Optimal. Leaf size=84 \[ \frac {3 F_1\left (\frac {5}{3};\frac {5}{2},1;\frac {8}{3};-i \tan (c+d x),i \tan (c+d x)\right ) \sqrt {1+i \tan (c+d x)} \tan ^{\frac {5}{3}}(c+d x)}{5 a d \sqrt {a+i a \tan (c+d x)}} \]
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Rubi [A]
time = 0.10, antiderivative size = 84, 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 = {3645, 129, 525,
524} \begin {gather*} \frac {3 \sqrt {1+i \tan (c+d x)} \tan ^{\frac {5}{3}}(c+d x) F_1\left (\frac {5}{3};\frac {5}{2},1;\frac {8}{3};-i \tan (c+d x),i \tan (c+d x)\right )}{5 a d \sqrt {a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 129
Rule 524
Rule 525
Rule 3645
Rubi steps
\begin {align*} \int \frac {\tan ^{\frac {2}{3}}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx &=\frac {\left (i a^2\right ) \text {Subst}\left (\int \frac {\left (-\frac {i x}{a}\right )^{2/3}}{(a+x)^{5/2} \left (-a^2+a x\right )} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {\left (3 a^3\right ) \text {Subst}\left (\int \frac {x^4}{\left (a+i a x^3\right )^{5/2} \left (-a^2+i a^2 x^3\right )} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{d}\\ &=-\frac {\left (3 a \sqrt {1+i \tan (c+d x)}\right ) \text {Subst}\left (\int \frac {x^4}{\left (1+i x^3\right )^{5/2} \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 {5}{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 a d \sqrt {a+i a \tan (c+d x)}}\\ \end {align*}
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Mathematica [F]
time = 12.19, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tan ^{\frac {2}{3}}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 1.03, size = 0, normalized size = 0.00 \[\int \frac {\tan ^{\frac {2}{3}}\left (d x +c \right )}{\left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}\, dx\]
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\tan ^{\frac {2}{3}}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {tan}\left (c+d\,x\right )}^{2/3}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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