Optimal. Leaf size=141 \[ -\frac {(-1)^{3/4} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{8 a^3 d}+\frac {\sqrt {\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac {i \sqrt {\cot (c+d x)}}{6 a d (i a+a \cot (c+d x))^2}+\frac {\sqrt {\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )} \]
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Rubi [A]
time = 0.20, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {3754, 3639,
3677, 12, 3630, 3614, 214} \begin {gather*} \frac {\sqrt {\cot (c+d x)}}{8 d \left (a^3 \cot (c+d x)+i a^3\right )}-\frac {(-1)^{3/4} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{8 a^3 d}+\frac {i \sqrt {\cot (c+d x)}}{6 a d (a \cot (c+d x)+i a)^2}+\frac {\sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 214
Rule 3614
Rule 3630
Rule 3639
Rule 3677
Rule 3754
Rubi steps
\begin {align*} \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx &=\int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(i a+a \cot (c+d x))^3} \, dx\\ &=\frac {\sqrt {\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac {\int \frac {-\frac {i a}{2}+\frac {7}{2} a \cot (c+d x)}{\sqrt {\cot (c+d x)} (i a+a \cot (c+d x))^2} \, dx}{6 a^2}\\ &=\frac {\sqrt {\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac {i \sqrt {\cot (c+d x)}}{6 a d (i a+a \cot (c+d x))^2}+\frac {\int -\frac {6 i a^2 \sqrt {\cot (c+d x)}}{i a+a \cot (c+d x)} \, dx}{24 a^4}\\ &=\frac {\sqrt {\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac {i \sqrt {\cot (c+d x)}}{6 a d (i a+a \cot (c+d x))^2}-\frac {i \int \frac {\sqrt {\cot (c+d x)}}{i a+a \cot (c+d x)} \, dx}{4 a^2}\\ &=\frac {\sqrt {\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac {i \sqrt {\cot (c+d x)}}{6 a d (i a+a \cot (c+d x))^2}+\frac {\sqrt {\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac {i \int \frac {-\frac {a}{2}+\frac {1}{2} i a \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{8 a^4}\\ &=\frac {\sqrt {\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac {i \sqrt {\cot (c+d x)}}{6 a d (i a+a \cot (c+d x))^2}+\frac {\sqrt {\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac {i \text {Subst}\left (\int \frac {1}{\frac {a}{2}+\frac {1}{2} i a x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{16 a^2 d}\\ &=-\frac {(-1)^{3/4} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{8 a^3 d}+\frac {\sqrt {\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac {i \sqrt {\cot (c+d x)}}{6 a d (i a+a \cot (c+d x))^2}+\frac {\sqrt {\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 1.67, size = 154, normalized size = 1.09 \begin {gather*} \frac {i \sqrt {\cot (c+d x)} \csc ^3(c+d x) \left (5 \cos (c+d x)-5 \cos (3 (c+d x))+3 i \sin (c+d x)-3 i \sin (3 (c+d x))+6 \tanh ^{-1}\left (\sqrt {\frac {-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right ) (\cos (3 (c+d x))+i \sin (3 (c+d x))) \sqrt {i \tan (c+d x)}\right )}{48 a^3 d (i+\cot (c+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 = 15.08, size = 5851, normalized size = 41.50
method | result | size |
default | \(\text {Expression too large to display}\) | \(5851\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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. 307 vs. \(2 (113) = 226\).
time = 1.68, size = 307, normalized size = 2.18 \begin {gather*} \frac {{\left (12 \, a^{3} d \sqrt {-\frac {i}{64 \, a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-2 \, {\left (8 \, {\left (i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{3} 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}} \sqrt {-\frac {i}{64 \, a^{6} d^{2}}} - i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) - 12 \, a^{3} d \sqrt {-\frac {i}{64 \, a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-2 \, {\left (8 \, {\left (-i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{3} 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}} \sqrt {-\frac {i}{64 \, a^{6} d^{2}}} - i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\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}} {\left (-4 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 4 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + i \, e^{\left (2 i \, d x + 2 i \, c\right )} - i\right )}\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{48 \, a^{3} d} \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*} \frac {i \int \frac {1}{\tan ^{3}{\left (c + d x \right )} \cot ^{\frac {3}{2}}{\left (c + d x \right )} - 3 i \tan ^{2}{\left (c + d x \right )} \cot ^{\frac {3}{2}}{\left (c + d x \right )} - 3 \tan {\left (c + d x \right )} \cot ^{\frac {3}{2}}{\left (c + d x \right )} + i \cot ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx}{a^{3}} \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 \frac {1}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,{\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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