Optimal. Leaf size=131 \[ -\frac {\sqrt {3} \tan ^{-1}\left (\frac {b^{2/3}-2 (b \tan (c+d x))^{2/3}}{\sqrt {3} b^{2/3}}\right )}{2 \sqrt [3]{b} d}+\frac {\log \left (b^{2/3}+(b \tan (c+d x))^{2/3}\right )}{2 \sqrt [3]{b} d}-\frac {\log \left (-b^{2/3} (b \tan (c+d x))^{2/3}+b^{4/3}+(b \tan (c+d x))^{4/3}\right )}{4 \sqrt [3]{b} d} \]
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Rubi [A] time = 0.10, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3476, 329, 275, 200, 31, 634, 617, 204, 628} \[ -\frac {\sqrt {3} \tan ^{-1}\left (\frac {b^{2/3}-2 (b \tan (c+d x))^{2/3}}{\sqrt {3} b^{2/3}}\right )}{2 \sqrt [3]{b} d}+\frac {\log \left (b^{2/3}+(b \tan (c+d x))^{2/3}\right )}{2 \sqrt [3]{b} d}-\frac {\log \left (-b^{2/3} (b \tan (c+d x))^{2/3}+b^{4/3}+(b \tan (c+d x))^{4/3}\right )}{4 \sqrt [3]{b} d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 275
Rule 329
Rule 617
Rule 628
Rule 634
Rule 3476
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{b \tan (c+d x)}} \, dx &=\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{x} \left (b^2+x^2\right )} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac {(3 b) \operatorname {Subst}\left (\int \frac {x}{b^2+x^6} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{d}\\ &=\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{b^2+x^3} \, dx,x,(b \tan (c+d x))^{2/3}\right )}{2 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{b^{2/3}+x} \, dx,x,(b \tan (c+d x))^{2/3}\right )}{2 \sqrt [3]{b} d}+\frac {\operatorname {Subst}\left (\int \frac {2 b^{2/3}-x}{b^{4/3}-b^{2/3} x+x^2} \, dx,x,(b \tan (c+d x))^{2/3}\right )}{2 \sqrt [3]{b} d}\\ &=\frac {\log \left (b^{2/3}+(b \tan (c+d x))^{2/3}\right )}{2 \sqrt [3]{b} d}-\frac {\operatorname {Subst}\left (\int \frac {-b^{2/3}+2 x}{b^{4/3}-b^{2/3} x+x^2} \, dx,x,(b \tan (c+d x))^{2/3}\right )}{4 \sqrt [3]{b} d}+\frac {\left (3 \sqrt [3]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{b^{4/3}-b^{2/3} x+x^2} \, dx,x,(b \tan (c+d x))^{2/3}\right )}{4 d}\\ &=\frac {\log \left (b^{2/3}+(b \tan (c+d x))^{2/3}\right )}{2 \sqrt [3]{b} d}-\frac {\log \left (b^{4/3}-b^{2/3} (b \tan (c+d x))^{2/3}+(b \tan (c+d x))^{4/3}\right )}{4 \sqrt [3]{b} d}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 (b \tan (c+d x))^{2/3}}{b^{2/3}}\right )}{2 \sqrt [3]{b} d}\\ &=-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 (b \tan (c+d x))^{2/3}}{b^{2/3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{b} d}+\frac {\log \left (b^{2/3}+(b \tan (c+d x))^{2/3}\right )}{2 \sqrt [3]{b} d}-\frac {\log \left (b^{4/3}-b^{2/3} (b \tan (c+d x))^{2/3}+(b \tan (c+d x))^{4/3}\right )}{4 \sqrt [3]{b} d}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 100, normalized size = 0.76 \[ \frac {\sqrt [3]{\tan (c+d x)} \left (2 \sqrt {3} \tan ^{-1}\left (\frac {2 \tan ^{\frac {2}{3}}(c+d x)-1}{\sqrt {3}}\right )+2 \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )-\log \left (\tan ^{\frac {4}{3}}(c+d x)-\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{4 d \sqrt [3]{b \tan (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 299, normalized size = 2.28 \[ \left [\frac {\sqrt {3} b \sqrt {-\frac {1}{b^{\frac {2}{3}}}} \log \left (\frac {2 \, \sqrt {3} \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} b \sqrt {-\frac {1}{b^{\frac {2}{3}}}} \tan \left (d x + c\right ) + 2 \, b \tan \left (d x + c\right )^{2} - \sqrt {3} b^{\frac {4}{3}} \sqrt {-\frac {1}{b^{\frac {2}{3}}}} + \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} {\left (\sqrt {3} b^{\frac {2}{3}} \sqrt {-\frac {1}{b^{\frac {2}{3}}}} - 3 \, b^{\frac {1}{3}}\right )} - b}{\tan \left (d x + c\right )^{2} + 1}\right ) - b^{\frac {2}{3}} \log \left (\left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} b \tan \left (d x + c\right ) - \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} b^{\frac {2}{3}} + b^{\frac {4}{3}}\right ) + 2 \, b^{\frac {2}{3}} \log \left (\left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + b^{\frac {2}{3}}\right )}{4 \, b d}, \frac {2 \, \sqrt {3} b^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} b^{\frac {2}{3}} - b^{\frac {4}{3}}\right )}}{3 \, b^{\frac {4}{3}}}\right ) - b^{\frac {2}{3}} \log \left (\left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} b \tan \left (d x + c\right ) - \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} b^{\frac {2}{3}} + b^{\frac {4}{3}}\right ) + 2 \, b^{\frac {2}{3}} \log \left (\left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + b^{\frac {2}{3}}\right )}{4 \, b d}\right ] \]
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 \[ \int \frac {1}{\left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 114, normalized size = 0.87 \[ \frac {b \ln \left (\left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}+\left (b^{2}\right )^{\frac {1}{3}}\right )}{2 d \left (b^{2}\right )^{\frac {2}{3}}}-\frac {b \ln \left (\left (b \tan \left (d x +c \right )\right )^{\frac {4}{3}}-\left (b^{2}\right )^{\frac {1}{3}} \left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}+\left (b^{2}\right )^{\frac {2}{3}}\right )}{4 d \left (b^{2}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}}{\left (b^{2}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{2 d \left (b^{2}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.77, size = 99, normalized size = 0.76 \[ \frac {2 \, \sqrt {3} b^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} - b^{\frac {2}{3}}\right )}}{3 \, b^{\frac {2}{3}}}\right ) - b^{\frac {2}{3}} \log \left (\left (b \tan \left (d x + c\right )\right )^{\frac {4}{3}} - \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} b^{\frac {2}{3}} + b^{\frac {4}{3}}\right ) + 2 \, b^{\frac {2}{3}} \log \left (\left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + b^{\frac {2}{3}}\right )}{4 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.73, size = 128, normalized size = 0.98 \[ \frac {\ln \left ({\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{2/3}+b^{2/3}\right )}{2\,b^{1/3}\,d}+\frac {\ln \left (\frac {81\,b^{11/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{d^3}+\frac {162\,b^3\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{2/3}}{d^3}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{4\,b^{1/3}\,d}-\frac {\ln \left (\frac {81\,b^{11/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{d^3}-\frac {162\,b^3\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{2/3}}{d^3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{4\,b^{1/3}\,d} \]
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 \[ \int \frac {1}{\sqrt [3]{b \tan {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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