Optimal. Leaf size=90 \[ \frac {4 i c^3}{5 f (a+i a \tan (e+f x))^5}+\frac {i c^3}{3 a^2 f (a+i a \tan (e+f x))^3}-\frac {i a^3 c^3}{f \left (a^2+i a^2 \tan (e+f x)\right )^4} \]
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Rubi [A]
time = 0.09, antiderivative size = 90, 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 c^3}{3 a^2 f (a+i a \tan (e+f x))^3}-\frac {i a^3 c^3}{f \left (a^2+i a^2 \tan (e+f x)\right )^4}+\frac {4 i c^3}{5 f (a+i a \tan (e+f x))^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3568
Rule 3603
Rubi steps
\begin {align*} \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^5} \, dx &=\left (a^3 c^3\right ) \int \frac {\sec ^6(e+f x)}{(a+i a \tan (e+f x))^8} \, dx\\ &=-\frac {\left (i c^3\right ) \text {Subst}\left (\int \frac {(a-x)^2}{(a+x)^6} \, dx,x,i a \tan (e+f x)\right )}{a^2 f}\\ &=-\frac {\left (i c^3\right ) \text {Subst}\left (\int \left (\frac {4 a^2}{(a+x)^6}-\frac {4 a}{(a+x)^5}+\frac {1}{(a+x)^4}\right ) \, dx,x,i a \tan (e+f x)\right )}{a^2 f}\\ &=\frac {4 i c^3}{5 f (a+i a \tan (e+f x))^5}-\frac {i c^3}{a f (a+i a \tan (e+f x))^4}+\frac {i c^3}{3 a^2 f (a+i a \tan (e+f x))^3}\\ \end {align*}
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Mathematica [A]
time = 1.34, size = 58, normalized size = 0.64 \begin {gather*} \frac {c^3 (15+16 \cos (2 (e+f x))+4 i \sin (2 (e+f x))) (i \cos (8 (e+f x))+\sin (8 (e+f x)))}{240 a^5 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 52, normalized size = 0.58
method | result | size |
derivativedivides | \(\frac {c^{3} \left (-\frac {1}{3 \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {4}{5 \left (\tan \left (f x +e \right )-i\right )^{5}}-\frac {i}{\left (\tan \left (f x +e \right )-i\right )^{4}}\right )}{f \,a^{5}}\) | \(52\) |
default | \(\frac {c^{3} \left (-\frac {1}{3 \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {4}{5 \left (\tan \left (f x +e \right )-i\right )^{5}}-\frac {i}{\left (\tan \left (f x +e \right )-i\right )^{4}}\right )}{f \,a^{5}}\) | \(52\) |
risch | \(\frac {i c^{3} {\mathrm e}^{-6 i \left (f x +e \right )}}{24 a^{5} f}+\frac {i c^{3} {\mathrm e}^{-8 i \left (f x +e \right )}}{16 a^{5} f}+\frac {i c^{3} {\mathrm e}^{-10 i \left (f x +e \right )}}{40 a^{5} f}\) | \(65\) |
norman | \(\frac {\frac {c^{3} \tan \left (f x +e \right )}{a f}+\frac {2 i c^{3}}{15 a f}-\frac {19 c^{3} \left (\tan ^{3}\left (f x +e \right )\right )}{3 a f}+\frac {77 c^{3} \left (\tan ^{5}\left (f x +e \right )\right )}{15 a f}-\frac {c^{3} \left (\tan ^{7}\left (f x +e \right )\right )}{3 a f}-\frac {10 i c^{3} \left (\tan ^{2}\left (f x +e \right )\right )}{3 a f}-\frac {2 i c^{3} \left (\tan ^{6}\left (f x +e \right )\right )}{a f}+\frac {22 i c^{3} \left (\tan ^{4}\left (f x +e \right )\right )}{3 a f}}{\left (1+\tan ^{2}\left (f x +e \right )\right )^{5} a^{4}}\) | \(163\) |
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 = 1.53, size = 54, normalized size = 0.60 \begin {gather*} \frac {{\left (10 i \, c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 15 i \, c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 6 i \, c^{3}\right )} e^{\left (-10 i \, f x - 10 i \, e\right )}}{240 \, a^{5} 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. 151 vs. \(2 (75) = 150\).
time = 0.36, size = 151, normalized size = 1.68 \begin {gather*} \begin {cases} \frac {\left (640 i a^{10} c^{3} f^{2} e^{18 i e} e^{- 6 i f x} + 960 i a^{10} c^{3} f^{2} e^{16 i e} e^{- 8 i f x} + 384 i a^{10} c^{3} f^{2} e^{14 i e} e^{- 10 i f x}\right ) e^{- 24 i e}}{15360 a^{15} f^{3}} & \text {for}\: a^{15} f^{3} e^{24 i e} \neq 0 \\\frac {x \left (c^{3} e^{4 i e} + 2 c^{3} e^{2 i e} + c^{3}\right ) e^{- 10 i e}}{4 a^{5}} & \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. 174 vs. \(2 (77) = 154\).
time = 1.09, size = 174, normalized size = 1.93 \begin {gather*} -\frac {2 \, {\left (15 \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} - 30 i \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 140 \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 170 i \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 282 \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 170 i \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 140 \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 30 i \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 15 \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{15 \, a^{5} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - i\right )}^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.86, size = 88, normalized size = 0.98 \begin {gather*} \frac {c^3\,\left (-{\mathrm {tan}\left (e+f\,x\right )}^2\,5{}\mathrm {i}+5\,\mathrm {tan}\left (e+f\,x\right )+2{}\mathrm {i}\right )}{15\,a^5\,f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^5\,1{}\mathrm {i}+5\,{\mathrm {tan}\left (e+f\,x\right )}^4-{\mathrm {tan}\left (e+f\,x\right )}^3\,10{}\mathrm {i}-10\,{\mathrm {tan}\left (e+f\,x\right )}^2+\mathrm {tan}\left (e+f\,x\right )\,5{}\mathrm {i}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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