Optimal. Leaf size=221 \[ -\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{a^{3/2} d}-\frac {7 \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}+\frac {13 i \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}+\frac {5 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \sqrt {a+i a \tan (c+d x)}}+\frac {\cot ^{\frac {3}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}} \]
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Rubi [A] time = 0.65, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4241, 3559, 3596, 3598, 12, 3544, 205} \[ -\frac {7 \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}+\frac {13 i \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{a^{3/2} d}+\frac {5 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \sqrt {a+i a \tan (c+d x)}}+\frac {\cot ^{\frac {3}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 3544
Rule 3559
Rule 3596
Rule 3598
Rule 4241
Rubi steps
\begin {align*} \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx &=\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx\\ &=\frac {\cot ^{\frac {3}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\frac {9 a}{2}-3 i a \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx}{3 a^2}\\ &=\frac {\cot ^{\frac {3}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {5 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \sqrt {a+i a \tan (c+d x)}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\sqrt {a+i a \tan (c+d x)} \left (\frac {63 a^2}{4}-15 i a^2 \tan (c+d x)\right )}{\tan ^{\frac {5}{2}}(c+d x)} \, dx}{3 a^4}\\ &=\frac {\cot ^{\frac {3}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {5 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \sqrt {a+i a \tan (c+d x)}}-\frac {7 \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}+\frac {\left (2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\sqrt {a+i a \tan (c+d x)} \left (-\frac {117 i a^3}{8}-\frac {63}{4} a^3 \tan (c+d x)\right )}{\tan ^{\frac {3}{2}}(c+d x)} \, dx}{9 a^5}\\ &=\frac {\cot ^{\frac {3}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {5 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \sqrt {a+i a \tan (c+d x)}}+\frac {13 i \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}-\frac {7 \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}+\frac {\left (4 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int -\frac {9 a^4 \sqrt {a+i a \tan (c+d x)}}{16 \sqrt {\tan (c+d x)}} \, dx}{9 a^6}\\ &=\frac {\cot ^{\frac {3}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {5 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \sqrt {a+i a \tan (c+d x)}}+\frac {13 i \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}-\frac {7 \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}-\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {\tan (c+d x)}} \, dx}{4 a^2}\\ &=\frac {\cot ^{\frac {3}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {5 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \sqrt {a+i a \tan (c+d x)}}+\frac {13 i \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}-\frac {7 \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}+\frac {\left (i \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{-i a-2 a^2 x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{2 d}\\ &=-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \tanh ^{-1}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{a^{3/2} d}+\frac {\cot ^{\frac {3}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {5 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \sqrt {a+i a \tan (c+d x)}}+\frac {13 i \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}-\frac {7 \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}\\ \end {align*}
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Mathematica [A] time = 1.71, size = 186, normalized size = 0.84 \[ \frac {i e^{-4 i (c+d x)} \sqrt {\frac {a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {\cot (c+d x)} \left (18 e^{2 i (c+d x)}-87 e^{4 i (c+d x)}+52 e^{6 i (c+d x)}+3 e^{3 i (c+d x)} \left (-1+e^{2 i (c+d x)}\right )^{3/2} \tanh ^{-1}\left (\frac {e^{i (c+d x)}}{\sqrt {-1+e^{2 i (c+d x)}}}\right )+1\right )}{6 \sqrt {2} a^2 d \left (-1+e^{2 i (c+d x)}\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.93, size = 416, normalized size = 1.88 \[ \frac {\sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} {\left (52 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 87 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 18 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} - 3 \, {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \sqrt {-\frac {i}{2 \, a^{3} d^{2}}} \log \left ({\left (\sqrt {2} {\left (4 i \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - 4 i \, a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {-\frac {i}{2 \, a^{3} d^{2}}} + 4 i \, a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + 3 \, {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \sqrt {-\frac {i}{2 \, a^{3} d^{2}}} \log \left ({\left (\sqrt {2} {\left (-4 i \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + 4 i \, a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {-\frac {i}{2 \, a^{3} d^{2}}} + 4 i \, a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right )}{12 \, {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )}} \]
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 \[ \int \frac {\cot \left (d x + c\right )^{\frac {5}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.08, size = 445, normalized size = 2.01 \[ \frac {\left (\frac {1}{12}+\frac {i}{12}\right ) \left (\frac {\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )^{\frac {5}{2}} \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (-15 i \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+39 i \sin \left (d x +c \right )-4 i \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {2}\right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}-3 i \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {2}\right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {2}+4 \left (\cos ^{5}\left (d x +c \right )\right )-4 \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )-3 i \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {2}\right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}+21 i \cos \left (d x +c \right )-3 \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {2}\right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {2}+13 \left (\cos ^{3}\left (d x +c \right )\right )-15 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-13 i \left (\cos ^{3}\left (d x +c \right )\right )-4 i \left (\cos ^{5}\left (d x +c \right )\right )-21 \cos \left (d x +c \right )+39 \sin \left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{2} a^{2}} \]
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 \[ \int \frac {{\mathrm {cot}\left (c+d\,x\right )}^{5/2}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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