Optimal. Leaf size=245 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{b^{4/3} d}+\frac {\text {ArcTan}\left (\sqrt {3}-\frac {2 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{2 b^{4/3} d}-\frac {\text {ArcTan}\left (\sqrt {3}+\frac {2 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{2 b^{4/3} d}-\frac {\sqrt {3} \log \left (b^{2/3}-\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 b^{4/3} d}+\frac {\sqrt {3} \log \left (b^{2/3}+\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 b^{4/3} d}-\frac {3}{b d \sqrt [3]{b \tan (c+d x)}} \]
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Rubi [A]
time = 0.30, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3555, 3557,
335, 301, 648, 632, 210, 642, 209} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{b^{4/3} d}+\frac {\text {ArcTan}\left (\sqrt {3}-\frac {2 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{2 b^{4/3} d}-\frac {\text {ArcTan}\left (\frac {2 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}+\sqrt {3}\right )}{2 b^{4/3} d}-\frac {\sqrt {3} \log \left (b^{2/3}-\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 b^{4/3} d}+\frac {\sqrt {3} \log \left (b^{2/3}+\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 b^{4/3} d}-\frac {3}{b d \sqrt [3]{b \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 210
Rule 301
Rule 335
Rule 632
Rule 642
Rule 648
Rule 3555
Rule 3557
Rubi steps
\begin {align*} \int \frac {1}{(b \tan (c+d x))^{4/3}} \, dx &=-\frac {3}{b d \sqrt [3]{b \tan (c+d x)}}-\frac {\int (b \tan (c+d x))^{2/3} \, dx}{b^2}\\ &=-\frac {3}{b d \sqrt [3]{b \tan (c+d x)}}-\frac {\text {Subst}\left (\int \frac {x^{2/3}}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=-\frac {3}{b d \sqrt [3]{b \tan (c+d x)}}-\frac {3 \text {Subst}\left (\int \frac {x^4}{b^2+x^6} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{b d}\\ &=-\frac {3}{b d \sqrt [3]{b \tan (c+d x)}}-\frac {\text {Subst}\left (\int \frac {-\frac {\sqrt [3]{b}}{2}+\frac {\sqrt {3} x}{2}}{b^{2/3}-\sqrt {3} \sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{b^{4/3} d}-\frac {\text {Subst}\left (\int \frac {-\frac {\sqrt [3]{b}}{2}-\frac {\sqrt {3} x}{2}}{b^{2/3}+\sqrt {3} \sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{b^{4/3} d}-\frac {\text {Subst}\left (\int \frac {1}{b^{2/3}+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{b d}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{b^{4/3} d}-\frac {3}{b d \sqrt [3]{b \tan (c+d x)}}-\frac {\sqrt {3} \text {Subst}\left (\int \frac {-\sqrt {3} \sqrt [3]{b}+2 x}{b^{2/3}-\sqrt {3} \sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{4 b^{4/3} d}+\frac {\sqrt {3} \text {Subst}\left (\int \frac {\sqrt {3} \sqrt [3]{b}+2 x}{b^{2/3}+\sqrt {3} \sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{4 b^{4/3} d}-\frac {\text {Subst}\left (\int \frac {1}{b^{2/3}-\sqrt {3} \sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{4 b d}-\frac {\text {Subst}\left (\int \frac {1}{b^{2/3}+\sqrt {3} \sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{4 b d}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{b^{4/3} d}-\frac {\sqrt {3} \log \left (b^{2/3}-\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 b^{4/3} d}+\frac {\sqrt {3} \log \left (b^{2/3}+\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 b^{4/3} d}-\frac {3}{b d \sqrt [3]{b \tan (c+d x)}}-\frac {\text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b \tan (c+d x)}}{\sqrt {3} \sqrt [3]{b}}\right )}{2 \sqrt {3} b^{4/3} d}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b \tan (c+d x)}}{\sqrt {3} \sqrt [3]{b}}\right )}{2 \sqrt {3} b^{4/3} d}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{b^{4/3} d}+\frac {\tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}-\frac {6 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )\right )}{2 b^{4/3} d}-\frac {\tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}+\frac {6 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )\right )}{2 b^{4/3} d}-\frac {\sqrt {3} \log \left (b^{2/3}-\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 b^{4/3} d}+\frac {\sqrt {3} \log \left (b^{2/3}+\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 b^{4/3} d}-\frac {3}{b d \sqrt [3]{b \tan (c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.07, size = 38, normalized size = 0.16 \begin {gather*} -\frac {3 \, _2F_1\left (-\frac {1}{6},1;\frac {5}{6};-\tan ^2(c+d x)\right )}{b d \sqrt [3]{b \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 212, normalized size = 0.87
method | result | size |
derivativedivides | \(\frac {3 b \left (-\frac {\frac {\sqrt {3}\, \left (b^{2}\right )^{\frac {5}{6}} \ln \left (\left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}-\sqrt {3}\, \left (b^{2}\right )^{\frac {1}{6}} \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b^{2}\right )^{\frac {1}{3}}\right )}{12 b^{2}}+\frac {\arctan \left (\frac {2 \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{\left (b^{2}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 \left (b^{2}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (b^{2}\right )^{\frac {5}{6}} \ln \left (\left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}+\sqrt {3}\, \left (b^{2}\right )^{\frac {1}{6}} \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b^{2}\right )^{\frac {1}{3}}\right )}{12 b^{2}}+\frac {\arctan \left (\frac {2 \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{\left (b^{2}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 \left (b^{2}\right )^{\frac {1}{6}}}+\frac {\arctan \left (\frac {\left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{\left (b^{2}\right )^{\frac {1}{6}}}\right )}{3 \left (b^{2}\right )^{\frac {1}{6}}}}{b^{2}}-\frac {1}{b^{2} \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}\right )}{d}\) | \(212\) |
default | \(\frac {3 b \left (-\frac {\frac {\sqrt {3}\, \left (b^{2}\right )^{\frac {5}{6}} \ln \left (\left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}-\sqrt {3}\, \left (b^{2}\right )^{\frac {1}{6}} \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b^{2}\right )^{\frac {1}{3}}\right )}{12 b^{2}}+\frac {\arctan \left (\frac {2 \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{\left (b^{2}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 \left (b^{2}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (b^{2}\right )^{\frac {5}{6}} \ln \left (\left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}+\sqrt {3}\, \left (b^{2}\right )^{\frac {1}{6}} \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b^{2}\right )^{\frac {1}{3}}\right )}{12 b^{2}}+\frac {\arctan \left (\frac {2 \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{\left (b^{2}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 \left (b^{2}\right )^{\frac {1}{6}}}+\frac {\arctan \left (\frac {\left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{\left (b^{2}\right )^{\frac {1}{6}}}\right )}{3 \left (b^{2}\right )^{\frac {1}{6}}}}{b^{2}}-\frac {1}{b^{2} \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}\right )}{d}\) | \(212\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 182, normalized size = 0.74 \begin {gather*} \frac {\frac {\sqrt {3} \log \left (\sqrt {3} \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} b^{\frac {1}{3}} + \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + b^{\frac {2}{3}}\right )}{b^{\frac {1}{3}}} - \frac {\sqrt {3} \log \left (-\sqrt {3} \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} b^{\frac {1}{3}} + \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + b^{\frac {2}{3}}\right )}{b^{\frac {1}{3}}} - \frac {2 \, \arctan \left (\frac {\sqrt {3} b^{\frac {1}{3}} + 2 \, \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right )}{b^{\frac {1}{3}}} - \frac {2 \, \arctan \left (-\frac {\sqrt {3} b^{\frac {1}{3}} - 2 \, \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right )}{b^{\frac {1}{3}}} - \frac {4 \, \arctan \left (\frac {\left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right )}{b^{\frac {1}{3}}} - \frac {12}{\left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}}{4 \, b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 701 vs.
\(2 (187) = 374\).
time = 0.41, size = 701, normalized size = 2.86 \begin {gather*} \frac {12 \, \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {2}{3}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 4 \, {\left (b^{2} d \cos \left (d x + c\right )^{2} - b^{2} d\right )} \left (\frac {1}{b^{8} d^{6}}\right )^{\frac {1}{6}} \arctan \left (2 \, \sqrt {\sqrt {3} b^{7} d^{5} \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {1}{3}} \left (\frac {1}{b^{8} d^{6}}\right )^{\frac {5}{6}} + b^{6} d^{4} \left (\frac {1}{b^{8} d^{6}}\right )^{\frac {2}{3}} + \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {2}{3}}} b d \left (\frac {1}{b^{8} d^{6}}\right )^{\frac {1}{6}} - 2 \, b d \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {1}{3}} \left (\frac {1}{b^{8} d^{6}}\right )^{\frac {1}{6}} - \sqrt {3}\right ) + 4 \, {\left (b^{2} d \cos \left (d x + c\right )^{2} - b^{2} d\right )} \left (\frac {1}{b^{8} d^{6}}\right )^{\frac {1}{6}} \arctan \left (2 \, \sqrt {-\sqrt {3} b^{7} d^{5} \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {1}{3}} \left (\frac {1}{b^{8} d^{6}}\right )^{\frac {5}{6}} + b^{6} d^{4} \left (\frac {1}{b^{8} d^{6}}\right )^{\frac {2}{3}} + \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {2}{3}}} b d \left (\frac {1}{b^{8} d^{6}}\right )^{\frac {1}{6}} - 2 \, b d \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {1}{3}} \left (\frac {1}{b^{8} d^{6}}\right )^{\frac {1}{6}} + \sqrt {3}\right ) + 8 \, {\left (b^{2} d \cos \left (d x + c\right )^{2} - b^{2} d\right )} \left (\frac {1}{b^{8} d^{6}}\right )^{\frac {1}{6}} \arctan \left (\sqrt {b^{6} d^{4} \left (\frac {1}{b^{8} d^{6}}\right )^{\frac {2}{3}} + \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {2}{3}}} b d \left (\frac {1}{b^{8} d^{6}}\right )^{\frac {1}{6}} - b d \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {1}{3}} \left (\frac {1}{b^{8} d^{6}}\right )^{\frac {1}{6}}\right ) + {\left (\sqrt {3} b^{2} d \cos \left (d x + c\right )^{2} - \sqrt {3} b^{2} d\right )} \left (\frac {1}{b^{8} d^{6}}\right )^{\frac {1}{6}} \log \left (\sqrt {3} b^{7} d^{5} \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {1}{3}} \left (\frac {1}{b^{8} d^{6}}\right )^{\frac {5}{6}} + b^{6} d^{4} \left (\frac {1}{b^{8} d^{6}}\right )^{\frac {2}{3}} + \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {2}{3}}\right ) - {\left (\sqrt {3} b^{2} d \cos \left (d x + c\right )^{2} - \sqrt {3} b^{2} d\right )} \left (\frac {1}{b^{8} d^{6}}\right )^{\frac {1}{6}} \log \left (-\sqrt {3} b^{7} d^{5} \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {1}{3}} \left (\frac {1}{b^{8} d^{6}}\right )^{\frac {5}{6}} + b^{6} d^{4} \left (\frac {1}{b^{8} d^{6}}\right )^{\frac {2}{3}} + \left (\frac {b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{\frac {2}{3}}\right )}{4 \, {\left (b^{2} d \cos \left (d x + c\right )^{2} - b^{2} d\right )}} \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 \frac {1}{\left (b \tan {\left (c + d x \right )}\right )^{\frac {4}{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.57, size = 227, normalized size = 0.93 \begin {gather*} \frac {1}{4} \, b {\left (\frac {\sqrt {3} {\left | b \right |}^{\frac {5}{3}} \log \left (\sqrt {3} \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} {\left | b \right |}^{\frac {1}{3}} + \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + {\left | b \right |}^{\frac {2}{3}}\right )}{b^{4} d} - \frac {\sqrt {3} {\left | b \right |}^{\frac {5}{3}} \log \left (-\sqrt {3} \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} {\left | b \right |}^{\frac {1}{3}} + \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + {\left | b \right |}^{\frac {2}{3}}\right )}{b^{4} d} - \frac {2 \, {\left | b \right |}^{\frac {5}{3}} \arctan \left (\frac {\sqrt {3} {\left | b \right |}^{\frac {1}{3}} + 2 \, \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{{\left | b \right |}^{\frac {1}{3}}}\right )}{b^{4} d} - \frac {2 \, {\left | b \right |}^{\frac {5}{3}} \arctan \left (-\frac {\sqrt {3} {\left | b \right |}^{\frac {1}{3}} - 2 \, \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{{\left | b \right |}^{\frac {1}{3}}}\right )}{b^{4} d} - \frac {4 \, {\left | b \right |}^{\frac {5}{3}} \arctan \left (\frac {\left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{{\left | b \right |}^{\frac {1}{3}}}\right )}{b^{4} d} - \frac {12}{\left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} b^{2} d}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.55, size = 278, normalized size = 1.13 \begin {gather*} -\frac {3}{b\,d\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}-\frac {{\left (-1\right )}^{1/6}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}{b^{1/3}}\right )\,1{}\mathrm {i}}{b^{4/3}\,d}-\frac {{\left (-1\right )}^{1/6}\,\ln \left (972\,b^{12}\,d^6-972\,{\left (-1\right )}^{1/6}\,b^{35/3}\,d^6\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b^{4/3}\,d}-\frac {{\left (-1\right )}^{1/6}\,\ln \left (972\,b^{12}\,d^6-972\,{\left (-1\right )}^{1/6}\,b^{35/3}\,d^6\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b^{4/3}\,d}+\frac {{\left (-1\right )}^{1/6}\,\ln \left (972\,b^{12}\,d^6+1944\,{\left (-1\right )}^{1/6}\,b^{35/3}\,d^6\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b^{4/3}\,d}+\frac {{\left (-1\right )}^{1/6}\,\ln \left (972\,b^{12}\,d^6+1944\,{\left (-1\right )}^{1/6}\,b^{35/3}\,d^6\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b^{4/3}\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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