Optimal. Leaf size=181 \[ \frac {3 i \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {\cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {13 \cot (c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {7 \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d} \]
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Rubi [A]
time = 0.38, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {3640, 3677,
3679, 3681, 3561, 212, 3680, 65, 214} \begin {gather*} \frac {3 i \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {7 \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}+\frac {13 \cot (c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {\cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 212
Rule 214
Rule 3561
Rule 3640
Rule 3677
Rule 3679
Rule 3680
Rule 3681
Rubi steps
\begin {align*} \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx &=\frac {\cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {\int \frac {\cot ^2(c+d x) \left (4 a-\frac {5}{2} i a \tan (c+d x)\right )}{\sqrt {a+i a \tan (c+d x)}} \, dx}{3 a^2}\\ &=\frac {\cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {13 \cot (c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {\int \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)} \left (\frac {21 a^2}{2}-\frac {39}{4} i a^2 \tan (c+d x)\right ) \, dx}{3 a^4}\\ &=\frac {\cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {13 \cot (c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {7 \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}+\frac {\int \cot (c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {9 i a^3}{2}-\frac {21}{4} a^3 \tan (c+d x)\right ) \, dx}{3 a^5}\\ &=\frac {\cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {13 \cot (c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {7 \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}-\frac {(3 i) \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)} \, dx}{2 a^3}-\frac {\int \sqrt {a+i a \tan (c+d x)} \, dx}{4 a^2}\\ &=\frac {\cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {13 \cot (c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {7 \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}+\frac {i \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{2 a d}-\frac {(3 i) \text {Subst}\left (\int \frac {1}{x \sqrt {a+i a x}} \, dx,x,\tan (c+d x)\right )}{2 a d}\\ &=\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {\cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {13 \cot (c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {7 \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}-\frac {3 \text {Subst}\left (\int \frac {1}{i-\frac {i x^2}{a}} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{a^2 d}\\ &=\frac {3 i \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {\cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {13 \cot (c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {7 \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}\\ \end {align*}
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Mathematica [A]
time = 2.01, size = 214, normalized size = 1.18 \begin {gather*} -\frac {e^{-4 i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \left (\sqrt {1+e^{2 i (c+d x)}} \left (-1-15 e^{2 i (c+d x)}+28 e^{4 i (c+d x)}\right )-3 e^{3 i (c+d x)} \left (-1+e^{2 i (c+d x)}\right ) \sinh ^{-1}\left (e^{i (c+d x)}\right )-18 \sqrt {2} e^{3 i (c+d x)} \left (-1+e^{2 i (c+d x)}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} e^{i (c+d x)}}{\sqrt {1+e^{2 i (c+d x)}}}\right )\right ) \csc (2 (c+d x))}{12 a d \sqrt {a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1398 vs. \(2 (145 ) = 290\).
time = 0.91, size = 1399, normalized size = 7.73
method | result | size |
default | \(\text {Expression too large to display}\) | \(1399\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 184, normalized size = 1.02 \begin {gather*} -\frac {i \, a {\left (\frac {4 \, {\left (21 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} - 13 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a - 2 \, a^{2}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2} - {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{3}} + \frac {3 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{a^{\frac {5}{2}}} + \frac {36 \, \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}}}\right )}}{24 \, 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. 601 vs. \(2 (138) = 276\).
time = 0.38, size = 601, normalized size = 3.32 \begin {gather*} -\frac {3 \, \sqrt {\frac {1}{2}} {\left (-i \, a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} + i \, a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \sqrt {\frac {1}{a^{3} d^{2}}} \log \left (4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{3} d^{2}}} + a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + 3 \, \sqrt {\frac {1}{2}} {\left (i \, a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - i \, a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \sqrt {\frac {1}{a^{3} d^{2}}} \log \left (-4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{3} d^{2}}} - a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + 9 \, {\left (-i \, a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} + i \, a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \sqrt {\frac {1}{a^{3} d^{2}}} \log \left (16 \, {\left (3 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 2 \, \sqrt {2} {\left (a^{3} d e^{\left (3 i \, d x + 3 i \, c\right )} + a^{3} d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{3} d^{2}}} + a^{2}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) + 9 \, {\left (i \, a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - i \, a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \sqrt {\frac {1}{a^{3} d^{2}}} \log \left (16 \, {\left (3 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 2 \, \sqrt {2} {\left (a^{3} d e^{\left (3 i \, d x + 3 i \, c\right )} + a^{3} d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{3} d^{2}}} + a^{2}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) - \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-28 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 13 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 16 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )}}{12 \, {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\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 {\cot ^{2}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}\, 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 = 4.02, size = 178, normalized size = 0.98 \begin {gather*} -\frac {\frac {\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\,13{}\mathrm {i}}{6\,d}+\frac {a\,1{}\mathrm {i}}{3\,d}-\frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2\,7{}\mathrm {i}}{2\,a\,d}}{a\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}-{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}}-\frac {\mathrm {atan}\left (\frac {\sqrt {-a^3}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{a^2}\right )\,\sqrt {-a^3}\,3{}\mathrm {i}}{a^3\,d}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {-a^3}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{2\,a^2}\right )\,\sqrt {-a^3}\,1{}\mathrm {i}}{4\,a^3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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