Optimal. Leaf size=106 \[ \frac {8 \sqrt [4]{-1} a^3 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}+\frac {8 a^3 \sqrt {\cot (c+d x)}}{d}-\frac {8 i a^3 \cot ^{\frac {3}{2}}(c+d x)}{5 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x) \left (i a^3+a^3 \cot (c+d x)\right )}{5 d} \]
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Rubi [A]
time = 0.15, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3754, 3637,
3673, 3609, 3614, 214} \begin {gather*} -\frac {8 i a^3 \cot ^{\frac {3}{2}}(c+d x)}{5 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x) \left (a^3 \cot (c+d x)+i a^3\right )}{5 d}+\frac {8 a^3 \sqrt {\cot (c+d x)}}{d}+\frac {8 \sqrt [4]{-1} a^3 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 3609
Rule 3614
Rule 3637
Rule 3673
Rule 3754
Rubi steps
\begin {align*} \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^3 \, dx &=\int \sqrt {\cot (c+d x)} (i a+a \cot (c+d x))^3 \, dx\\ &=-\frac {2 \cot ^{\frac {3}{2}}(c+d x) \left (i a^3+a^3 \cot (c+d x)\right )}{5 d}-\frac {1}{5} (2 i a) \int \sqrt {\cot (c+d x)} (-4 i a-6 a \cot (c+d x)) (i a+a \cot (c+d x)) \, dx\\ &=-\frac {8 i a^3 \cot ^{\frac {3}{2}}(c+d x)}{5 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x) \left (i a^3+a^3 \cot (c+d x)\right )}{5 d}-\frac {1}{5} (2 i a) \int \sqrt {\cot (c+d x)} \left (10 a^2-10 i a^2 \cot (c+d x)\right ) \, dx\\ &=\frac {8 a^3 \sqrt {\cot (c+d x)}}{d}-\frac {8 i a^3 \cot ^{\frac {3}{2}}(c+d x)}{5 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x) \left (i a^3+a^3 \cot (c+d x)\right )}{5 d}-\frac {1}{5} (2 i a) \int \frac {10 i a^2+10 a^2 \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx\\ &=\frac {8 a^3 \sqrt {\cot (c+d x)}}{d}-\frac {8 i a^3 \cot ^{\frac {3}{2}}(c+d x)}{5 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x) \left (i a^3+a^3 \cot (c+d x)\right )}{5 d}+\frac {\left (80 i a^5\right ) \text {Subst}\left (\int \frac {1}{-10 i a^2+10 a^2 x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d}\\ &=\frac {8 \sqrt [4]{-1} a^3 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}+\frac {8 a^3 \sqrt {\cot (c+d x)}}{d}-\frac {8 i a^3 \cot ^{\frac {3}{2}}(c+d x)}{5 d}-\frac {2 \cot ^{\frac {3}{2}}(c+d x) \left (i a^3+a^3 \cot (c+d x)\right )}{5 d}\\ \end {align*}
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Mathematica [A]
time = 2.21, size = 147, normalized size = 1.39 \begin {gather*} -\frac {a^3 e^{-3 i c} \sqrt {\cot (c+d x)} (\cos (3 (c+d x))+i \sin (3 (c+d x))) \left (\csc ^2(c+d x) (-19+21 \cos (2 (c+d x))+5 i \sin (2 (c+d x)))+40 \tanh ^{-1}\left (\sqrt {\frac {-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right ) \sqrt {i \tan (c+d x)}\right )}{5 d (\cos (d x)+i \sin (d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 14.35, size = 1482, normalized size = 13.98
method | result | size |
default | \(\text {Expression too large to display}\) | \(1482\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 158, normalized size = 1.49 \begin {gather*} \frac {5 \, {\left (\left (2 i - 2\right ) \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \left (2 i - 2\right ) \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \left (i + 1\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \left (i + 1\right ) \, \sqrt {2} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a^{3} + \frac {40 \, a^{3}}{\sqrt {\tan \left (d x + c\right )}} - \frac {10 i \, a^{3}}{\tan \left (d x + c\right )^{\frac {3}{2}}} - \frac {2 \, a^{3}}{\tan \left (d x + c\right )^{\frac {5}{2}}}}{5 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 340 vs. \(2 (86) = 172\).
time = 0.65, size = 340, normalized size = 3.21 \begin {gather*} -\frac {5 \, \sqrt {\frac {64 i \, a^{6}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {{\left (8 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {\frac {64 i \, a^{6}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, a^{3}}\right ) - 5 \, \sqrt {\frac {64 i \, a^{6}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {{\left (8 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt {\frac {64 i \, a^{6}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, a^{3}}\right ) - 16 \, {\left (13 \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 19 \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 8 \, a^{3}\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}}{20 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {cot}\left (c+d\,x\right )}^{7/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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