Optimal. Leaf size=354 \[ -\frac {4 i \tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} d}-\frac {i \sqrt {3} \tan ^{-1}\left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{4 \sqrt [3]{2} a^{4/3} d}+\frac {2 i \log (\tan (c+d x))}{3 a^{4/3} d}-\frac {2 i \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{a^{4/3} d}-\frac {3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{8 \sqrt [3]{2} a^{4/3} d}-\frac {i \log (\cos (c+d x))}{8 \sqrt [3]{2} a^{4/3} d}+\frac {x}{8 \sqrt [3]{2} a^{4/3}}-\frac {19 i}{4 a d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {11 i}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}} \]
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Rubi [A] time = 0.71, antiderivative size = 354, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {3561, 3596, 3600, 3481, 55, 617, 204, 31, 3599} \[ -\frac {4 i \tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} d}-\frac {i \sqrt {3} \tan ^{-1}\left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{4 \sqrt [3]{2} a^{4/3} d}+\frac {2 i \log (\tan (c+d x))}{3 a^{4/3} d}-\frac {2 i \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{a^{4/3} d}-\frac {3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{8 \sqrt [3]{2} a^{4/3} d}-\frac {i \log (\cos (c+d x))}{8 \sqrt [3]{2} a^{4/3} d}+\frac {x}{8 \sqrt [3]{2} a^{4/3}}-\frac {19 i}{4 a d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {11 i}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 55
Rule 204
Rule 617
Rule 3481
Rule 3561
Rule 3596
Rule 3599
Rule 3600
Rubi steps
\begin {align*} \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx &=-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}+\frac {\int \frac {\cot (c+d x) \left (-\frac {4 i a}{3}-\frac {7}{3} a \tan (c+d x)\right )}{(a+i a \tan (c+d x))^{4/3}} \, dx}{a}\\ &=-\frac {11 i}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}+\frac {3 \int \frac {\cot (c+d x) \left (-\frac {32 i a^2}{9}-\frac {44}{9} a^2 \tan (c+d x)\right )}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx}{8 a^3}\\ &=-\frac {11 i}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}-\frac {19 i}{4 a d \sqrt [3]{a+i a \tan (c+d x)}}+\frac {9 \int \cot (c+d x) (a+i a \tan (c+d x))^{2/3} \left (-\frac {64 i a^3}{27}-\frac {76}{27} a^3 \tan (c+d x)\right ) \, dx}{16 a^5}\\ &=-\frac {11 i}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}-\frac {19 i}{4 a d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {(4 i) \int \cot (c+d x) (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{2/3} \, dx}{3 a^3}-\frac {\int (a+i a \tan (c+d x))^{2/3} \, dx}{4 a^2}\\ &=-\frac {11 i}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}-\frac {19 i}{4 a d \sqrt [3]{a+i a \tan (c+d x)}}+\frac {i \operatorname {Subst}\left (\int \frac {1}{(a-x) \sqrt [3]{a+x}} \, dx,x,i a \tan (c+d x)\right )}{4 a d}-\frac {(4 i) \operatorname {Subst}\left (\int \frac {1}{x \sqrt [3]{a+i a x}} \, dx,x,\tan (c+d x)\right )}{3 a d}\\ &=\frac {x}{8 \sqrt [3]{2} a^{4/3}}-\frac {i \log (\cos (c+d x))}{8 \sqrt [3]{2} a^{4/3} d}+\frac {2 i \log (\tan (c+d x))}{3 a^{4/3} d}-\frac {11 i}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}-\frac {19 i}{4 a d \sqrt [3]{a+i a \tan (c+d x)}}+\frac {(2 i) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{a^{4/3} d}+\frac {(3 i) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{8 \sqrt [3]{2} a^{4/3} d}-\frac {(3 i) \operatorname {Subst}\left (\int \frac {1}{2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{8 a d}-\frac {(2 i) \operatorname {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{a d}\\ &=\frac {x}{8 \sqrt [3]{2} a^{4/3}}-\frac {i \log (\cos (c+d x))}{8 \sqrt [3]{2} a^{4/3} d}+\frac {2 i \log (\tan (c+d x))}{3 a^{4/3} d}-\frac {2 i \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{a^{4/3} d}-\frac {3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{8 \sqrt [3]{2} a^{4/3} d}-\frac {11 i}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}-\frac {19 i}{4 a d \sqrt [3]{a+i a \tan (c+d x)}}+\frac {(4 i) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}\right )}{a^{4/3} d}+\frac {(3 i) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}\right )}{4 \sqrt [3]{2} a^{4/3} d}\\ &=\frac {x}{8 \sqrt [3]{2} a^{4/3}}-\frac {4 i \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{4/3} d}-\frac {i \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} a^{4/3} d}-\frac {i \log (\cos (c+d x))}{8 \sqrt [3]{2} a^{4/3} d}+\frac {2 i \log (\tan (c+d x))}{3 a^{4/3} d}-\frac {2 i \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{a^{4/3} d}-\frac {3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{8 \sqrt [3]{2} a^{4/3} d}-\frac {11 i}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac {\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}-\frac {19 i}{4 a d \sqrt [3]{a+i a \tan (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 2.07, size = 233, normalized size = 0.66 \[ \frac {6 e^{4 i (c+d x)} \left (-1+e^{2 i (c+d x)}\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right )+64 e^{4 i (c+d x)} \left (-1+e^{2 i (c+d x)}\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {2 e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right )+63 e^{2 i (c+d x)}-35 e^{4 i (c+d x)}-95 e^{6 i (c+d x)}+3}{8 a d \left (-1+e^{2 i (c+d x)}\right ) \left (1+e^{2 i (c+d x)}\right )^2 (\tan (c+d x)-i) \sqrt [3]{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 792, normalized size = 2.24 \[ \text {result too large to display} \]
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 \[ \int \frac {\cot \left (d x + c\right )^{2}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.81, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{2}\left (d x +c \right )}{\left (a +i a \tan \left (d x +c \right )\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 310, normalized size = 0.88 \[ -\frac {i \, a {\left (\frac {6 \, {\left (38 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} - 27 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a - 3 \, a^{2}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{3}} a^{2} - {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}} a^{3}} + \frac {6 \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{a^{\frac {7}{3}}} - \frac {3 \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right )}{a^{\frac {7}{3}}} + \frac {6 \cdot 2^{\frac {2}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}{a^{\frac {7}{3}}} + \frac {64 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {7}{3}}} - \frac {32 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {7}{3}}} + \frac {64 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {7}{3}}}\right )}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.99, size = 893, normalized size = 2.52 \[ \text {result too large to display} \]
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 \[ \int \frac {\cot ^{2}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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