Optimal. Leaf size=128 \[ -\frac{\log \left (\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}+1\right )}{2 b}+\frac{\log \left (\frac{\sin ^{\frac{4}{3}}(a+b x)}{\cos ^{\frac{4}{3}}(a+b x)}-\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}+1\right )}{4 b}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}}{\sqrt{3}}\right )}{2 b} \]
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Rubi [A] time = 0.15127, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {2574, 275, 292, 31, 634, 618, 204, 628} \[ -\frac{\log \left (\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}+1\right )}{2 b}+\frac{\log \left (\frac{\sin ^{\frac{4}{3}}(a+b x)}{\cos ^{\frac{4}{3}}(a+b x)}-\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}+1\right )}{4 b}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}}{\sqrt{3}}\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 2574
Rule 275
Rule 292
Rule 31
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}} \, dx &=\frac{3 \operatorname{Subst}\left (\int \frac{x^3}{1+x^6} \, dx,x,\frac{\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{b}\\ &=\frac{3 \operatorname{Subst}\left (\int \frac{x}{1+x^3} \, dx,x,\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}\right )}{2 b}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}\right )}{2 b}+\frac{\operatorname{Subst}\left (\int \frac{1+x}{1-x+x^2} \, dx,x,\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}\right )}{2 b}\\ &=-\frac{\log \left (1+\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}\right )}{2 b}+\frac{\operatorname{Subst}\left (\int \frac{-1+2 x}{1-x+x^2} \, dx,x,\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}\right )}{4 b}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{1-x+x^2} \, dx,x,\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}\right )}{4 b}\\ &=-\frac{\log \left (1+\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}\right )}{2 b}+\frac{\log \left (1-\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}+\frac{\sin ^{\frac{4}{3}}(a+b x)}{\cos ^{\frac{4}{3}}(a+b x)}\right )}{4 b}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+\frac{2 \sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}\right )}{2 b}\\ &=-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}}{\sqrt{3}}\right )}{2 b}-\frac{\log \left (1+\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}\right )}{2 b}+\frac{\log \left (1-\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}+\frac{\sin ^{\frac{4}{3}}(a+b x)}{\cos ^{\frac{4}{3}}(a+b x)}\right )}{4 b}\\ \end{align*}
Mathematica [C] time = 0.040584, size = 57, normalized size = 0.45 \[ \frac{3 \sin ^{\frac{4}{3}}(a+b x) \cos ^2(a+b x)^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\sin ^2(a+b x)\right )}{4 b \cos ^{\frac{4}{3}}(a+b x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.096, size = 0, normalized size = 0. \begin{align*} \int{\sqrt [3]{\sin \left ( bx+a \right ) }{\frac{1}{\sqrt [3]{\cos \left ( bx+a \right ) }}}}\, dx \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{\sin \left (b x + a\right )^{\frac{1}{3}}}{\cos \left (b x + a\right )^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.24578, size = 424, normalized size = 3.31 \begin{align*} \frac{2 \, \sqrt{3} \arctan \left (-\frac{\sqrt{3} \cos \left (b x + a\right ) - 2 \, \sqrt{3} \cos \left (b x + a\right )^{\frac{1}{3}} \sin \left (b x + a\right )^{\frac{2}{3}}}{3 \, \cos \left (b x + a\right )}\right ) - 2 \, \log \left (\frac{\cos \left (b x + a\right )^{\frac{1}{3}} \sin \left (b x + a\right )^{\frac{2}{3}} + \cos \left (b x + a\right )}{\cos \left (b x + a\right )}\right ) + \log \left (\frac{\cos \left (b x + a\right )^{2} - \cos \left (b x + a\right )^{\frac{4}{3}} \sin \left (b x + a\right )^{\frac{2}{3}} + \cos \left (b x + a\right )^{\frac{2}{3}} \sin \left (b x + a\right )^{\frac{4}{3}}}{\cos \left (b x + a\right )^{2}}\right )}{4 \, b} \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{\sqrt [3]{\sin{\left (a + b x \right )}}}{\sqrt [3]{\cos{\left (a + b x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (b x + a\right )^{\frac{1}{3}}}{\cos \left (b x + a\right )^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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