Optimal. Leaf size=329 \[ -\frac {1}{4} (a-i b)^{5/3} x-\frac {1}{4} (a+i b)^{5/3} x+\frac {i \sqrt {3} (a-i b)^{5/3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-i b}}}{\sqrt {3}}\right )}{2 d}-\frac {i \sqrt {3} (a+i b)^{5/3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+i b}}}{\sqrt {3}}\right )}{2 d}+\frac {i (a-i b)^{5/3} \log (\cos (c+d x))}{4 d}-\frac {i (a+i b)^{5/3} \log (\cos (c+d x))}{4 d}+\frac {3 i (a-i b)^{5/3} \log \left (\sqrt [3]{a-i b}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 d}-\frac {3 i (a+i b)^{5/3} \log \left (\sqrt [3]{a+i b}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 d}+\frac {3 b (a+b \tan (c+d x))^{2/3}}{2 d} \]
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Rubi [A]
time = 0.26, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3563, 3620,
3618, 57, 631, 210, 31} \begin {gather*} \frac {i \sqrt {3} (a-i b)^{5/3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-i b}}}{\sqrt {3}}\right )}{2 d}-\frac {i \sqrt {3} (a+i b)^{5/3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+i b}}}{\sqrt {3}}\right )}{2 d}+\frac {3 b (a+b \tan (c+d x))^{2/3}}{2 d}+\frac {3 i (a-i b)^{5/3} \log \left (-\sqrt [3]{a+b \tan (c+d x)}+\sqrt [3]{a-i b}\right )}{4 d}-\frac {3 i (a+i b)^{5/3} \log \left (-\sqrt [3]{a+b \tan (c+d x)}+\sqrt [3]{a+i b}\right )}{4 d}+\frac {i (a-i b)^{5/3} \log (\cos (c+d x))}{4 d}-\frac {i (a+i b)^{5/3} \log (\cos (c+d x))}{4 d}-\frac {1}{4} x (a-i b)^{5/3}-\frac {1}{4} x (a+i b)^{5/3} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 57
Rule 210
Rule 631
Rule 3563
Rule 3618
Rule 3620
Rubi steps
\begin {align*} \int (a+b \tan (c+d x))^{5/3} \, dx &=\frac {3 b (a+b \tan (c+d x))^{2/3}}{2 d}+\int \frac {a^2-b^2+2 a b \tan (c+d x)}{\sqrt [3]{a+b \tan (c+d x)}} \, dx\\ &=\frac {3 b (a+b \tan (c+d x))^{2/3}}{2 d}+\frac {1}{2} (a-i b)^2 \int \frac {1+i \tan (c+d x)}{\sqrt [3]{a+b \tan (c+d x)}} \, dx+\frac {1}{2} (a+i b)^2 \int \frac {1-i \tan (c+d x)}{\sqrt [3]{a+b \tan (c+d x)}} \, dx\\ &=\frac {3 b (a+b \tan (c+d x))^{2/3}}{2 d}+\frac {\left (i (a-i b)^2\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 d}-\frac {\left (i (a+i b)^2\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 d}\\ &=-\frac {1}{4} (a-i b)^{5/3} x-\frac {1}{4} (a+i b)^{5/3} x+\frac {i (a-i b)^{5/3} \log (\cos (c+d x))}{4 d}-\frac {i (a+i b)^{5/3} \log (\cos (c+d x))}{4 d}+\frac {3 b (a+b \tan (c+d x))^{2/3}}{2 d}-\frac {\left (3 i (a-i b)^{5/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a-i b}-x} \, dx,x,\sqrt [3]{a+b \tan (c+d x)}\right )}{4 d}+\frac {\left (3 i (a-i b)^2\right ) \text {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 d}+\frac {\left (3 i (a+i b)^{5/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a+i b}-x} \, dx,x,\sqrt [3]{a+b \tan (c+d x)}\right )}{4 d}-\frac {\left (3 i (a+i b)^2\right ) \text {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 d}\\ &=-\frac {1}{4} (a-i b)^{5/3} x-\frac {1}{4} (a+i b)^{5/3} x+\frac {i (a-i b)^{5/3} \log (\cos (c+d x))}{4 d}-\frac {i (a+i b)^{5/3} \log (\cos (c+d x))}{4 d}+\frac {3 i (a-i b)^{5/3} \log \left (\sqrt [3]{a-i b}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 d}-\frac {3 i (a+i b)^{5/3} \log \left (\sqrt [3]{a+i b}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 d}+\frac {3 b (a+b \tan (c+d x))^{2/3}}{2 d}-\frac {\left (3 i (a-i b)^{5/3}\right ) \text {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 d}+\frac {\left (3 i (a+i b)^{5/3}\right ) \text {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 d}\\ &=-\frac {1}{4} (a-i b)^{5/3} x-\frac {1}{4} (a+i b)^{5/3} x+\frac {i \sqrt {3} (a-i b)^{5/3} \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}-\frac {i \sqrt {3} (a+i b)^{5/3} \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}+\frac {i (a-i b)^{5/3} \log (\cos (c+d x))}{4 d}-\frac {i (a+i b)^{5/3} \log (\cos (c+d x))}{4 d}+\frac {3 i (a-i b)^{5/3} \log \left (\sqrt [3]{a-i b}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 d}-\frac {3 i (a+i b)^{5/3} \log \left (\sqrt [3]{a+i b}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 d}+\frac {3 b (a+b \tan (c+d x))^{2/3}}{2 d}\\ \end {align*}
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Mathematica [A]
time = 1.04, size = 300, normalized size = 0.91 \begin {gather*} \frac {(i a+b) \left (2 \sqrt {3} (a-i b)^{2/3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-i b}}}{\sqrt {3}}\right )-(a-i b)^{2/3} \log (i+\tan (c+d x))+3 \left ((a-i b)^{2/3} \log \left (\sqrt [3]{a-i b}-\sqrt [3]{a+b \tan (c+d x)}\right )+(a+b \tan (c+d x))^{2/3}\right )\right )+(-i a+b) \left (2 \sqrt {3} (a+i b)^{2/3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+i b}}}{\sqrt {3}}\right )-(a+i b)^{2/3} \log (i-\tan (c+d x))+3 \left ((a+i b)^{2/3} \log \left (\sqrt [3]{a+i b}-\sqrt [3]{a+b \tan (c+d x)}\right )+(a+b \tan (c+d x))^{2/3}\right )\right )}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.22, size = 90, normalized size = 0.27
method | result | size |
derivativedivides | \(\frac {3 b \left (\frac {\left (a +b \tan \left (d x +c \right )\right )^{\frac {2}{3}}}{2}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-2 a \,\textit {\_Z}^{3}+a^{2}+b^{2}\right )}{\sum }\frac {\left (-2 a \,\textit {\_R}^{4}+\left (a^{2}+b^{2}\right ) \textit {\_R} \right ) \ln \left (\left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} a}\right )}{6}\right )}{d}\) | \(90\) |
default | \(\frac {3 b \left (\frac {\left (a +b \tan \left (d x +c \right )\right )^{\frac {2}{3}}}{2}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-2 a \,\textit {\_Z}^{3}+a^{2}+b^{2}\right )}{\sum }\frac {\left (-2 a \,\textit {\_R}^{4}+\left (a^{2}+b^{2}\right ) \textit {\_R} \right ) \ln \left (\left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} a}\right )}{6}\right )}{d}\) | \(90\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 \left (a + b \tan {\left (c + d x \right )}\right )^{\frac {5}{3}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.09, size = 1540, normalized size = 4.68 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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