Optimal. Leaf size=87 \[ \frac {i c^4 (a-i a \tan (e+f x))^4}{80 a^5 f (a+i a \tan (e+f x))^4}+\frac {i c^4 (1-i \tan (e+f x))^4}{10 f (a+i a \tan (e+f x))^5} \]
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Rubi [A] time = 0.12, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {3522, 3487, 45, 37} \[ \frac {i c^4 (a-i a \tan (e+f x))^4}{80 a^5 f (a+i a \tan (e+f x))^4}+\frac {i c^4 (1-i \tan (e+f x))^4}{10 f (a+i a \tan (e+f x))^5} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 3487
Rule 3522
Rubi steps
\begin {align*} \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^5} \, dx &=\left (a^4 c^4\right ) \int \frac {\sec ^8(e+f x)}{(a+i a \tan (e+f x))^9} \, dx\\ &=-\frac {\left (i c^4\right ) \operatorname {Subst}\left (\int \frac {(a-x)^3}{(a+x)^6} \, dx,x,i a \tan (e+f x)\right )}{a^3 f}\\ &=\frac {i c^4 (1-i \tan (e+f x))^4}{10 f (a+i a \tan (e+f x))^5}-\frac {\left (i c^4\right ) \operatorname {Subst}\left (\int \frac {(a-x)^3}{(a+x)^5} \, dx,x,i a \tan (e+f x)\right )}{10 a^4 f}\\ &=\frac {i c^4 (1-i \tan (e+f x))^4}{10 f (a+i a \tan (e+f x))^5}+\frac {i c^4 (a-i a \tan (e+f x))^4}{80 a^5 f (a+i a \tan (e+f x))^4}\\ \end {align*}
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Mathematica [A] time = 1.97, size = 53, normalized size = 0.61 \[ \frac {c^4 (9 \cos (e+f x)+i \sin (e+f x)) (\sin (9 (e+f x))+i \cos (9 (e+f x)))}{80 a^5 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 37, normalized size = 0.43 \[ \frac {{\left (5 i \, c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + 4 i \, c^{4}\right )} e^{\left (-10 i \, f x - 10 i \, e\right )}}{80 \, a^{5} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.99, size = 174, normalized size = 2.00 \[ -\frac {2 \, {\left (5 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} - 5 i \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 50 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 35 i \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 98 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 35 i \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 50 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 5 i \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 5 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{5 \, a^{5} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - i\right )}^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 66, normalized size = 0.76 \[ \frac {c^{4} \left (-\frac {3 i}{\left (\tan \left (f x +e \right )-i\right )^{4}}+\frac {8}{5 \left (\tan \left (f x +e \right )-i\right )^{5}}+\frac {i}{2 \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {2}{\left (\tan \left (f x +e \right )-i\right )^{3}}\right )}{f \,a^{5}} \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.87, size = 98, normalized size = 1.13 \[ \frac {c^4\,\left (-5\,{\mathrm {tan}\left (e+f\,x\right )}^3-{\mathrm {tan}\left (e+f\,x\right )}^2\,5{}\mathrm {i}+5\,\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )}{10\,a^5\,f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^5\,1{}\mathrm {i}+5\,{\mathrm {tan}\left (e+f\,x\right )}^4-{\mathrm {tan}\left (e+f\,x\right )}^3\,10{}\mathrm {i}-10\,{\mathrm {tan}\left (e+f\,x\right )}^2+\mathrm {tan}\left (e+f\,x\right )\,5{}\mathrm {i}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.62, size = 109, normalized size = 1.25 \[ \begin {cases} \frac {\left (20 i a^{5} c^{4} f e^{10 i e} e^{- 8 i f x} + 16 i a^{5} c^{4} f e^{8 i e} e^{- 10 i f x}\right ) e^{- 18 i e}}{320 a^{10} f^{2}} & \text {for}\: 320 a^{10} f^{2} e^{18 i e} \neq 0 \\\frac {x \left (c^{4} e^{2 i e} + c^{4}\right ) e^{- 10 i e}}{2 a^{5}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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