Optimal. Leaf size=99 \[ \frac {4 i a^3 (c-i c \tan (e+f x))^n}{f n}-\frac {4 i a^3 (c-i c \tan (e+f x))^{1+n}}{c f (1+n)}+\frac {i a^3 (c-i c \tan (e+f x))^{2+n}}{c^2 f (2+n)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.11, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3603, 3568, 45}
\begin {gather*} \frac {i a^3 (c-i c \tan (e+f x))^{n+2}}{c^2 f (n+2)}+\frac {4 i a^3 (c-i c \tan (e+f x))^n}{f n}-\frac {4 i a^3 (c-i c \tan (e+f x))^{n+1}}{c f (n+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 3568
Rule 3603
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^n \, dx &=\left (a^3 c^3\right ) \int \sec ^6(e+f x) (c-i c \tan (e+f x))^{-3+n} \, dx\\ &=\frac {\left (i a^3\right ) \text {Subst}\left (\int (c-x)^2 (c+x)^{-1+n} \, dx,x,-i c \tan (e+f x)\right )}{c^2 f}\\ &=\frac {\left (i a^3\right ) \text {Subst}\left (\int \left (4 c^2 (c+x)^{-1+n}-4 c (c+x)^n+(c+x)^{1+n}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c^2 f}\\ &=\frac {4 i a^3 (c-i c \tan (e+f x))^n}{f n}-\frac {4 i a^3 (c-i c \tan (e+f x))^{1+n}}{c f (1+n)}+\frac {i a^3 (c-i c \tan (e+f x))^{2+n}}{c^2 f (2+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 2.96, size = 110, normalized size = 1.11 \begin {gather*} \frac {i a^3 e^{n (-\log (c \sec (e+f x))+\log (c-i c \tan (e+f x)))} \sec ^2(e+f x) (c \sec (e+f x))^n \left (2 (2+n)+\left (4+3 n+n^2\right ) \cos (2 (e+f x))+i n (3+n) \sin (2 (e+f x))\right )}{f n (1+n) (2+n)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.29, size = 129, normalized size = 1.30
method | result | size |
norman | \(\frac {i a^{3} \left (n^{2}+5 n +8\right ) {\mathrm e}^{n \ln \left (c -i c \tan \left (f x +e \right )\right )}}{\left (1+n \right ) \left (2+n \right ) f n}-\frac {i a^{3} \left (\tan ^{2}\left (f x +e \right )\right ) {\mathrm e}^{n \ln \left (c -i c \tan \left (f x +e \right )\right )}}{f \left (2+n \right )}-\frac {2 a^{3} \left (3+n \right ) \tan \left (f x +e \right ) {\mathrm e}^{n \ln \left (c -i c \tan \left (f x +e \right )\right )}}{\left (1+n \right ) f \left (2+n \right )}\) | \(129\) |
risch | \(\text {Expression too large to display}\) | \(1052\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 576 vs. \(2 (90) = 180\).
time = 0.57, size = 576, normalized size = 5.82 \begin {gather*} \frac {2^{n + 3} a^{3} c^{n} \cos \left (n \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right ) - i \cdot 2^{n + 3} a^{3} c^{n} \sin \left (n \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right ) + 8 \, {\left (a^{3} c^{n} n + 2 \, a^{3} c^{n}\right )} 2^{n} \cos \left (-2 \, f x + n \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right ) - 2 \, e\right ) + 4 \, {\left (a^{3} c^{n} n^{2} + 3 \, a^{3} c^{n} n + 2 \, a^{3} c^{n}\right )} 2^{n} \cos \left (-4 \, f x + n \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right ) - 4 \, e\right ) + 8 \, {\left (-i \, a^{3} c^{n} n - 2 i \, a^{3} c^{n}\right )} 2^{n} \sin \left (-2 \, f x + n \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right ) - 2 \, e\right ) + 4 \, {\left (-i \, a^{3} c^{n} n^{2} - 3 i \, a^{3} c^{n} n - 2 i \, a^{3} c^{n}\right )} 2^{n} \sin \left (-4 \, f x + n \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right ) - 4 \, e\right )}{{\left ({\left (-i \, n^{3} - 3 i \, n^{2} - 2 i \, n\right )} {\left (\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )}^{\frac {1}{2} \, n} \cos \left (4 \, f x + 4 \, e\right ) + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} {\left (\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )}^{\frac {1}{2} \, n} \sin \left (4 \, f x + 4 \, e\right ) + {\left (-i \, n^{3} - 3 i \, n^{2} - 2 \, {\left (i \, n^{3} + 3 i \, n^{2} + 2 i \, n\right )} \cos \left (2 \, f x + 2 \, e\right ) + 2 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} \sin \left (2 \, f x + 2 \, e\right ) - 2 i \, n\right )} {\left (\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )}^{\frac {1}{2} \, n}\right )} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.15, size = 155, normalized size = 1.57 \begin {gather*} -\frac {4 \, {\left (-2 i \, a^{3} + {\left (-i \, a^{3} n^{2} - 3 i \, a^{3} n - 2 i \, a^{3}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, {\left (-i \, a^{3} n - 2 i \, a^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \left (\frac {2 \, c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{n}}{f n^{3} + 3 \, f n^{2} + 2 \, f n + {\left (f n^{3} + 3 \, f n^{2} + 2 \, f n\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, {\left (f n^{3} + 3 \, f n^{2} + 2 \, f n\right )} e^{\left (2 i \, f x + 2 i \, e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 979 vs. \(2 (80) = 160\).
time = 0.99, size = 979, normalized size = 9.89 \begin {gather*} \begin {cases} x \left (i a \tan {\left (e \right )} + a\right )^{3} \left (- i c \tan {\left (e \right )} + c\right )^{n} & \text {for}\: f = 0 \\\frac {2 a^{3} f x \tan ^{2}{\left (e + f x \right )}}{2 c^{2} f \tan ^{2}{\left (e + f x \right )} + 4 i c^{2} f \tan {\left (e + f x \right )} - 2 c^{2} f} + \frac {4 i a^{3} f x \tan {\left (e + f x \right )}}{2 c^{2} f \tan ^{2}{\left (e + f x \right )} + 4 i c^{2} f \tan {\left (e + f x \right )} - 2 c^{2} f} - \frac {2 a^{3} f x}{2 c^{2} f \tan ^{2}{\left (e + f x \right )} + 4 i c^{2} f \tan {\left (e + f x \right )} - 2 c^{2} f} + \frac {i a^{3} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )} \tan ^{2}{\left (e + f x \right )}}{2 c^{2} f \tan ^{2}{\left (e + f x \right )} + 4 i c^{2} f \tan {\left (e + f x \right )} - 2 c^{2} f} - \frac {2 a^{3} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )} \tan {\left (e + f x \right )}}{2 c^{2} f \tan ^{2}{\left (e + f x \right )} + 4 i c^{2} f \tan {\left (e + f x \right )} - 2 c^{2} f} - \frac {i a^{3} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 c^{2} f \tan ^{2}{\left (e + f x \right )} + 4 i c^{2} f \tan {\left (e + f x \right )} - 2 c^{2} f} - \frac {8 a^{3} \tan {\left (e + f x \right )}}{2 c^{2} f \tan ^{2}{\left (e + f x \right )} + 4 i c^{2} f \tan {\left (e + f x \right )} - 2 c^{2} f} - \frac {4 i a^{3}}{2 c^{2} f \tan ^{2}{\left (e + f x \right )} + 4 i c^{2} f \tan {\left (e + f x \right )} - 2 c^{2} f} & \text {for}\: n = -2 \\- \frac {4 a^{3} f x \tan {\left (e + f x \right )}}{c f \tan {\left (e + f x \right )} + i c f} - \frac {4 i a^{3} f x}{c f \tan {\left (e + f x \right )} + i c f} - \frac {2 i a^{3} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )} \tan {\left (e + f x \right )}}{c f \tan {\left (e + f x \right )} + i c f} + \frac {2 a^{3} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{c f \tan {\left (e + f x \right )} + i c f} + \frac {a^{3} \tan ^{2}{\left (e + f x \right )}}{c f \tan {\left (e + f x \right )} + i c f} + \frac {5 a^{3}}{c f \tan {\left (e + f x \right )} + i c f} & \text {for}\: n = -1 \\4 a^{3} x + \frac {2 i a^{3} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} - \frac {i a^{3} \tan ^{2}{\left (e + f x \right )}}{2 f} - \frac {3 a^{3} \tan {\left (e + f x \right )}}{f} & \text {for}\: n = 0 \\- \frac {i a^{3} n^{2} \left (- i c \tan {\left (e + f x \right )} + c\right )^{n} \tan ^{2}{\left (e + f x \right )}}{f n^{3} + 3 f n^{2} + 2 f n} - \frac {2 a^{3} n^{2} \left (- i c \tan {\left (e + f x \right )} + c\right )^{n} \tan {\left (e + f x \right )}}{f n^{3} + 3 f n^{2} + 2 f n} + \frac {i a^{3} n^{2} \left (- i c \tan {\left (e + f x \right )} + c\right )^{n}}{f n^{3} + 3 f n^{2} + 2 f n} - \frac {i a^{3} n \left (- i c \tan {\left (e + f x \right )} + c\right )^{n} \tan ^{2}{\left (e + f x \right )}}{f n^{3} + 3 f n^{2} + 2 f n} - \frac {6 a^{3} n \left (- i c \tan {\left (e + f x \right )} + c\right )^{n} \tan {\left (e + f x \right )}}{f n^{3} + 3 f n^{2} + 2 f n} + \frac {5 i a^{3} n \left (- i c \tan {\left (e + f x \right )} + c\right )^{n}}{f n^{3} + 3 f n^{2} + 2 f n} + \frac {8 i a^{3} \left (- i c \tan {\left (e + f x \right )} + c\right )^{n}}{f n^{3} + 3 f n^{2} + 2 f n} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.68, size = 230, normalized size = 2.32 \begin {gather*} \frac {2\,a^3\,{\left (\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}\right )}^n\,\left (n\,7{}\mathrm {i}+\cos \left (2\,e+2\,f\,x\right )\,16{}\mathrm {i}+\cos \left (4\,e+4\,f\,x\right )\,4{}\mathrm {i}-2\,n^2\,\sin \left (2\,e+2\,f\,x\right )-n^2\,\sin \left (4\,e+4\,f\,x\right )+n\,\cos \left (2\,e+2\,f\,x\right )\,10{}\mathrm {i}+n\,\cos \left (4\,e+4\,f\,x\right )\,3{}\mathrm {i}-6\,n\,\sin \left (2\,e+2\,f\,x\right )-3\,n\,\sin \left (4\,e+4\,f\,x\right )+n^2\,1{}\mathrm {i}+n^2\,\cos \left (2\,e+2\,f\,x\right )\,2{}\mathrm {i}+n^2\,\cos \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}+12{}\mathrm {i}\right )}{f\,n\,\left (4\,\cos \left (2\,e+2\,f\,x\right )+\cos \left (4\,e+4\,f\,x\right )+3\right )\,\left (n^2+3\,n+2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________