Optimal. Leaf size=336 \[ -\frac{3 b}{d \left (a^2+b^2\right ) \sqrt [3]{a+b \tan (c+d x)}}+\frac{i \sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-i b}}}{\sqrt{3}}\right )}{2 d (a-i b)^{4/3}}-\frac{i \sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+i b}}}{\sqrt{3}}\right )}{2 d (a+i b)^{4/3}}+\frac{3 i \log \left (-\sqrt [3]{a+b \tan (c+d x)}+\sqrt [3]{a-i b}\right )}{4 d (a-i b)^{4/3}}-\frac{3 i \log \left (-\sqrt [3]{a+b \tan (c+d x)}+\sqrt [3]{a+i b}\right )}{4 d (a+i b)^{4/3}}+\frac{i \log (\cos (c+d x))}{4 d (a-i b)^{4/3}}-\frac{i \log (\cos (c+d x))}{4 d (a+i b)^{4/3}}-\frac{x}{4 (a-i b)^{4/3}}-\frac{x}{4 (a+i b)^{4/3}} \]
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Rubi [A] time = 0.357756, antiderivative size = 336, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3483, 3539, 3537, 55, 617, 204, 31} \[ -\frac{3 b}{d \left (a^2+b^2\right ) \sqrt [3]{a+b \tan (c+d x)}}+\frac{i \sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-i b}}}{\sqrt{3}}\right )}{2 d (a-i b)^{4/3}}-\frac{i \sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+i b}}}{\sqrt{3}}\right )}{2 d (a+i b)^{4/3}}+\frac{3 i \log \left (-\sqrt [3]{a+b \tan (c+d x)}+\sqrt [3]{a-i b}\right )}{4 d (a-i b)^{4/3}}-\frac{3 i \log \left (-\sqrt [3]{a+b \tan (c+d x)}+\sqrt [3]{a+i b}\right )}{4 d (a+i b)^{4/3}}+\frac{i \log (\cos (c+d x))}{4 d (a-i b)^{4/3}}-\frac{i \log (\cos (c+d x))}{4 d (a+i b)^{4/3}}-\frac{x}{4 (a-i b)^{4/3}}-\frac{x}{4 (a+i b)^{4/3}} \]
Antiderivative was successfully verified.
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Rule 3483
Rule 3539
Rule 3537
Rule 55
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{(a+b \tan (c+d x))^{4/3}} \, dx &=-\frac{3 b}{\left (a^2+b^2\right ) d \sqrt [3]{a+b \tan (c+d x)}}+\frac{\int \frac{a-b \tan (c+d x)}{\sqrt [3]{a+b \tan (c+d x)}} \, dx}{a^2+b^2}\\ &=-\frac{3 b}{\left (a^2+b^2\right ) d \sqrt [3]{a+b \tan (c+d x)}}+\frac{\int \frac{1+i \tan (c+d x)}{\sqrt [3]{a+b \tan (c+d x)}} \, dx}{2 (a-i b)}+\frac{\int \frac{1-i \tan (c+d x)}{\sqrt [3]{a+b \tan (c+d x)}} \, dx}{2 (a+i b)}\\ &=-\frac{3 b}{\left (a^2+b^2\right ) d \sqrt [3]{a+b \tan (c+d x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt [3]{a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 (i a-b) d}-\frac{\operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt [3]{a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 (i a+b) d}\\ &=-\frac{x}{4 (a-i b)^{4/3}}-\frac{x}{4 (a+i b)^{4/3}}+\frac{i \log (\cos (c+d x))}{4 (a-i b)^{4/3} d}-\frac{i \log (\cos (c+d x))}{4 (a+i b)^{4/3} d}-\frac{3 b}{\left (a^2+b^2\right ) d \sqrt [3]{a+b \tan (c+d x)}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{(a+i b)^{2/3}+\sqrt [3]{a+i b} x+x^2} \, dx,x,\sqrt [3]{a+b \tan (c+d x)}\right )}{4 (i a-b) d}-\frac{(3 i) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a-i b}-x} \, dx,x,\sqrt [3]{a+b \tan (c+d x)}\right )}{4 (a-i b)^{4/3} d}+\frac{(3 i) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a+i b}-x} \, dx,x,\sqrt [3]{a+b \tan (c+d x)}\right )}{4 (a+i b)^{4/3} d}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{(a-i b)^{2/3}+\sqrt [3]{a-i b} x+x^2} \, dx,x,\sqrt [3]{a+b \tan (c+d x)}\right )}{4 (i a+b) d}\\ &=-\frac{x}{4 (a-i b)^{4/3}}-\frac{x}{4 (a+i b)^{4/3}}+\frac{i \log (\cos (c+d x))}{4 (a-i b)^{4/3} d}-\frac{i \log (\cos (c+d x))}{4 (a+i b)^{4/3} d}+\frac{3 i \log \left (\sqrt [3]{a-i b}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 (a-i b)^{4/3} d}-\frac{3 i \log \left (\sqrt [3]{a+i b}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 (a+i b)^{4/3} d}-\frac{3 b}{\left (a^2+b^2\right ) d \sqrt [3]{a+b \tan (c+d x)}}-\frac{(3 i) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-i b}}\right )}{2 (a-i b)^{4/3} d}+\frac{(3 i) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+i b}}\right )}{2 (a+i b)^{4/3} d}\\ &=-\frac{x}{4 (a-i b)^{4/3}}-\frac{x}{4 (a+i b)^{4/3}}+\frac{i \sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-i b}}}{\sqrt{3}}\right )}{2 (a-i b)^{4/3} d}-\frac{i \sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+i b}}}{\sqrt{3}}\right )}{2 (a+i b)^{4/3} d}+\frac{i \log (\cos (c+d x))}{4 (a-i b)^{4/3} d}-\frac{i \log (\cos (c+d x))}{4 (a+i b)^{4/3} d}+\frac{3 i \log \left (\sqrt [3]{a-i b}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 (a-i b)^{4/3} d}-\frac{3 i \log \left (\sqrt [3]{a+i b}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 (a+i b)^{4/3} d}-\frac{3 b}{\left (a^2+b^2\right ) d \sqrt [3]{a+b \tan (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.209676, size = 106, normalized size = 0.32 \[ \frac{3 i \left ((a+i b) \, _2F_1\left (-\frac{1}{3},1;\frac{2}{3};\frac{a+b \tan (c+d x)}{a-i b}\right )-(a-i b) \, _2F_1\left (-\frac{1}{3},1;\frac{2}{3};\frac{a+b \tan (c+d x)}{a+i b}\right )\right )}{2 d \left (a^2+b^2\right ) \sqrt [3]{a+b \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.019, size = 102, normalized size = 0.3 \begin{align*} -3\,{\frac{b}{ \left ({a}^{2}+{b}^{2} \right ) d\sqrt [3]{a+b\tan \left ( dx+c \right ) }}}-{\frac{b}{2\, \left ({a}^{2}+{b}^{2} \right ) d}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}-2\,{{\it \_Z}}^{3}a+{a}^{2}+{b}^{2} \right ) }{\frac{{{\it \_R}}^{4}-2\,{\it \_R}\,a}{{{\it \_R}}^{5}-{{\it \_R}}^{2}a}\ln \left ( \sqrt [3]{a+b\tan \left ( dx+c \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{1}{{\left (b \tan \left (d x + c\right ) + 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \tan{\left (c + d x \right )}\right )^{\frac{4}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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