Optimal. Leaf size=27 \[ -\frac {i (a+i a \tan (c+d x))^4}{4 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3568, 32}
\begin {gather*} -\frac {i (a+i a \tan (c+d x))^4}{4 a d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 32
Rule 3568
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+i a \tan (c+d x))^3 \, dx &=-\frac {i \text {Subst}\left (\int (a+x)^3 \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=-\frac {i (a+i a \tan (c+d x))^4}{4 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(84\) vs. \(2(27)=54\).
time = 0.48, size = 84, normalized size = 3.11 \begin {gather*} \frac {a^3 \sec (c) \sec ^4(c+d x) (3 i \cos (c)+2 i \cos (c+2 d x)+2 i \cos (3 c+2 d x)-3 \sin (c)+2 \sin (c+2 d x)-2 \sin (3 c+2 d x)+\sin (3 c+4 d x))}{4 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 72 vs. \(2 (23 ) = 46\).
time = 0.23, size = 73, normalized size = 2.70
method | result | size |
risch | \(\frac {4 i a^{3} \left (4 \,{\mathrm e}^{6 i \left (d x +c \right )}+6 \,{\mathrm e}^{4 i \left (d x +c \right )}+4 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}\) | \(58\) |
derivativedivides | \(\frac {-\frac {i a^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4 \cos \left (d x +c \right )^{4}}-\frac {a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{3}}+\frac {3 i a^{3}}{2 \cos \left (d x +c \right )^{2}}+a^{3} \tan \left (d x +c \right )}{d}\) | \(73\) |
default | \(\frac {-\frac {i a^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4 \cos \left (d x +c \right )^{4}}-\frac {a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{3}}+\frac {3 i a^{3}}{2 \cos \left (d x +c \right )^{2}}+a^{3} \tan \left (d x +c \right )}{d}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 21, normalized size = 0.78 \begin {gather*} -\frac {i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4}}{4 \, a 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. 101 vs. \(2 (21) = 42\).
time = 0.36, size = 101, normalized size = 3.74 \begin {gather*} -\frac {4 \, {\left (-4 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 6 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 4 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{3}\right )}}{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} \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*} - i a^{3} \left (\int i \sec ^{2}{\left (c + d x \right )}\, dx + \int \left (- 3 \tan {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\right )\, dx + \int \tan ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \left (- 3 i \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\right )\, dx\right ) \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. 56 vs. \(2 (21) = 42\).
time = 0.58, size = 56, normalized size = 2.07 \begin {gather*} -\frac {i \, a^{3} \tan \left (d x + c\right )^{4} + 4 \, a^{3} \tan \left (d x + c\right )^{3} - 6 i \, a^{3} \tan \left (d x + c\right )^{2} - 4 \, a^{3} \tan \left (d x + c\right )}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.27, size = 56, normalized size = 2.07 \begin {gather*} \frac {-\frac {a^3\,{\mathrm {tan}\left (c+d\,x\right )}^4\,1{}\mathrm {i}}{4}-a^3\,{\mathrm {tan}\left (c+d\,x\right )}^3+\frac {a^3\,{\mathrm {tan}\left (c+d\,x\right )}^2\,3{}\mathrm {i}}{2}+a^3\,\mathrm {tan}\left (c+d\,x\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________