Optimal. Leaf size=148 \[ -\frac{\sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt{3}}\right )}{f \sqrt [3]{c-i d}}-\frac{3 \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c-i d}\right )}{2 f \sqrt [3]{c-i d}}-\frac{\log (\cos (e+f x))}{2 f \sqrt [3]{c-i d}}-\frac{i x}{2 \sqrt [3]{c-i d}} \]
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Rubi [A] time = 0.125532, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {3537, 55, 617, 204, 31} \[ -\frac{\sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt{3}}\right )}{f \sqrt [3]{c-i d}}-\frac{3 \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c-i d}\right )}{2 f \sqrt [3]{c-i d}}-\frac{\log (\cos (e+f x))}{2 f \sqrt [3]{c-i d}}-\frac{i x}{2 \sqrt [3]{c-i d}} \]
Antiderivative was successfully verified.
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Rule 3537
Rule 55
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{i-\tan (e+f x)}{\sqrt [3]{c+d \tan (e+f x)}} \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{1}{(1+i x) \sqrt [3]{c-d x}} \, dx,x,-\tan (e+f x)\right )}{f}\\ &=-\frac{i x}{2 \sqrt [3]{c-i d}}-\frac{\log (\cos (e+f x))}{2 \sqrt [3]{c-i d} f}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{(c-i d)^{2/3}+\sqrt [3]{c-i d} x+x^2} \, dx,x,\sqrt [3]{c+d \tan (e+f x)}\right )}{2 f}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{c-i d}-x} \, dx,x,\sqrt [3]{c+d \tan (e+f x)}\right )}{2 \sqrt [3]{c-i d} f}\\ &=-\frac{i x}{2 \sqrt [3]{c-i d}}-\frac{\log (\cos (e+f x))}{2 \sqrt [3]{c-i d} f}-\frac{3 \log \left (\sqrt [3]{c-i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 \sqrt [3]{c-i d} f}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}\right )}{\sqrt [3]{c-i d} f}\\ &=-\frac{i x}{2 \sqrt [3]{c-i d}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt{3}}\right )}{\sqrt [3]{c-i d} f}-\frac{\log (\cos (e+f x))}{2 \sqrt [3]{c-i d} f}-\frac{3 \log \left (\sqrt [3]{c-i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 \sqrt [3]{c-i d} f}\\ \end{align*}
Mathematica [C] time = 1.72742, size = 109, normalized size = 0.74 \[ \frac{3 \left (c-\frac{i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}\right )^{2/3} \text{Hypergeometric2F1}\left (\frac{2}{3},1,\frac{5}{3},\frac{i c+\frac{d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}{d+i c}\right )}{2 f (c-i d)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.086, size = 42, normalized size = 0.3 \begin{align*} -{\frac{1}{f}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}+id-c \right ) }{\frac{1}{{\it \_R}}\ln \left ( \sqrt [3]{c+d\tan \left ( fx+e \right ) }-{\it \_R} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{\tan \left (f x + e\right ) - i}{{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.18713, size = 757, normalized size = 5.11 \begin{align*} \frac{1}{2} \,{\left (i \, \sqrt{3} - 1\right )} \left (-\frac{i}{{\left (i \, c + d\right )} f^{3}}\right )^{\frac{1}{3}} \log \left (\frac{1}{2} \,{\left (\sqrt{3}{\left (i \, c + d\right )} f^{2} +{\left (c - i \, d\right )} f^{2}\right )} \left (-\frac{i}{{\left (i \, c + d\right )} f^{3}}\right )^{\frac{2}{3}} + \left (\frac{{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac{1}{3}}\right ) + \frac{1}{2} \,{\left (-i \, \sqrt{3} - 1\right )} \left (-\frac{i}{{\left (i \, c + d\right )} f^{3}}\right )^{\frac{1}{3}} \log \left (\frac{1}{2} \,{\left (\sqrt{3}{\left (-i \, c - d\right )} f^{2} +{\left (c - i \, d\right )} f^{2}\right )} \left (-\frac{i}{{\left (i \, c + d\right )} f^{3}}\right )^{\frac{2}{3}} + \left (\frac{{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac{1}{3}}\right ) + \left (-\frac{i}{{\left (i \, c + d\right )} f^{3}}\right )^{\frac{1}{3}} \log \left (-{\left (c - i \, d\right )} f^{2} \left (-\frac{i}{{\left (i \, c + d\right )} f^{3}}\right )^{\frac{2}{3}} + \left (\frac{{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac{1}{3}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{- \tan{\left (e + f x \right )} + i}{\sqrt [3]{c + d \tan{\left (e + f x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.57748, size = 1229, normalized size = 8.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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