Optimal. Leaf size=250 \[ -\frac{3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}-\frac{\sqrt{3} \log \left (\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}-\frac{\sqrt{3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1\right )}{4 b}+\frac{\sqrt{3} \log \left (\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}+\frac{\sqrt{3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1\right )}{4 b}+\frac{\tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{2 b}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\sqrt{3}\right )}{2 b}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{b} \]
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Rubi [A] time = 0.325825, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {2567, 2574, 295, 634, 618, 204, 628, 203} \[ -\frac{3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}-\frac{\sqrt{3} \log \left (\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}-\frac{\sqrt{3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1\right )}{4 b}+\frac{\sqrt{3} \log \left (\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}+\frac{\sqrt{3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+1\right )}{4 b}+\frac{\tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{2 b}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\sqrt{3}\right )}{2 b}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2567
Rule 2574
Rule 295
Rule 634
Rule 618
Rule 204
Rule 628
Rule 203
Rubi steps
\begin{align*} \int \frac{\cos ^{\frac{4}{3}}(a+b x)}{\sin ^{\frac{4}{3}}(a+b x)} \, dx &=-\frac{3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}-\int \frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)} \, dx\\ &=-\frac{3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}-\frac{3 \operatorname{Subst}\left (\int \frac{x^4}{1+x^6} \, dx,x,\frac{\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{b}\\ &=-\frac{3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{b}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{2}+\frac{\sqrt{3} x}{2}}{1-\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{b}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{2}-\frac{\sqrt{3} x}{2}}{1+\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{b}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{b}-\frac{3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{1-\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{4 b}-\frac{\operatorname{Subst}\left (\int \frac{1}{1+\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{4 b}-\frac{\sqrt{3} \operatorname{Subst}\left (\int \frac{-\sqrt{3}+2 x}{1-\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{4 b}+\frac{\sqrt{3} \operatorname{Subst}\left (\int \frac{\sqrt{3}+2 x}{1+\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{4 b}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{b}-\frac{\sqrt{3} \log \left (1-\frac{\sqrt{3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}\right )}{4 b}+\frac{\sqrt{3} \log \left (1+\frac{\sqrt{3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}\right )}{4 b}-\frac{3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,-\sqrt{3}+\frac{2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{2 b}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{3}+\frac{2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{2 b}\\ &=\frac{\tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{2 b}-\frac{\tan ^{-1}\left (\sqrt{3}+\frac{2 \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{2 b}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}\right )}{b}-\frac{\sqrt{3} \log \left (1-\frac{\sqrt{3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}\right )}{4 b}+\frac{\sqrt{3} \log \left (1+\frac{\sqrt{3} \sqrt [3]{\sin (a+b x)}}{\sqrt [3]{\cos (a+b x)}}+\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}\right )}{4 b}-\frac{3 \sqrt [3]{\cos (a+b x)}}{b \sqrt [3]{\sin (a+b x)}}\\ \end{align*}
Mathematica [C] time = 0.0337652, size = 55, normalized size = 0.22 \[ -\frac{3 \cos ^2(a+b x)^{5/6} \, _2F_1\left (-\frac{1}{6},-\frac{1}{6};\frac{5}{6};\sin ^2(a+b x)\right )}{b \sqrt [3]{\sin (a+b x)} \cos ^{\frac{5}{3}}(a+b x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \cos \left ( bx+a \right ) \right ) ^{{\frac{4}{3}}} \left ( \sin \left ( bx+a \right ) \right ) ^{-{\frac{4}{3}}}}\, 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{\cos \left (b x + a\right )^{\frac{4}{3}}}{\sin \left (b x + a\right )^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\cos \left (b x + a\right )^{\frac{4}{3}}}{\sin \left (b x + a\right )^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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