Optimal. Leaf size=200 \[ -\frac {\sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}-\frac {\left (\frac {1}{8}+\frac {3 i}{8}\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a d}+\frac {\left (\frac {1}{8}+\frac {3 i}{8}\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a d}+\frac {\left (\frac {1}{4}-\frac {3 i}{4}\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a d}-\frac {\left (\frac {1}{4}-\frac {3 i}{4}\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a d} \]
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Rubi [A] time = 0.20, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {3673, 3552, 3534, 1168, 1162, 617, 204, 1165, 628} \[ -\frac {\sqrt {\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}-\frac {\left (\frac {1}{8}+\frac {3 i}{8}\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a d}+\frac {\left (\frac {1}{8}+\frac {3 i}{8}\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a d}+\frac {\left (\frac {1}{4}-\frac {3 i}{4}\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a d}-\frac {\left (\frac {1}{4}-\frac {3 i}{4}\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a d} \]
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rule 3534
Rule 3552
Rule 3673
Rubi steps
\begin {align*} \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))} \, dx &=\int \frac {1}{\sqrt {\cot (c+d x)} (i a+a \cot (c+d x))} \, dx\\ &=-\frac {\sqrt {\cot (c+d x)}}{2 d (i a+a \cot (c+d x))}-\frac {\int \frac {\frac {3 i a}{2}-\frac {1}{2} a \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{2 a^2}\\ &=-\frac {\sqrt {\cot (c+d x)}}{2 d (i a+a \cot (c+d x))}-\frac {\operatorname {Subst}\left (\int \frac {-\frac {3 i a}{2}+\frac {a x^2}{2}}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a^2 d}\\ &=-\frac {\sqrt {\cot (c+d x)}}{2 d (i a+a \cot (c+d x))}--\frac {\left (\frac {1}{4}+\frac {3 i}{4}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a d}-\frac {\left (\frac {1}{4}-\frac {3 i}{4}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a d}\\ &=-\frac {\sqrt {\cot (c+d x)}}{2 d (i a+a \cot (c+d x))}-\frac {\left (\frac {1}{8}-\frac {3 i}{8}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a d}-\frac {\left (\frac {1}{8}-\frac {3 i}{8}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a d}-\frac {\left (\frac {1}{8}+\frac {3 i}{8}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{\sqrt {2} a d}-\frac {\left (\frac {1}{8}+\frac {3 i}{8}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{\sqrt {2} a d}\\ &=-\frac {\sqrt {\cot (c+d x)}}{2 d (i a+a \cot (c+d x))}-\frac {\left (\frac {1}{8}+\frac {3 i}{8}\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a d}+\frac {\left (\frac {1}{8}+\frac {3 i}{8}\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a d}--\frac {\left (\frac {1}{4}-\frac {3 i}{4}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a d}-\frac {\left (\frac {1}{4}-\frac {3 i}{4}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a d}\\ &=\frac {\left (\frac {1}{4}-\frac {3 i}{4}\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a d}-\frac {\left (\frac {1}{4}-\frac {3 i}{4}\right ) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a d}-\frac {\sqrt {\cot (c+d x)}}{2 d (i a+a \cot (c+d x))}-\frac {\left (\frac {1}{8}+\frac {3 i}{8}\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a d}+\frac {\left (\frac {1}{8}+\frac {3 i}{8}\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt {2} a d}\\ \end {align*}
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Mathematica [A] time = 1.07, size = 174, normalized size = 0.87 \[ \frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \sqrt {\sin (2 (c+d x))} \sqrt {\cot (c+d x)} \left (-\frac {2-2 i}{\sqrt {\sin (2 (c+d x))}}-(1-2 i) \sec (c+d x) \log \left (\sin (c+d x)+\sqrt {\sin (2 (c+d x))}+\cos (c+d x)\right )+(2+i) \csc (c+d x) \log \left (\sin (c+d x)+\sqrt {\sin (2 (c+d x))}+\cos (c+d x)\right )+(1+2 i) \sin ^{-1}(\cos (c+d x)-\sin (c+d x)) (\csc (c+d x)+i \sec (c+d x))\right )}{a d (\cot (c+d x)+i)} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.11, size = 467, normalized size = 2.34 \[ -\frac {{\left (a d \sqrt {-\frac {i}{4 \, a^{2} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left ({\left ({\left (4 i \, a d e^{\left (2 i \, d x + 2 i \, c\right )} - 4 i \, a d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {-\frac {i}{4 \, a^{2} d^{2}}} + 2 i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) - a d \sqrt {-\frac {i}{4 \, a^{2} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left ({\left ({\left (-4 i \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + 4 i \, a d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {-\frac {i}{4 \, a^{2} d^{2}}} + 2 i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) + a d \sqrt {\frac {i}{a^{2} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-\frac {{\left ({\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} - a d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {i}{a^{2} d^{2}}} + 1\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{a d}\right ) - a d \sqrt {\frac {i}{a^{2} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (\frac {{\left ({\left (a d e^{\left (2 i \, d x + 2 i \, c\right )} - a d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {i}{a^{2} d^{2}}} - 1\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{a d}\right ) - \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} {\left (i \, e^{\left (2 i \, d x + 2 i \, c\right )} - i\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, a d} \]
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 \[ \int \frac {1}{{\left (i \, a \tan \left (d x + c\right ) + a\right )} \cot \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.27, size = 1836, normalized size = 9.18 \[ \text {result too large to display} \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]
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 \[ - \frac {i \int \frac {1}{\tan {\left (c + d x \right )} \cot ^{\frac {3}{2}}{\left (c + d x \right )} - i \cot ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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