Optimal. Leaf size=50 \[ \frac {i c^3 \left (a^2-i a^2 \tan (e+f x)\right )^3}{6 f \left (a^3+i a^3 \tan (e+f x)\right )^3} \]
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Rubi [A] time = 0.11, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3522, 3487, 37} \[ \frac {i c^3 \left (a^2-i a^2 \tan (e+f x)\right )^3}{6 f \left (a^3+i a^3 \tan (e+f x)\right )^3} \]
Antiderivative was successfully verified.
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Rule 37
Rule 3487
Rule 3522
Rubi steps
\begin {align*} \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^3} \, dx &=\left (a^3 c^3\right ) \int \frac {\sec ^6(e+f x)}{(a+i a \tan (e+f x))^6} \, dx\\ &=-\frac {\left (i c^3\right ) \operatorname {Subst}\left (\int \frac {(a-x)^2}{(a+x)^4} \, dx,x,i a \tan (e+f x)\right )}{a^2 f}\\ &=\frac {i c^3 (1-i \tan (e+f x))^3}{6 f (a+i a \tan (e+f x))^3}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 34, normalized size = 0.68 \[ \frac {c^3 (\sin (6 (e+f x))+i \cos (6 (e+f x)))}{6 a^3 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 20, normalized size = 0.40 \[ \frac {i \, c^{3} e^{\left (-6 i \, f x - 6 i \, e\right )}}{6 \, a^{3} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.55, size = 72, normalized size = 1.44 \[ -\frac {2 \, {\left (3 \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 10 \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3 \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{3 \, a^{3} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - i\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 50, normalized size = 1.00 \[ \frac {c^{3} \left (\frac {1}{\tan \left (f x +e \right )-i}+\frac {2 i}{\left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {4}{3 \left (\tan \left (f x +e \right )-i\right )^{3}}\right )}{f \,a^{3}} \]
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 = 4.82, size = 59, normalized size = 1.18 \[ -\frac {c^3\,\left ({\mathrm {tan}\left (e+f\,x\right )}^2\,1{}\mathrm {i}-\frac {1}{3}{}\mathrm {i}\right )}{a^3\,f\,\left (-{\mathrm {tan}\left (e+f\,x\right )}^3\,1{}\mathrm {i}-3\,{\mathrm {tan}\left (e+f\,x\right )}^2+\mathrm {tan}\left (e+f\,x\right )\,3{}\mathrm {i}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.32, size = 53, normalized size = 1.06 \[ \begin {cases} \frac {i c^{3} e^{- 6 i e} e^{- 6 i f x}}{6 a^{3} f} & \text {for}\: 6 a^{3} f e^{6 i e} \neq 0 \\\frac {c^{3} x e^{- 6 i e}}{a^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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