Optimal. Leaf size=50 \[ -\frac {i a^3 \left (c^2+i c^2 \tan (e+f x)\right )^3}{6 f \left (c^3-i c^3 \tan (e+f x)\right )^3} \]
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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} \[ -\frac {i a^3 \left (c^2+i c^2 \tan (e+f x)\right )^3}{6 f \left (c^3-i c^3 \tan (e+f x)\right )^3} \]
Antiderivative was successfully verified.
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Rule 37
Rule 3487
Rule 3522
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^3}{(c-i c \tan (e+f x))^3} \, dx &=\left (a^3 c^3\right ) \int \frac {\sec ^6(e+f x)}{(c-i c \tan (e+f x))^6} \, dx\\ &=\frac {\left (i a^3\right ) \operatorname {Subst}\left (\int \frac {(c-x)^2}{(c+x)^4} \, dx,x,-i c \tan (e+f x)\right )}{c^2 f}\\ &=-\frac {i a^3 (c+i c \tan (e+f x))^3}{6 f \left (c^2-i c^2 \tan (e+f x)\right )^3}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 34, normalized size = 0.68 \[ \frac {a^3 (\sin (6 (e+f x))-i \cos (6 (e+f x)))}{6 c^3 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 20, normalized size = 0.40 \[ -\frac {i \, a^{3} e^{\left (6 i \, f x + 6 i \, e\right )}}{6 \, c^{3} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.65, size = 72, normalized size = 1.44 \[ -\frac {2 \, {\left (3 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 10 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{3 \, c^{3} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 50, normalized size = 1.00 \[ \frac {a^{3} \left (-\frac {2 i}{\left (\tan \left (f x +e \right )+i\right )^{2}}+\frac {1}{\tan \left (f x +e \right )+i}-\frac {4}{3 \left (\tan \left (f x +e \right )+i\right )^{3}}\right )}{f \,c^{3}} \]
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 = 4.71, size = 55, normalized size = 1.10 \[ -\frac {a^3\,\left ({\mathrm {tan}\left (e+f\,x\right )}^2-\frac {1}{3}\right )}{c^3\,f\,\left (-{\mathrm {tan}\left (e+f\,x\right )}^3-{\mathrm {tan}\left (e+f\,x\right )}^2\,3{}\mathrm {i}+3\,\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.32, size = 48, normalized size = 0.96 \[ \begin {cases} - \frac {i a^{3} e^{6 i e} e^{6 i f x}}{6 c^{3} f} & \text {for}\: 6 c^{3} f \neq 0 \\\frac {a^{3} x e^{6 i e}}{c^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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