Optimal. Leaf size=82 \[ \frac {a^3 c^4 \tan ^5(e+f x)}{5 f}+\frac {2 a^3 c^4 \tan ^3(e+f x)}{3 f}+\frac {a^3 c^4 \tan (e+f x)}{f}-\frac {i a^3 c^4 \sec ^6(e+f x)}{6 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 82, 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 = {3522, 3486, 3767} \[ \frac {a^3 c^4 \tan ^5(e+f x)}{5 f}+\frac {2 a^3 c^4 \tan ^3(e+f x)}{3 f}+\frac {a^3 c^4 \tan (e+f x)}{f}-\frac {i a^3 c^4 \sec ^6(e+f x)}{6 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3486
Rule 3522
Rule 3767
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^4 \, dx &=\left (a^3 c^3\right ) \int \sec ^6(e+f x) (c-i c \tan (e+f x)) \, dx\\ &=-\frac {i a^3 c^4 \sec ^6(e+f x)}{6 f}+\left (a^3 c^4\right ) \int \sec ^6(e+f x) \, dx\\ &=-\frac {i a^3 c^4 \sec ^6(e+f x)}{6 f}-\frac {\left (a^3 c^4\right ) \operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (e+f x)\right )}{f}\\ &=-\frac {i a^3 c^4 \sec ^6(e+f x)}{6 f}+\frac {a^3 c^4 \tan (e+f x)}{f}+\frac {2 a^3 c^4 \tan ^3(e+f x)}{3 f}+\frac {a^3 c^4 \tan ^5(e+f x)}{5 f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 4.24, size = 63, normalized size = 0.77 \[ \frac {a^3 c^4 \sec (e) \sec ^6(e+f x) (15 \sin (e+2 f x)+6 \sin (3 e+4 f x)+\sin (5 e+6 f x)-10 \sin (e)-10 i \cos (e))}{60 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 120, normalized size = 1.46 \[ \frac {240 i \, a^{3} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} + 96 i \, a^{3} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + 16 i \, a^{3} c^{4}}{15 \, {\left (f e^{\left (12 i \, f x + 12 i \, e\right )} + 6 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 15 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 20 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 15 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 6 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.49, size = 128, normalized size = 1.56 \[ \frac {240 i \, a^{3} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} + 96 i \, a^{3} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + 16 i \, a^{3} c^{4}}{15 \, {\left (f e^{\left (12 i \, f x + 12 i \, e\right )} + 6 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 15 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 20 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 15 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 6 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 71, normalized size = 0.87 \[ \frac {a^{3} c^{4} \left (\tan \left (f x +e \right )-\frac {i \left (\tan ^{6}\left (f x +e \right )\right )}{6}+\frac {\left (\tan ^{5}\left (f x +e \right )\right )}{5}-\frac {i \left (\tan ^{4}\left (f x +e \right )\right )}{2}+\frac {2 \left (\tan ^{3}\left (f x +e \right )\right )}{3}-\frac {i \left (\tan ^{2}\left (f x +e \right )\right )}{2}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.82, size = 100, normalized size = 1.22 \[ \frac {-10 i \, a^{3} c^{4} \tan \left (f x + e\right )^{6} + 12 \, a^{3} c^{4} \tan \left (f x + e\right )^{5} - 30 i \, a^{3} c^{4} \tan \left (f x + e\right )^{4} + 40 \, a^{3} c^{4} \tan \left (f x + e\right )^{3} - 30 i \, a^{3} c^{4} \tan \left (f x + e\right )^{2} + 60 \, a^{3} c^{4} \tan \left (f x + e\right )}{60 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.55, size = 117, normalized size = 1.43 \[ \frac {a^3\,c^4\,\sin \left (e+f\,x\right )\,\left (30\,{\cos \left (e+f\,x\right )}^5-{\cos \left (e+f\,x\right )}^4\,\sin \left (e+f\,x\right )\,15{}\mathrm {i}+20\,{\cos \left (e+f\,x\right )}^3\,{\sin \left (e+f\,x\right )}^2-{\cos \left (e+f\,x\right )}^2\,{\sin \left (e+f\,x\right )}^3\,15{}\mathrm {i}+6\,\cos \left (e+f\,x\right )\,{\sin \left (e+f\,x\right )}^4-{\sin \left (e+f\,x\right )}^5\,5{}\mathrm {i}\right )}{30\,f\,{\cos \left (e+f\,x\right )}^6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.71, size = 184, normalized size = 2.24 \[ \frac {- 240 a^{3} c^{4} e^{4 i e} e^{4 i f x} - 96 a^{3} c^{4} e^{2 i e} e^{2 i f x} - 16 a^{3} c^{4}}{15 i f e^{12 i e} e^{12 i f x} + 90 i f e^{10 i e} e^{10 i f x} + 225 i f e^{8 i e} e^{8 i f x} + 300 i f e^{6 i e} e^{6 i f x} + 225 i f e^{4 i e} e^{4 i f x} + 90 i f e^{2 i e} e^{2 i f x} + 15 i f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________