Optimal. Leaf size=83 \[ \frac {\sqrt [3]{1+i \tan (c+d x)} \tan ^{m+1}(c+d x) F_1\left (m+1;\frac {4}{3},1;m+2;-i \tan (c+d x),i \tan (c+d x)\right )}{d (m+1) \sqrt [3]{a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.10, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {3564, 135, 133} \[ \frac {\sqrt [3]{1+i \tan (c+d x)} \tan ^{m+1}(c+d x) F_1\left (m+1;\frac {4}{3},1;m+2;-i \tan (c+d x),i \tan (c+d x)\right )}{d (m+1) \sqrt [3]{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 133
Rule 135
Rule 3564
Rubi steps
\begin {align*} \int \frac {\tan ^m(c+d x)}{\sqrt [3]{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 )^m}{(a+x)^{4/3} \left (-a^2+a x\right )} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=\frac {\left (i a \sqrt [3]{1+i \tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {i x}{a}\right )^m}{\left (1+\frac {x}{a}\right )^{4/3} \left (-a^2+a x\right )} \, dx,x,i a \tan (c+d x)\right )}{d \sqrt [3]{a+i a \tan (c+d x)}}\\ &=\frac {F_1\left (1+m;\frac {4}{3},1;2+m;-i \tan (c+d x),i \tan (c+d x)\right ) \sqrt [3]{1+i \tan (c+d x)} \tan ^{1+m}(c+d x)}{d (1+m) \sqrt [3]{a+i a \tan (c+d x)}}\\ \end {align*}
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Mathematica [F] time = 180.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]
Verification is Not applicable to the result.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {2^{\frac {2}{3}} \left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{m} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {2}{3}} {\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (-\frac {2}{3} i \, d x - \frac {2}{3} i \, c\right )}}{2 \, a}, x\right ) \]
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 \[ \int \frac {\tan \left (d x + c\right )^{m}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.85, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{m}\left (d x +c \right )}{\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {\tan \left (d x + c\right )^{m}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {tan}\left (c+d\,x\right )}^m}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}} \,d x \]
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 \[ \int \frac {\tan ^{m}{\left (c + d x \right )}}{\sqrt [3]{i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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