Optimal. Leaf size=131 \[ -\frac {\sqrt {3} \sqrt [3]{b} \text {ArcTan}\left (\frac {b^{2/3}-2 (b \tan (c+d x))^{2/3}}{\sqrt {3} b^{2/3}}\right )}{2 d}-\frac {\sqrt [3]{b} \log \left (b^{2/3}+(b \tan (c+d x))^{2/3}\right )}{2 d}+\frac {\sqrt [3]{b} \log \left (b^{4/3}-b^{2/3} (b \tan (c+d x))^{2/3}+(b \tan (c+d x))^{4/3}\right )}{4 d} \]
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Rubi [A]
time = 0.08, 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 = {3557, 335, 281,
298, 31, 648, 631, 210, 642} \begin {gather*} -\frac {\sqrt {3} \sqrt [3]{b} \text {ArcTan}\left (\frac {b^{2/3}-2 (b \tan (c+d x))^{2/3}}{\sqrt {3} b^{2/3}}\right )}{2 d}-\frac {\sqrt [3]{b} \log \left (b^{2/3}+(b \tan (c+d x))^{2/3}\right )}{2 d}+\frac {\sqrt [3]{b} \log \left (-b^{2/3} (b \tan (c+d x))^{2/3}+b^{4/3}+(b \tan (c+d x))^{4/3}\right )}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 281
Rule 298
Rule 335
Rule 631
Rule 642
Rule 648
Rule 3557
Rubi steps
\begin {align*} \int \sqrt [3]{b \tan (c+d x)} \, dx &=\frac {b \text {Subst}\left (\int \frac {\sqrt [3]{x}}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac {(3 b) \text {Subst}\left (\int \frac {x^3}{b^2+x^6} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{d}\\ &=\frac {(3 b) \text {Subst}\left (\int \frac {x}{b^2+x^3} \, dx,x,(b \tan (c+d x))^{2/3}\right )}{2 d}\\ &=-\frac {\sqrt [3]{b} \text {Subst}\left (\int \frac {1}{b^{2/3}+x} \, dx,x,(b \tan (c+d x))^{2/3}\right )}{2 d}+\frac {\sqrt [3]{b} \text {Subst}\left (\int \frac {b^{2/3}+x}{b^{4/3}-b^{2/3} x+x^2} \, dx,x,(b \tan (c+d x))^{2/3}\right )}{2 d}\\ &=-\frac {\sqrt [3]{b} \log \left (b^{2/3}+(b \tan (c+d x))^{2/3}\right )}{2 d}+\frac {\sqrt [3]{b} \text {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 d}+\frac {(3 b) \text {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 {\sqrt [3]{b} \log \left (b^{2/3}+(b \tan (c+d x))^{2/3}\right )}{2 d}+\frac {\sqrt [3]{b} \log \left (b^{4/3}-b^{2/3} (b \tan (c+d x))^{2/3}+(b \tan (c+d x))^{4/3}\right )}{4 d}+\frac {\left (3 \sqrt [3]{b}\right ) \text {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 d}\\ &=-\frac {\sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {1-\frac {2 (b \tan (c+d x))^{2/3}}{b^{2/3}}}{\sqrt {3}}\right )}{2 d}-\frac {\sqrt [3]{b} \log \left (b^{2/3}+(b \tan (c+d x))^{2/3}\right )}{2 d}+\frac {\sqrt [3]{b} \log \left (b^{4/3}-b^{2/3} (b \tan (c+d x))^{2/3}+(b \tan (c+d x))^{4/3}\right )}{4 d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.05, size = 40, normalized size = 0.31 \begin {gather*} \frac {3 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\tan ^2(c+d x)\right ) (b \tan (c+d x))^{4/3}}{4 b d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 108, normalized size = 0.82
method | result | size |
derivativedivides | \(\frac {3 b \left (-\frac {\ln \left (\left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}+\left (b^{2}\right )^{\frac {1}{3}}\right )}{6 \left (b^{2}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (b \tan \left (d x +c \right )\right )^{\frac {4}{3}}-\left (b^{2}\right )^{\frac {1}{3}} \left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}+\left (b^{2}\right )^{\frac {2}{3}}\right )}{12 \left (b^{2}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}}{\left (b^{2}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{6 \left (b^{2}\right )^{\frac {1}{3}}}\right )}{d}\) | \(108\) |
default | \(\frac {3 b \left (-\frac {\ln \left (\left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}+\left (b^{2}\right )^{\frac {1}{3}}\right )}{6 \left (b^{2}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (b \tan \left (d x +c \right )\right )^{\frac {4}{3}}-\left (b^{2}\right )^{\frac {1}{3}} \left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}+\left (b^{2}\right )^{\frac {2}{3}}\right )}{12 \left (b^{2}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}}{\left (b^{2}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{6 \left (b^{2}\right )^{\frac {1}{3}}}\right )}{d}\) | \(108\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 98, normalized size = 0.75 \begin {gather*} \frac {2 \, \sqrt {3} b^{\frac {4}{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 {4}{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 {4}{3}} \log \left (\left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + b^{\frac {2}{3}}\right )}{4 \, b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 124, normalized size = 0.95 \begin {gather*} \frac {2 \, \sqrt {3} \left (-b\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} \left (-b\right )^{\frac {1}{3}} + \sqrt {3} b}{3 \, b}\right ) - \left (-b\right )^{\frac {1}{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}} \left (-b\right )^{\frac {2}{3}} - \left (-b\right )^{\frac {1}{3}} b\right ) + 2 \, \left (-b\right )^{\frac {1}{3}} \log \left (\left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + \left (-b\right )^{\frac {2}{3}}\right )}{4 \, d} \end {gather*}
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 \begin {gather*} \int \sqrt [3]{b \tan {\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 127, normalized size = 0.97 \begin {gather*} \frac {1}{4} \, b {\left (\frac {2 \, \sqrt {3} {\left | b \right |}^{\frac {4}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} - {\left | b \right |}^{\frac {2}{3}}\right )}}{3 \, {\left | b \right |}^{\frac {2}{3}}}\right )}{b^{2} d} + \frac {{\left | b \right |}^{\frac {4}{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}} {\left | b \right |}^{\frac {2}{3}} + {\left | b \right |}^{\frac {4}{3}}\right )}{b^{2} d} - \frac {2 \, {\left | b \right |}^{\frac {4}{3}} \log \left (\left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + {\left | b \right |}^{\frac {2}{3}}\right )}{b^{2} d}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.63, size = 146, normalized size = 1.11 \begin {gather*} \frac {{\left (-b\right )}^{1/3}\,\ln \left (81\,{\left (-b\right )}^{16/3}\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{2/3}+81\,b^6\right )}{2\,d}-\frac {{\left (-b\right )}^{1/3}\,\ln \left (\frac {81\,b^6}{d^4}-\frac {81\,{\left (-b\right )}^{16/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{2/3}}{d^4}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,d}+\frac {{\left (-b\right )}^{1/3}\,\ln \left (\frac {81\,b^6}{d^4}+\frac {162\,{\left (-b\right )}^{16/3}\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{2/3}}{d^4}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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