Optimal. Leaf size=378 \[ \frac {i \tan ^{-1}\left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a-i a \tan (e+f x)}}{\sqrt {3} \sqrt [3]{a}}\right ) (d \sec (e+f x))^{2/3}}{2^{2/3} \sqrt {3} a^{2/3} f \sqrt [3]{a-i a \tan (e+f x)} \sqrt [3]{a+i a \tan (e+f x)}}-\frac {x (d \sec (e+f x))^{2/3}}{6\ 2^{2/3} a^{2/3} \sqrt [3]{a-i a \tan (e+f x)} \sqrt [3]{a+i a \tan (e+f x)}}-\frac {i (d \sec (e+f x))^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a-i a \tan (e+f x)}\right )}{2\ 2^{2/3} a^{2/3} f \sqrt [3]{a-i a \tan (e+f x)} \sqrt [3]{a+i a \tan (e+f x)}}-\frac {i (d \sec (e+f x))^{2/3} \log (\cos (e+f x))}{6\ 2^{2/3} a^{2/3} f \sqrt [3]{a-i a \tan (e+f x)} \sqrt [3]{a+i a \tan (e+f x)}}+\frac {i (d \sec (e+f x))^{2/3}}{2 f (a+i a \tan (e+f x))^{4/3}} \]
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Rubi [A] time = 0.34, antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3505, 3522, 3487, 51, 57, 617, 204, 31} \[ \frac {i \tan ^{-1}\left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a-i a \tan (e+f x)}}{\sqrt {3} \sqrt [3]{a}}\right ) (d \sec (e+f x))^{2/3}}{2^{2/3} \sqrt {3} a^{2/3} f \sqrt [3]{a-i a \tan (e+f x)} \sqrt [3]{a+i a \tan (e+f x)}}-\frac {x (d \sec (e+f x))^{2/3}}{6\ 2^{2/3} a^{2/3} \sqrt [3]{a-i a \tan (e+f x)} \sqrt [3]{a+i a \tan (e+f x)}}-\frac {i (d \sec (e+f x))^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a-i a \tan (e+f x)}\right )}{2\ 2^{2/3} a^{2/3} f \sqrt [3]{a-i a \tan (e+f x)} \sqrt [3]{a+i a \tan (e+f x)}}-\frac {i (d \sec (e+f x))^{2/3} \log (\cos (e+f x))}{6\ 2^{2/3} a^{2/3} f \sqrt [3]{a-i a \tan (e+f x)} \sqrt [3]{a+i a \tan (e+f x)}}+\frac {i (d \sec (e+f x))^{2/3}}{2 f (a+i a \tan (e+f x))^{4/3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 51
Rule 57
Rule 204
Rule 617
Rule 3487
Rule 3505
Rule 3522
Rubi steps
\begin {align*} \int \frac {(d \sec (e+f x))^{2/3}}{(a+i a \tan (e+f x))^{4/3}} \, dx &=\frac {(d \sec (e+f x))^{2/3} \int \frac {\sqrt [3]{a-i a \tan (e+f x)}}{a+i a \tan (e+f x)} \, dx}{\sqrt [3]{a-i a \tan (e+f x)} \sqrt [3]{a+i a \tan (e+f x)}}\\ &=\frac {(d \sec (e+f x))^{2/3} \int \cos ^2(e+f x) (a-i a \tan (e+f x))^{4/3} \, dx}{a^2 \sqrt [3]{a-i a \tan (e+f x)} \sqrt [3]{a+i a \tan (e+f x)}}\\ &=\frac {\left (i a (d \sec (e+f x))^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x)^2 (a+x)^{2/3}} \, dx,x,-i a \tan (e+f x)\right )}{f \sqrt [3]{a-i a \tan (e+f x)} \sqrt [3]{a+i a \tan (e+f x)}}\\ &=\frac {i (d \sec (e+f x))^{2/3}}{2 f (a+i a \tan (e+f x))^{4/3}}+\frac {\left (i (d \sec (e+f x))^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x) (a+x)^{2/3}} \, dx,x,-i a \tan (e+f x)\right )}{3 f \sqrt [3]{a-i a \tan (e+f x)} \sqrt [3]{a+i a \tan (e+f x)}}\\ &=\frac {i (d \sec (e+f x))^{2/3}}{2 f (a+i a \tan (e+f x))^{4/3}}-\frac {x (d \sec (e+f x))^{2/3}}{6\ 2^{2/3} a^{2/3} \sqrt [3]{a-i a \tan (e+f x)} \sqrt [3]{a+i a \tan (e+f x)}}-\frac {i \log (\cos (e+f x)) (d \sec (e+f x))^{2/3}}{6\ 2^{2/3} a^{2/3} f \sqrt [3]{a-i a \tan (e+f x)} \sqrt [3]{a+i a \tan (e+f x)}}+\frac {\left (i (d \sec (e+f x))^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a-i a \tan (e+f x)}\right )}{2\ 2^{2/3} a^{2/3} f \sqrt [3]{a-i a \tan (e+f x)} \sqrt [3]{a+i a \tan (e+f x)}}+\frac {\left (i (d \sec (e+f x))^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a-i a \tan (e+f x)}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} f \sqrt [3]{a-i a \tan (e+f x)} \sqrt [3]{a+i a \tan (e+f x)}}\\ &=\frac {i (d \sec (e+f x))^{2/3}}{2 f (a+i a \tan (e+f x))^{4/3}}-\frac {x (d \sec (e+f x))^{2/3}}{6\ 2^{2/3} a^{2/3} \sqrt [3]{a-i a \tan (e+f x)} \sqrt [3]{a+i a \tan (e+f x)}}-\frac {i \log (\cos (e+f x)) (d \sec (e+f x))^{2/3}}{6\ 2^{2/3} a^{2/3} f \sqrt [3]{a-i a \tan (e+f x)} \sqrt [3]{a+i a \tan (e+f x)}}-\frac {i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a-i a \tan (e+f x)}\right ) (d \sec (e+f x))^{2/3}}{2\ 2^{2/3} a^{2/3} f \sqrt [3]{a-i a \tan (e+f x)} \sqrt [3]{a+i a \tan (e+f x)}}-\frac {\left (i (d \sec (e+f x))^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} \sqrt [3]{a-i a \tan (e+f x)}}{\sqrt [3]{a}}\right )}{2^{2/3} a^{2/3} f \sqrt [3]{a-i a \tan (e+f x)} \sqrt [3]{a+i a \tan (e+f x)}}\\ &=\frac {i (d \sec (e+f x))^{2/3}}{2 f (a+i a \tan (e+f x))^{4/3}}-\frac {x (d \sec (e+f x))^{2/3}}{6\ 2^{2/3} a^{2/3} \sqrt [3]{a-i a \tan (e+f x)} \sqrt [3]{a+i a \tan (e+f x)}}+\frac {i \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{a-i a \tan (e+f x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right ) (d \sec (e+f x))^{2/3}}{2^{2/3} \sqrt {3} a^{2/3} f \sqrt [3]{a-i a \tan (e+f x)} \sqrt [3]{a+i a \tan (e+f x)}}-\frac {i \log (\cos (e+f x)) (d \sec (e+f x))^{2/3}}{6\ 2^{2/3} a^{2/3} f \sqrt [3]{a-i a \tan (e+f x)} \sqrt [3]{a+i a \tan (e+f x)}}-\frac {i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a-i a \tan (e+f x)}\right ) (d \sec (e+f x))^{2/3}}{2\ 2^{2/3} a^{2/3} f \sqrt [3]{a-i a \tan (e+f x)} \sqrt [3]{a+i a \tan (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 1.32, size = 220, normalized size = 0.58 \[ \frac {e^{-i (e+f x)} (d \sec (e+f x))^{5/3} \left (-2 f x e^{2 i (e+f x)} \sqrt [3]{1+e^{2 i (e+f x)}}+3 i e^{2 i (e+f x)}-3 i e^{2 i (e+f x)} \sqrt [3]{1+e^{2 i (e+f x)}} \log \left (1-\sqrt [3]{1+e^{2 i (e+f x)}}\right )-2 i \sqrt {3} e^{2 i (e+f x)} \sqrt [3]{1+e^{2 i (e+f x)}} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+e^{2 i (e+f x)}}}{\sqrt {3}}\right )+3 i\right )}{12 d f (a+i a \tan (e+f x))^{4/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 515, normalized size = 1.36 \[ \frac {{\left (4 \, a^{2} f \left (\frac {i \, d^{2}}{108 \, a^{4} f^{3}}\right )^{\frac {1}{3}} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left ({\left (-12 i \, a^{2} f \left (\frac {i \, d^{2}}{108 \, a^{4} f^{3}}\right )^{\frac {1}{3}} e^{\left (2 i \, f x + 2 i \, e\right )} + 2 \cdot 2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac {2}{3}} \left (\frac {d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac {2}{3}} {\left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}\right ) + 2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac {2}{3}} \left (\frac {d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac {2}{3}} {\left (i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 2 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + i\right )} e^{\left (2 i \, f x + 2 i \, e\right )} - 2 \, {\left (-i \, \sqrt {3} a^{2} f + a^{2} f\right )} \left (\frac {i \, d^{2}}{108 \, a^{4} f^{3}}\right )^{\frac {1}{3}} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left ({\left (2 \cdot 2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac {2}{3}} \left (\frac {d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac {2}{3}} {\left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (6 \, \sqrt {3} a^{2} f + 6 i \, a^{2} f\right )} \left (\frac {i \, d^{2}}{108 \, a^{4} f^{3}}\right )^{\frac {1}{3}} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}\right ) - 2 \, {\left (i \, \sqrt {3} a^{2} f + a^{2} f\right )} \left (\frac {i \, d^{2}}{108 \, a^{4} f^{3}}\right )^{\frac {1}{3}} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left ({\left (2 \cdot 2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac {2}{3}} \left (\frac {d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac {2}{3}} {\left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )} e^{\left (2 i \, f x + 2 i \, e\right )} - {\left (6 \, \sqrt {3} a^{2} f - 6 i \, a^{2} f\right )} \left (\frac {i \, d^{2}}{108 \, a^{4} f^{3}}\right )^{\frac {1}{3}} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}\right )\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{4 \, a^{2} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \sec \left (f x + e\right )\right )^{\frac {2}{3}}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.78, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \sec \left (f x +e \right )\right )^{\frac {2}{3}}}{\left (a +i a \tan \left (f x +e \right )\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.95, size = 1907, normalized size = 5.04 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{2/3}}{{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{4/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \sec {\left (e + f x \right )}\right )^{\frac {2}{3}}}{\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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