Optimal. Leaf size=402 \[ -\frac {\sqrt {3} \sqrt [3]{c} \tan ^{-1}\left (\frac {2 \sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c}}{\sqrt {3} \sqrt [3]{c}}\right )}{f}+\frac {\sqrt {3} \sqrt [3]{c-i d} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt {3}}\right )}{2 f}+\frac {\sqrt {3} \sqrt [3]{c+i d} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )}{2 f}+\frac {3 \sqrt [3]{c} \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 f}-\frac {3 \sqrt [3]{c-i d} \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c-i d}\right )}{4 f}-\frac {3 \sqrt [3]{c+i d} \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c+i d}\right )}{4 f}-\frac {\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}-\frac {\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}-\frac {1}{4} i x \sqrt [3]{c-i d}+\frac {1}{4} i x \sqrt [3]{c+i d}-\frac {\sqrt [3]{c} \log (\tan (e+f x))}{2 f} \]
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Rubi [A] time = 0.48, antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3574, 3528, 3539, 3537, 57, 617, 204, 31, 3634, 50} \[ -\frac {\sqrt {3} \sqrt [3]{c} \tan ^{-1}\left (\frac {2 \sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c}}{\sqrt {3} \sqrt [3]{c}}\right )}{f}+\frac {\sqrt {3} \sqrt [3]{c-i d} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt {3}}\right )}{2 f}+\frac {\sqrt {3} \sqrt [3]{c+i d} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )}{2 f}+\frac {3 \sqrt [3]{c} \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 f}-\frac {3 \sqrt [3]{c-i d} \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c-i d}\right )}{4 f}-\frac {3 \sqrt [3]{c+i d} \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c+i d}\right )}{4 f}-\frac {\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}-\frac {\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}-\frac {1}{4} i x \sqrt [3]{c-i d}+\frac {1}{4} i x \sqrt [3]{c+i d}-\frac {\sqrt [3]{c} \log (\tan (e+f x))}{2 f} \]
Antiderivative was successfully verified.
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Rule 31
Rule 50
Rule 57
Rule 204
Rule 617
Rule 3528
Rule 3537
Rule 3539
Rule 3574
Rule 3634
Rubi steps
\begin {align*} \int \cot (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx &=-\int \tan (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx+\int \cot (e+f x) \sqrt [3]{c+d \tan (e+f x)} \left (1+\tan ^2(e+f x)\right ) \, dx\\ &=-\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt [3]{c+d x}}{x} \, dx,x,\tan (e+f x)\right )}{f}-\int \frac {-d+c \tan (e+f x)}{(c+d \tan (e+f x))^{2/3}} \, dx\\ &=-\left (\frac {1}{2} (-i c-d) \int \frac {1+i \tan (e+f x)}{(c+d \tan (e+f x))^{2/3}} \, dx\right )-\frac {1}{2} (i c-d) \int \frac {1-i \tan (e+f x)}{(c+d \tan (e+f x))^{2/3}} \, dx+\frac {c \operatorname {Subst}\left (\int \frac {1}{x (c+d x)^{2/3}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {\sqrt [3]{c} \log (\tan (e+f x))}{2 f}-\frac {\left (3 \sqrt [3]{c}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d \tan (e+f x)}\right )}{2 f}-\frac {\left (3 c^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{c^{2/3}+\sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c+d \tan (e+f x)}\right )}{2 f}-\frac {(c-i d) \operatorname {Subst}\left (\int \frac {1}{(-1+x) (c-i d x)^{2/3}} \, dx,x,i \tan (e+f x)\right )}{2 f}-\frac {(c+i d) \operatorname {Subst}\left (\int \frac {1}{(-1+x) (c+i d x)^{2/3}} \, dx,x,-i \tan (e+f x)\right )}{2 f}\\ &=-\frac {1}{4} i \sqrt [3]{c-i d} x+\frac {1}{4} i \sqrt [3]{c+i d} x-\frac {\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}-\frac {\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}-\frac {\sqrt [3]{c} \log (\tan (e+f x))}{2 f}+\frac {3 \sqrt [3]{c} \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 f}+\frac {\left (3 \sqrt [3]{c}\right ) \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}}\right )}{f}+\frac {\left (3 \sqrt [3]{c-i d}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{c-i d}-x} \, dx,x,\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}+\frac {\left (3 (c-i d)^{2/3}\right ) \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 )}{4 f}+\frac {\left (3 \sqrt [3]{c+i d}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{c+i d}-x} \, dx,x,\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}+\frac {\left (3 (c+i d)^{2/3}\right ) \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 )}{4 f}\\ &=-\frac {1}{4} i \sqrt [3]{c-i d} x+\frac {1}{4} i \sqrt [3]{c+i d} x-\frac {\sqrt {3} \sqrt [3]{c} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{f}-\frac {\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}-\frac {\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}-\frac {\sqrt [3]{c} \log (\tan (e+f x))}{2 f}+\frac {3 \sqrt [3]{c} \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 f}-\frac {3 \sqrt [3]{c-i d} \log \left (\sqrt [3]{c-i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}-\frac {3 \sqrt [3]{c+i d} \log \left (\sqrt [3]{c+i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}-\frac {\left (3 \sqrt [3]{c-i d}\right ) \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 )}{2 f}-\frac {\left (3 \sqrt [3]{c+i d}\right ) \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 )}{2 f}\\ &=-\frac {1}{4} i \sqrt [3]{c-i d} x+\frac {1}{4} i \sqrt [3]{c+i d} x-\frac {\sqrt {3} \sqrt [3]{c} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{f}+\frac {\sqrt {3} \sqrt [3]{c-i d} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt {3}}\right )}{2 f}+\frac {\sqrt {3} \sqrt [3]{c+i d} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )}{2 f}-\frac {\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}-\frac {\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}-\frac {\sqrt [3]{c} \log (\tan (e+f x))}{2 f}+\frac {3 \sqrt [3]{c} \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 f}-\frac {3 \sqrt [3]{c-i d} \log \left (\sqrt [3]{c-i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}-\frac {3 \sqrt [3]{c+i d} \log \left (\sqrt [3]{c+i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}\\ \end {align*}
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Mathematica [A] time = 0.90, size = 744, normalized size = 1.85 \[ \frac {-2 \sqrt [3]{c} \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )-4 \sqrt {3} \sqrt [3]{c} \tan ^{-1}\left (\frac {2 \sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c}}{\sqrt {3} \sqrt [3]{c}}\right )+2 \sqrt {3} \sqrt [3]{c-i d} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt {3}}\right )+\frac {2 i \sqrt {3} d \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )}{(c+i d)^{2/3}}+\frac {2 \sqrt {3} c \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )}{(c+i d)^{2/3}}+4 \sqrt [3]{c} \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )+\frac {2 i d \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c-i d}\right )}{(c-i d)^{2/3}}-\frac {2 c \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c-i d}\right )}{(c-i d)^{2/3}}-\frac {2 i d \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c+i d}\right )}{(c+i d)^{2/3}}-\frac {2 c \log \left (-\sqrt [3]{c+d \tan (e+f x)}+\sqrt [3]{c+i d}\right )}{(c+i d)^{2/3}}-\frac {i d \log \left (\sqrt [3]{c-i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c-i d)^{2/3}\right )}{(c-i d)^{2/3}}+\frac {c \log \left (\sqrt [3]{c-i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c-i d)^{2/3}\right )}{(c-i d)^{2/3}}+\frac {i d \log \left (\sqrt [3]{c+i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c+i d)^{2/3}\right )}{(c+i d)^{2/3}}+\frac {c \log \left (\sqrt [3]{c+i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c+i d)^{2/3}\right )}{(c+i d)^{2/3}}}{4 f} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}} \cot \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.77, size = 0, normalized size = 0.00 \[ \int \cot \left (f x +e \right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {1}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}} \cot \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 15.56, size = 2133, normalized size = 5.31 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt [3]{c + d \tan {\left (e + f x \right )}} \cot {\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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