Optimal. Leaf size=329 \[ \frac {i \sqrt {3} (a-i b)^{5/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-i b}}}{\sqrt {3}}\right )}{2 d}-\frac {i \sqrt {3} (a+i b)^{5/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+i b}}}{\sqrt {3}}\right )}{2 d}+\frac {3 b (a+b \tan (c+d x))^{2/3}}{2 d}+\frac {3 i (a-i b)^{5/3} \log \left (-\sqrt [3]{a+b \tan (c+d x)}+\sqrt [3]{a-i b}\right )}{4 d}-\frac {3 i (a+i b)^{5/3} \log \left (-\sqrt [3]{a+b \tan (c+d x)}+\sqrt [3]{a+i b}\right )}{4 d}+\frac {i (a-i b)^{5/3} \log (\cos (c+d x))}{4 d}-\frac {i (a+i b)^{5/3} \log (\cos (c+d x))}{4 d}-\frac {1}{4} x (a-i b)^{5/3}-\frac {1}{4} x (a+i b)^{5/3} \]
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Rubi [A] time = 0.38, 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 = {3482, 3539, 3537, 55, 617, 204, 31} \[ \frac {i \sqrt {3} (a-i b)^{5/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-i b}}}{\sqrt {3}}\right )}{2 d}-\frac {i \sqrt {3} (a+i b)^{5/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+i b}}}{\sqrt {3}}\right )}{2 d}+\frac {3 b (a+b \tan (c+d x))^{2/3}}{2 d}+\frac {3 i (a-i b)^{5/3} \log \left (-\sqrt [3]{a+b \tan (c+d x)}+\sqrt [3]{a-i b}\right )}{4 d}-\frac {3 i (a+i b)^{5/3} \log \left (-\sqrt [3]{a+b \tan (c+d x)}+\sqrt [3]{a+i b}\right )}{4 d}+\frac {i (a-i b)^{5/3} \log (\cos (c+d x))}{4 d}-\frac {i (a+i b)^{5/3} \log (\cos (c+d x))}{4 d}-\frac {1}{4} x (a-i b)^{5/3}-\frac {1}{4} x (a+i b)^{5/3} \]
Antiderivative was successfully verified.
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Rule 31
Rule 55
Rule 204
Rule 617
Rule 3482
Rule 3537
Rule 3539
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 0.98, size = 300, normalized size = 0.91 \[ \frac {(b+i a) \left (2 \sqrt {3} (a-i b)^{2/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-i b}}}{\sqrt {3}}\right )-(a-i b)^{2/3} \log (\tan (c+d x)+i)+3 \left ((a+b \tan (c+d x))^{2/3}+(a-i b)^{2/3} \log \left (-\sqrt [3]{a+b \tan (c+d x)}+\sqrt [3]{a-i b}\right )\right )\right )+(b-i a) \left (2 \sqrt {3} (a+i b)^{2/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+i b}}}{\sqrt {3}}\right )-(a+i b)^{2/3} \log (-\tan (c+d x)+i)+3 \left ((a+b \tan (c+d x))^{2/3}+(a+i b)^{2/3} \log \left (-\sqrt [3]{a+b \tan (c+d x)}+\sqrt [3]{a+i b}\right )\right )\right )}{4 d} \]
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 \[ \int {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.22, size = 96, normalized size = 0.29 \[ \frac {3 b \left (a +b \tan \left (d x +c \right )\right )^{\frac {2}{3}}}{2 d}+\frac {b \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-2 a \,\textit {\_Z}^{3}+a^{2}+b^{2}\right )}{\sum }\frac {\left (2 a \,\textit {\_R}^{4}+\left (-a^{2}-b^{2}\right ) \textit {\_R} \right ) \ln \left (\left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} a}\right )}{2 d} \]
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 \[ \int {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{3}}\,{d x} \]
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 \[ \text {result too large to display} \]
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 \[ \int \left (a + b \tan {\left (c + d x \right )}\right )^{\frac {5}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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