\right )^{5} - 5 i \, a^{2} \tan \left (d x + c\right )^{4} - 10 i \, a^{2} \tan \left (d x + c\right )^{2} - 10 \, a^{2} \tan \left (d x + c\right )}{10 \, 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. 113 vs. \(2 (43) = 86\).
time = 0.34, size = 113, normalized size = 2.05 \begin {gather*} -\frac {8 \, {\left (-10 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} - 10 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 5 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{2}\right )}}{5 \, {\left (d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, 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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - a^{2} \left (\int \tan ^{2}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \left (- 2 i \tan {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\right )\, dx + \int \left (- \sec ^{4}{\left (c + d x \right )}\right )\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.56, size = 56, normalized size = 1.02 \begin {gather*} -\frac {2 \, a^{2} \tan \left (d x + c\right )^{5} - 5 i \, a^{2} \tan \left (d x + c\right )^{4} - 10 i \, a^{2} \tan \left (d x + c\right )^{2} - 10 \, a^{2} \tan \left (d x + c\right )}{10 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.22, size = 56, normalized size = 1.02 \begin {gather*} \frac {-\frac {a^2\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5}+\frac {a^2\,{\mathrm {tan}\left (c+d\,x\right )}^4\,1{}\mathrm {i}}{2}+a^2\,{\mathrm {tan}\left (c+d\,x\right )}^2\,1{}\mathrm {i}+a^2\,\mathrm {tan}\left (c+d\,x\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________