Optimal. Leaf size=466 \[ -\frac{\sqrt{3} a^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{\tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{b^{2/3} d \left (a^2+b^2\right )}+\frac{\sqrt{3} a \tan ^{-1}\left (\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right )}{2 d \left (a^2+b^2\right )}-\frac{b \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{\tan (c+d x)}\right )}{2 d \left (a^2+b^2\right )}+\frac{b \tan ^{-1}\left (2 \sqrt [3]{\tan (c+d x)}+\sqrt{3}\right )}{2 d \left (a^2+b^2\right )}+\frac{b \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{d \left (a^2+b^2\right )}-\frac{a \log \left (\tan ^{\frac{2}{3}}(c+d x)+1\right )}{2 d \left (a^2+b^2\right )}+\frac{a \log \left (\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right )}{4 d \left (a^2+b^2\right )}+\frac{\sqrt{3} b \log \left (\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt [3]{\tan (c+d x)}+1\right )}{4 d \left (a^2+b^2\right )}-\frac{\sqrt{3} b \log \left (\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt [3]{\tan (c+d x)}+1\right )}{4 d \left (a^2+b^2\right )}-\frac{3 a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{\tan (c+d x)}\right )}{2 b^{2/3} d \left (a^2+b^2\right )}+\frac{a^{5/3} \log (a+b \tan (c+d x))}{2 b^{2/3} d \left (a^2+b^2\right )} \]
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Rubi [A] time = 0.689228, antiderivative size = 466, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 18, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.783, Rules used = {3574, 3528, 3538, 3476, 329, 275, 200, 31, 634, 618, 204, 628, 295, 203, 3634, 50, 56, 617} \[ -\frac{\sqrt{3} a^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{\tan (c+d x)}}{\sqrt{3} \sqrt [3]{a}}\right )}{b^{2/3} d \left (a^2+b^2\right )}+\frac{\sqrt{3} a \tan ^{-1}\left (\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right )}{2 d \left (a^2+b^2\right )}-\frac{b \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{\tan (c+d x)}\right )}{2 d \left (a^2+b^2\right )}+\frac{b \tan ^{-1}\left (2 \sqrt [3]{\tan (c+d x)}+\sqrt{3}\right )}{2 d \left (a^2+b^2\right )}+\frac{b \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{d \left (a^2+b^2\right )}-\frac{a \log \left (\tan ^{\frac{2}{3}}(c+d x)+1\right )}{2 d \left (a^2+b^2\right )}+\frac{a \log \left (\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right )}{4 d \left (a^2+b^2\right )}+\frac{\sqrt{3} b \log \left (\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt [3]{\tan (c+d x)}+1\right )}{4 d \left (a^2+b^2\right )}-\frac{\sqrt{3} b \log \left (\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt [3]{\tan (c+d x)}+1\right )}{4 d \left (a^2+b^2\right )}-\frac{3 a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{\tan (c+d x)}\right )}{2 b^{2/3} d \left (a^2+b^2\right )}+\frac{a^{5/3} \log (a+b \tan (c+d x))}{2 b^{2/3} d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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Rule 3574
Rule 3528
Rule 3538
Rule 3476
Rule 329
Rule 275
Rule 200
Rule 31
Rule 634
Rule 618
Rule 204
Rule 628
Rule 295
Rule 203
Rule 3634
Rule 50
Rule 56
Rule 617
Rubi steps
\begin{align*} \int \frac{\tan ^{\frac{5}{3}}(c+d x)}{a+b \tan (c+d x)} \, dx &=\frac{\int \tan ^{\frac{5}{3}}(c+d x) (a-b \tan (c+d x)) \, dx}{a^2+b^2}+\frac{b^2 \int \frac{\tan ^{\frac{5}{3}}(c+d x) \left (1+\tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{a^2+b^2}\\ &=-\frac{3 b \tan ^{\frac{5}{3}}(c+d x)}{5 \left (a^2+b^2\right ) d}+\frac{\int \tan ^{\frac{2}{3}}(c+d x) (b+a \tan (c+d x)) \, dx}{a^2+b^2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{x^{5/3}}{a+b x} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=\frac{3 a \tan ^{\frac{2}{3}}(c+d x)}{2 \left (a^2+b^2\right ) d}+\frac{\int \frac{-a+b \tan (c+d x)}{\sqrt [3]{\tan (c+d x)}} \, dx}{a^2+b^2}-\frac{(a b) \operatorname{Subst}\left (\int \frac{x^{2/3}}{a+b x} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=-\frac{a \int \frac{1}{\sqrt [3]{\tan (c+d x)}} \, dx}{a^2+b^2}+\frac{b \int \tan ^{\frac{2}{3}}(c+d x) \, dx}{a^2+b^2}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=\frac{a^{5/3} \log (a+b \tan (c+d x))}{2 b^{2/3} \left (a^2+b^2\right ) d}-\frac{a \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}+\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{a^{2/3}}{b^{2/3}}-\frac{\sqrt [3]{a} x}{\sqrt [3]{b}}+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{2 b \left (a^2+b^2\right ) d}-\frac{\left (3 a^{5/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{2 b^{2/3} \left (a^2+b^2\right ) d}+\frac{b \operatorname{Subst}\left (\int \frac{x^{2/3}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=-\frac{3 a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{\tan (c+d x)}\right )}{2 b^{2/3} \left (a^2+b^2\right ) d}+\frac{a^{5/3} \log (a+b \tan (c+d x))}{2 b^{2/3} \left (a^2+b^2\right ) d}-\frac{(3 a) \operatorname{Subst}\left (\int \frac{x}{1+x^6} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}+\frac{\left (3 a^{5/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} \sqrt [3]{\tan (c+d x)}}{\sqrt [3]{a}}\right )}{b^{2/3} \left (a^2+b^2\right ) d}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{x^4}{1+x^6} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}\\ &=-\frac{\sqrt{3} a^{5/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{\tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{2/3} \left (a^2+b^2\right ) d}-\frac{3 a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{\tan (c+d x)}\right )}{2 b^{2/3} \left (a^2+b^2\right ) d}+\frac{a^{5/3} \log (a+b \tan (c+d x))}{2 b^{2/3} \left (a^2+b^2\right ) d}-\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{1+x^3} \, dx,x,\tan ^{\frac{2}{3}}(c+d x)\right )}{2 \left (a^2+b^2\right ) d}+\frac{b \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}+\frac{b \operatorname{Subst}\left (\int \frac{-\frac{1}{2}+\frac{\sqrt{3} x}{2}}{1-\sqrt{3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}+\frac{b \operatorname{Subst}\left (\int \frac{-\frac{1}{2}-\frac{\sqrt{3} x}{2}}{1+\sqrt{3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}\\ &=-\frac{\sqrt{3} a^{5/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{\tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{2/3} \left (a^2+b^2\right ) d}+\frac{b \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac{3 a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{\tan (c+d x)}\right )}{2 b^{2/3} \left (a^2+b^2\right ) d}+\frac{a^{5/3} \log (a+b \tan (c+d x))}{2 b^{2/3} \left (a^2+b^2\right ) d}-\frac{a \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\tan ^{\frac{2}{3}}(c+d x)\right )}{2 \left (a^2+b^2\right ) d}-\frac{a \operatorname{Subst}\left (\int \frac{2-x}{1-x+x^2} \, dx,x,\tan ^{\frac{2}{3}}(c+d x)\right )}{2 \left (a^2+b^2\right ) d}+\frac{b \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{4 \left (a^2+b^2\right ) d}+\frac{b \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{4 \left (a^2+b^2\right ) d}+\frac{\left (\sqrt{3} b\right ) \operatorname{Subst}\left (\int \frac{-\sqrt{3}+2 x}{1-\sqrt{3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{4 \left (a^2+b^2\right ) d}-\frac{\left (\sqrt{3} b\right ) \operatorname{Subst}\left (\int \frac{\sqrt{3}+2 x}{1+\sqrt{3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{4 \left (a^2+b^2\right ) d}\\ &=-\frac{\sqrt{3} a^{5/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{\tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{2/3} \left (a^2+b^2\right ) d}+\frac{b \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac{3 a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{\tan (c+d x)}\right )}{2 b^{2/3} \left (a^2+b^2\right ) d}-\frac{a \log \left (1+\tan ^{\frac{2}{3}}(c+d x)\right )}{2 \left (a^2+b^2\right ) d}+\frac{\sqrt{3} b \log \left (1-\sqrt{3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac{2}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}-\frac{\sqrt{3} b \log \left (1+\sqrt{3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac{2}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}+\frac{a^{5/3} \log (a+b \tan (c+d x))}{2 b^{2/3} \left (a^2+b^2\right ) d}+\frac{a \operatorname{Subst}\left (\int \frac{-1+2 x}{1-x+x^2} \, dx,x,\tan ^{\frac{2}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}-\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{1-x+x^2} \, dx,x,\tan ^{\frac{2}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}-\frac{b \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,-\sqrt{3}+2 \sqrt [3]{\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}-\frac{b \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{3}+2 \sqrt [3]{\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}\\ &=-\frac{b \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}+\frac{b \tan ^{-1}\left (\sqrt{3}+2 \sqrt [3]{\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}-\frac{\sqrt{3} a^{5/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{\tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{2/3} \left (a^2+b^2\right ) d}+\frac{b \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac{3 a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{\tan (c+d x)}\right )}{2 b^{2/3} \left (a^2+b^2\right ) d}-\frac{a \log \left (1+\tan ^{\frac{2}{3}}(c+d x)\right )}{2 \left (a^2+b^2\right ) d}+\frac{\sqrt{3} b \log \left (1-\sqrt{3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac{2}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}-\frac{\sqrt{3} b \log \left (1+\sqrt{3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac{2}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}+\frac{a^{5/3} \log (a+b \tan (c+d x))}{2 b^{2/3} \left (a^2+b^2\right ) d}+\frac{a \log \left (1-\tan ^{\frac{2}{3}}(c+d x)+\tan ^{\frac{4}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}+\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 \tan ^{\frac{2}{3}}(c+d x)\right )}{2 \left (a^2+b^2\right ) d}\\ &=-\frac{b \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}+\frac{b \tan ^{-1}\left (\sqrt{3}+2 \sqrt [3]{\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}-\frac{\sqrt{3} a^{5/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{\tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{2/3} \left (a^2+b^2\right ) d}+\frac{\sqrt{3} a \tan ^{-1}\left (\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right )}{2 \left (a^2+b^2\right ) d}+\frac{b \tan ^{-1}\left (\sqrt [3]{\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac{3 a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{\tan (c+d x)}\right )}{2 b^{2/3} \left (a^2+b^2\right ) d}-\frac{a \log \left (1+\tan ^{\frac{2}{3}}(c+d x)\right )}{2 \left (a^2+b^2\right ) d}+\frac{\sqrt{3} b \log \left (1-\sqrt{3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac{2}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}-\frac{\sqrt{3} b \log \left (1+\sqrt{3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac{2}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}+\frac{a^{5/3} \log (a+b \tan (c+d x))}{2 b^{2/3} \left (a^2+b^2\right ) d}+\frac{a \log \left (1-\tan ^{\frac{2}{3}}(c+d x)+\tan ^{\frac{4}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}\\ \end{align*}
Mathematica [C] time = 0.353421, size = 153, normalized size = 0.33 \[ \frac{30 a \tan ^{\frac{2}{3}}(c+d x) \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};-\frac{b \tan (c+d x)}{a}\right )+5 a \left (2 \sqrt{3} \tan ^{-1}\left (\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right )-2 \log \left (\tan ^{\frac{2}{3}}(c+d x)+1\right )+\log \left (\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right )\right )+12 b \tan ^{\frac{5}{3}}(c+d x) \, _2F_1\left (\frac{5}{6},1;\frac{11}{6};-\tan ^2(c+d x)\right )}{20 d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 541, normalized size = 1.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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{\tan \left (d x + c\right )^{\frac{5}{3}}}{b \tan \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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