Optimal. Leaf size=326 \[ \frac {3^{3/4} \sqrt {2+\sqrt {3}} a^2 \tan (c+d x) \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right ) \sqrt {\frac {\sqrt [3]{e} \sqrt [3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}{\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}\right )|-7-4 \sqrt {3}\right )}{2 d e (a-a \sec (c+d x)) \sqrt {a \sec (c+d x)+a} \sqrt {\frac {\sqrt [3]{e} \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}}}+\frac {3 a \tan (c+d x)}{2 d \sqrt {a \sec (c+d x)+a} (e \sec (c+d x))^{2/3}} \]
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Rubi [A] time = 0.24, antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3806, 51, 63, 218} \[ \frac {3^{3/4} \sqrt {2+\sqrt {3}} a^2 \tan (c+d x) \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right ) \sqrt {\frac {\sqrt [3]{e} \sqrt [3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}{\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}\right )|-7-4 \sqrt {3}\right )}{2 d e (a-a \sec (c+d x)) \sqrt {a \sec (c+d x)+a} \sqrt {\frac {\sqrt [3]{e} \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}}}+\frac {3 a \tan (c+d x)}{2 d \sqrt {a \sec (c+d x)+a} (e \sec (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 218
Rule 3806
Rubi steps
\begin {align*} \int \frac {\sqrt {a+a \sec (c+d x)}}{(e \sec (c+d x))^{2/3}} \, dx &=-\frac {\left (a^2 e \tan (c+d x)\right ) \operatorname {Subst}\left (\int \frac {1}{(e x)^{5/3} \sqrt {a-a x}} \, dx,x,\sec (c+d x)\right )}{d \sqrt {a-a \sec (c+d x)} \sqrt {a+a \sec (c+d x)}}\\ &=\frac {3 a \tan (c+d x)}{2 d (e \sec (c+d x))^{2/3} \sqrt {a+a \sec (c+d x)}}-\frac {\left (a^2 \tan (c+d x)\right ) \operatorname {Subst}\left (\int \frac {1}{(e x)^{2/3} \sqrt {a-a x}} \, dx,x,\sec (c+d x)\right )}{4 d \sqrt {a-a \sec (c+d x)} \sqrt {a+a \sec (c+d x)}}\\ &=\frac {3 a \tan (c+d x)}{2 d (e \sec (c+d x))^{2/3} \sqrt {a+a \sec (c+d x)}}-\frac {\left (3 a^2 \tan (c+d x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\frac {a x^3}{e}}} \, dx,x,\sqrt [3]{e \sec (c+d x)}\right )}{4 d e \sqrt {a-a \sec (c+d x)} \sqrt {a+a \sec (c+d x)}}\\ &=\frac {3 a \tan (c+d x)}{2 d (e \sec (c+d x))^{2/3} \sqrt {a+a \sec (c+d x)}}+\frac {3^{3/4} \sqrt {2+\sqrt {3}} a^2 F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}{\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}}\right )|-7-4 \sqrt {3}\right ) \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right ) \sqrt {\frac {e^{2/3}+\sqrt [3]{e} \sqrt [3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}} \tan (c+d x)}{2 d e (a-a \sec (c+d x)) \sqrt {a+a \sec (c+d x)} \sqrt {\frac {\sqrt [3]{e} \left (\sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{e}-\sqrt [3]{e \sec (c+d x)}\right )^2}}}\\ \end {align*}
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Mathematica [C] time = 0.17, size = 71, normalized size = 0.22 \[ \frac {2 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^{\frac {2}{3}}(c+d x) \sqrt {a (\sec (c+d x)+1)} \, _2F_1\left (\frac {1}{2},\frac {5}{3};\frac {3}{2};1-\sec (c+d x)\right )}{d (e \sec (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \sec \left (d x + c\right ) + a} \left (e \sec \left (d x + c\right )\right )^{\frac {1}{3}}}{e \sec \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {\sqrt {a \sec \left (d x + c\right ) + a}}{\left (e \sec \left (d x + c\right )\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.33, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a +a \sec \left (d x +c \right )}}{\left (e \sec \left (d x +c \right )\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a \sec \left (d x + c\right ) + a}}{\left (e \sec \left (d x + c\right )\right )^{\frac {2}{3}}}\,{d x} \]
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 \[ \int \frac {\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{2/3}} \,d x \]
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 \[ \int \frac {\sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )}}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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