Optimal. Leaf size=214 \[ \frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (A-i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{a^{3/2} d}-\frac {(25 A+7 i B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{6 a^2 d}+\frac {(A+i B) \sqrt {\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(11 A+5 i B) \sqrt {\cot (c+d x)}}{6 a d \sqrt {a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.73, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4241, 3596, 3598, 12, 3544, 205} \[ -\frac {(25 A+7 i B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{6 a^2 d}+\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (A-i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{a^{3/2} d}+\frac {(A+i B) \sqrt {\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(11 A+5 i B) \sqrt {\cot (c+d x)}}{6 a d \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 3544
Rule 3596
Rule 3598
Rule 4241
Rubi steps
\begin {align*} \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (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 {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx\\ &=\frac {(A+i B) \sqrt {\cot (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 {1}{2} a (7 A+i B)-2 a (i A-B) \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx}{3 a^2}\\ &=\frac {(A+i B) \sqrt {\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(11 A+5 i B) \sqrt {\cot (c+d x)}}{6 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 {1}{4} a^2 (25 A+7 i B)-\frac {1}{2} a^2 (11 i A-5 B) \tan (c+d x)\right )}{\tan ^{\frac {3}{2}}(c+d x)} \, dx}{3 a^4}\\ &=\frac {(A+i B) \sqrt {\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(11 A+5 i B) \sqrt {\cot (c+d x)}}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(25 A+7 i B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{6 a^2 d}+\frac {\left (2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {3 a^3 (i A+B) \sqrt {a+i a \tan (c+d x)}}{8 \sqrt {\tan (c+d x)}} \, dx}{3 a^5}\\ &=\frac {(A+i B) \sqrt {\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(11 A+5 i B) \sqrt {\cot (c+d x)}}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(25 A+7 i B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{6 a^2 d}+\frac {\left ((i A+B) \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 {(A+i B) \sqrt {\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(11 A+5 i B) \sqrt {\cot (c+d x)}}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(25 A+7 i B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{6 a^2 d}-\frac {\left (i (i A+B) \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 ) (i A+B) \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 {(A+i B) \sqrt {\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(11 A+5 i B) \sqrt {\cot (c+d x)}}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(25 A+7 i B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{6 a^2 d}\\ \end {align*}
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Mathematica [A] time = 5.10, size = 195, normalized size = 0.91 \[ -\frac {\cot ^{\frac {3}{2}}(c+d x) \left (-3 (A-i B) e^{3 i (c+d x)} \sqrt {-1+e^{2 i (c+d x)}} \tanh ^{-1}\left (\frac {e^{i (c+d x)}}{\sqrt {-1+e^{2 i (c+d x)}}}\right )+A \left (-13 e^{2 i (c+d x)}+38 e^{4 i (c+d x)}-1\right )+i B \left (-7 e^{2 i (c+d x)}+8 e^{4 i (c+d x)}-1\right )\right )}{3 a d \left (1+e^{2 i (c+d x)}\right )^2 (\cot (c+d x)+i) \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.78, size = 466, normalized size = 2.18 \[ \frac {{\left (3 \, \sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{3} d^{2}}} e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (\frac {{\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (4 i \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - 4 i \, a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{3} d^{2}}} - 4 \, {\left (A - i \, B\right )} a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{i \, A + B}\right ) - 3 \, \sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{3} d^{2}}} e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (\frac {{\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (-4 i \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + 4 i \, a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{3} d^{2}}} - 4 \, {\left (A - i \, B\right )} a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{i \, A + B}\right ) - \sqrt {2} {\left (2 \, {\left (19 \, A + 4 i \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - {\left (13 \, A + 7 i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - A - i \, B\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{12 \, a^{2} d} \]
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 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \cot \left (d x + c\right )^{\frac {3}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 4.16, size = 648, normalized size = 3.03 \[ \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 {3}{2}} \sin \left (d x +c \right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (25 A +7 B +4 A \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )-4 B \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+11 A \cos \left (d x +c \right ) \sin \left (d x +c \right )-5 B \cos \left (d x +c \right ) \sin \left (d x +c \right )-3 i A \sqrt {2}\, \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 )-9 A \left (\cos ^{2}\left (d x +c \right )\right )+4 i A \left (\cos ^{4}\left (d x +c \right )\right )+11 i A \cos \left (d x +c \right ) \sin \left (d x +c \right )+4 i A \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 A \sqrt {2}\, \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 )}}+5 i B \cos \left (d x +c \right ) \sin \left (d x +c \right )+7 i B +4 i B \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )-3 i B \sqrt {2}\, \cos \left (d x +c \right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \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 )-4 A \left (\cos ^{4}\left (d x +c \right )\right )+9 i A \left (\cos ^{2}\left (d x +c \right )\right )+3 A \sqrt {2}\, \cos \left (d x +c \right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \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 )-3 i B \sqrt {2}\, \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 )}}-3 B \sqrt {2}\, \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 )-4 i B \left (\cos ^{4}\left (d x +c \right )\right )-3 i B \left (\cos ^{2}\left (d x +c \right )\right )-3 B \left (\cos ^{2}\left (d x +c \right )\right )-25 i A -4 B \left (\cos ^{4}\left (d x +c \right )\right )\right )}{d \cos \left (d x +c \right ) 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 \[ \int \frac {{\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]
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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