Optimal. Leaf size=81 \[ \frac {3 \sqrt {1+i \tan (c+d x)} \tan ^{\frac {5}{3}}(c+d x) F_1\left (\frac {5}{3};\frac {3}{2},1;\frac {8}{3};-i \tan (c+d x),i \tan (c+d x)\right )}{5 d \sqrt {a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.14, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3564, 130, 511, 510} \[ \frac {3 \sqrt {1+i \tan (c+d x)} \tan ^{\frac {5}{3}}(c+d x) F_1\left (\frac {5}{3};\frac {3}{2},1;\frac {8}{3};-i \tan (c+d x),i \tan (c+d x)\right )}{5 d \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 130
Rule 510
Rule 511
Rule 3564
Rubi steps
\begin {align*} \int \frac {\tan ^{\frac {2}{3}}(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx &=\frac {\left (i a^2\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {i x}{a}\right )^{2/3}}{(a+x)^{3/2} \left (-a^2+a x\right )} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (a+i a x^3\right )^{3/2} \left (-a^2+i a^2 x^3\right )} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{d}\\ &=-\frac {\left (3 a^2 \sqrt {1+i \tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (1+i x^3\right )^{3/2} \left (-a^2+i a^2 x^3\right )} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{d \sqrt {a+i a \tan (c+d x)}}\\ &=\frac {3 F_1\left (\frac {5}{3};\frac {3}{2},1;\frac {8}{3};-i \tan (c+d x),i \tan (c+d x)\right ) \sqrt {1+i \tan (c+d x)} \tan ^{\frac {5}{3}}(c+d x)}{5 d \sqrt {a+i a \tan (c+d x)}}\\ \end {align*}
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Mathematica [F] time = 2.62, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{\frac {2}{3}}(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.09, size = 0, normalized size = 0.00 \[ \frac {\sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {2}{3}} {\left (2 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 4 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 2 i\right )} + {\left (a d e^{\left (3 i \, d x + 3 i \, c\right )} - 4 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + 4 \, a d e^{\left (i \, d x + i \, c\right )}\right )} {\rm integral}\left (\frac {\sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {2}{3}} {\left (-3 i \, e^{\left (5 i \, d x + 5 i \, c\right )} + 66 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 32 i \, e^{\left (3 i \, d x + 3 i \, c\right )} + 64 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 29 i \, e^{\left (i \, d x + i \, c\right )} - 2 i\right )}}{6 \, {\left (a d e^{\left (5 i \, d x + 5 i \, c\right )} - 6 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + 11 \, a d e^{\left (3 i \, d x + 3 i \, c\right )} - 2 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} - 12 \, a d e^{\left (i \, d x + i \, c\right )} + 8 \, a d\right )}}, x\right )}{a d e^{\left (3 i \, d x + 3 i \, c\right )} - 4 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + 4 \, a d e^{\left (i \, d x + i \, c\right )}} \]
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 \[ \int \frac {\tan \left (d x + c\right )^{\frac {2}{3}}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.49, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{\frac {2}{3}}\left (d x +c \right )}{\sqrt {a +i a \tan \left (d x +c \right )}}\, dx \]
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 \[ \int \frac {\tan \left (d x + c\right )^{\frac {2}{3}}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {tan}\left (c+d\,x\right )}^{2/3}}{\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}} \,d x \]
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 \[ \int \frac {\tan ^{\frac {2}{3}}{\left (c + d x \right )}}{\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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