Optimal. Leaf size=127 \[ \frac {2 a^3 (i A+B)}{3 c^6 f (i+\tan (e+f x))^6}-\frac {4 a^3 (A-2 i B)}{5 c^6 f (i+\tan (e+f x))^5}-\frac {a^3 (i A+5 B)}{4 c^6 f (i+\tan (e+f x))^4}-\frac {i a^3 B}{3 c^6 f (i+\tan (e+f x))^3} \]
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Rubi [A]
time = 0.12, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {3669, 78}
\begin {gather*} -\frac {a^3 (5 B+i A)}{4 c^6 f (\tan (e+f x)+i)^4}-\frac {4 a^3 (A-2 i B)}{5 c^6 f (\tan (e+f x)+i)^5}+\frac {2 a^3 (B+i A)}{3 c^6 f (\tan (e+f x)+i)^6}-\frac {i a^3 B}{3 c^6 f (\tan (e+f x)+i)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 3669
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^6} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {(a+i a x)^2 (A+B x)}{(c-i c x)^7} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \text {Subst}\left (\int \left (-\frac {4 i a^2 (A-i B)}{c^7 (i+x)^7}+\frac {4 a^2 (A-2 i B)}{c^7 (i+x)^6}+\frac {a^2 (i A+5 B)}{c^7 (i+x)^5}+\frac {i a^2 B}{c^7 (i+x)^4}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {2 a^3 (i A+B)}{3 c^6 f (i+\tan (e+f x))^6}-\frac {4 a^3 (A-2 i B)}{5 c^6 f (i+\tan (e+f x))^5}-\frac {a^3 (i A+5 B)}{4 c^6 f (i+\tan (e+f x))^4}-\frac {i a^3 B}{3 c^6 f (i+\tan (e+f x))^3}\\ \end {align*}
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Mathematica [A]
time = 2.25, size = 112, normalized size = 0.88 \begin {gather*} \frac {a^3 (3 (-27 i A+B) \cos (e+f x)+10 (-3 i A+B) \cos (3 (e+f x))-(A+3 i B) (9 \sin (e+f x)+10 \sin (3 (e+f x)))) (\cos (9 e+12 f x)+i \sin (9 e+12 f x))}{960 c^6 f (\cos (f x)+i \sin (f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 90, normalized size = 0.71
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {i B}{3 \left (i+\tan \left (f x +e \right )\right )^{3}}-\frac {-4 i A -4 B}{6 \left (i+\tan \left (f x +e \right )\right )^{6}}-\frac {-8 i B +4 A}{5 \left (i+\tan \left (f x +e \right )\right )^{5}}-\frac {i A +5 B}{4 \left (i+\tan \left (f x +e \right )\right )^{4}}\right )}{f \,c^{6}}\) | \(90\) |
default | \(\frac {a^{3} \left (-\frac {i B}{3 \left (i+\tan \left (f x +e \right )\right )^{3}}-\frac {-4 i A -4 B}{6 \left (i+\tan \left (f x +e \right )\right )^{6}}-\frac {-8 i B +4 A}{5 \left (i+\tan \left (f x +e \right )\right )^{5}}-\frac {i A +5 B}{4 \left (i+\tan \left (f x +e \right )\right )^{4}}\right )}{f \,c^{6}}\) | \(90\) |
risch | \(-\frac {a^{3} {\mathrm e}^{12 i \left (f x +e \right )} B}{96 c^{6} f}-\frac {i a^{3} {\mathrm e}^{12 i \left (f x +e \right )} A}{96 c^{6} f}-\frac {{\mathrm e}^{10 i \left (f x +e \right )} B \,a^{3}}{80 c^{6} f}-\frac {3 i {\mathrm e}^{10 i \left (f x +e \right )} A \,a^{3}}{80 c^{6} f}+\frac {{\mathrm e}^{8 i \left (f x +e \right )} B \,a^{3}}{64 c^{6} f}-\frac {3 i {\mathrm e}^{8 i \left (f x +e \right )} A \,a^{3}}{64 c^{6} f}+\frac {a^{3} {\mathrm e}^{6 i \left (f x +e \right )} B}{48 c^{6} f}-\frac {i a^{3} {\mathrm e}^{6 i \left (f x +e \right )} A}{48 c^{6} f}\) | \(174\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.01, size = 93, normalized size = 0.73 \begin {gather*} -\frac {10 \, {\left (i \, A + B\right )} a^{3} e^{\left (12 i \, f x + 12 i \, e\right )} + 12 \, {\left (3 i \, A + B\right )} a^{3} e^{\left (10 i \, f x + 10 i \, e\right )} + 15 \, {\left (3 i \, A - B\right )} a^{3} e^{\left (8 i \, f x + 8 i \, e\right )} + 20 \, {\left (i \, A - B\right )} a^{3} e^{\left (6 i \, f x + 6 i \, e\right )}}{960 \, c^{6} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 332 vs. \(2 (105) = 210\).
time = 0.63, size = 332, normalized size = 2.61 \begin {gather*} \begin {cases} \frac {\left (- 491520 i A a^{3} c^{18} f^{3} e^{6 i e} + 491520 B a^{3} c^{18} f^{3} e^{6 i e}\right ) e^{6 i f x} + \left (- 1105920 i A a^{3} c^{18} f^{3} e^{8 i e} + 368640 B a^{3} c^{18} f^{3} e^{8 i e}\right ) e^{8 i f x} + \left (- 884736 i A a^{3} c^{18} f^{3} e^{10 i e} - 294912 B a^{3} c^{18} f^{3} e^{10 i e}\right ) e^{10 i f x} + \left (- 245760 i A a^{3} c^{18} f^{3} e^{12 i e} - 245760 B a^{3} c^{18} f^{3} e^{12 i e}\right ) e^{12 i f x}}{23592960 c^{24} f^{4}} & \text {for}\: c^{24} f^{4} \neq 0 \\\frac {x \left (A a^{3} e^{12 i e} + 3 A a^{3} e^{10 i e} + 3 A a^{3} e^{8 i e} + A a^{3} e^{6 i e} - i B a^{3} e^{12 i e} - i B a^{3} e^{10 i e} + i B a^{3} e^{8 i e} + i B a^{3} e^{6 i e}\right )}{8 c^{6}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 345 vs. \(2 (107) = 214\).
time = 0.96, size = 345, normalized size = 2.72 \begin {gather*} -\frac {2 \, {\left (15 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 45 i \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 15 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 215 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} - 390 i \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} + 90 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} + 738 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 24 i \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 746 i \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 158 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 738 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 24 i \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 390 i \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 90 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 215 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 45 i \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 15 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 15 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{15 \, c^{6} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.09, size = 140, normalized size = 1.10 \begin {gather*} -\frac {-\frac {a^3\,\left (-B+A\,7{}\mathrm {i}\right )}{60}+\frac {a^3\,\mathrm {tan}\left (e+f\,x\right )\,\left (18\,A-B\,6{}\mathrm {i}\right )}{60}+\frac {B\,a^3\,{\mathrm {tan}\left (e+f\,x\right )}^3\,1{}\mathrm {i}}{3}+\frac {a^3\,{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (15\,B+A\,15{}\mathrm {i}\right )}{60}}{c^6\,f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^6+{\mathrm {tan}\left (e+f\,x\right )}^5\,6{}\mathrm {i}-15\,{\mathrm {tan}\left (e+f\,x\right )}^4-{\mathrm {tan}\left (e+f\,x\right )}^3\,20{}\mathrm {i}+15\,{\mathrm {tan}\left (e+f\,x\right )}^2+\mathrm {tan}\left (e+f\,x\right )\,6{}\mathrm {i}-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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