Optimal. Leaf size=284 \[ \frac {(A+5 i B) \sqrt {\cot (c+d x)}}{8 a^2 d (\cot (c+d x)+i)}+\frac {((1-3 i) A-(9-5 i) B) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{32 \sqrt {2} a^2 d}+\frac {\left (\frac {1}{32}+\frac {i}{32}\right ) ((1+2 i) A+(2-7 i) B) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a^2 d}-\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) ((2+i) A+(7-2 i) B) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^2 d}+\frac {((1+3 i) A+(9+5 i) B) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{16 \sqrt {2} a^2 d}+\frac {(-B+i A) \sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2} \]
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Rubi [A] time = 0.60, antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3581, 3596, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac {(A+5 i B) \sqrt {\cot (c+d x)}}{8 a^2 d (\cot (c+d x)+i)}+\frac {((1-3 i) A-(9-5 i) B) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{32 \sqrt {2} a^2 d}+\frac {\left (\frac {1}{32}+\frac {i}{32}\right ) ((1+2 i) A+(2-7 i) B) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a^2 d}-\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) ((2+i) A+(7-2 i) B) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^2 d}+\frac {((1+3 i) A+(9+5 i) B) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{16 \sqrt {2} a^2 d}+\frac {(-B+i A) \sqrt {\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rule 3534
Rule 3581
Rule 3596
Rubi steps
\begin {align*} \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx &=\int \frac {B+A \cot (c+d x)}{\sqrt {\cot (c+d x)} (i a+a \cot (c+d x))^2} \, dx\\ &=\frac {(i A-B) \sqrt {\cot (c+d x)}}{4 d (i a+a \cot (c+d x))^2}+\frac {\int \frac {\frac {1}{2} a (A-7 i B)-\frac {3}{2} a (i A-B) \cot (c+d x)}{\sqrt {\cot (c+d x)} (i a+a \cot (c+d x))} \, dx}{4 a^2}\\ &=\frac {(A+5 i B) \sqrt {\cot (c+d x)}}{8 a^2 d (i+\cot (c+d x))}+\frac {(i A-B) \sqrt {\cot (c+d x)}}{4 d (i a+a \cot (c+d x))^2}+\frac {\int \frac {-\frac {3}{2} a^2 (i A+3 B)-\frac {1}{2} a^2 (A+5 i B) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{8 a^4}\\ &=\frac {(A+5 i B) \sqrt {\cot (c+d x)}}{8 a^2 d (i+\cot (c+d x))}+\frac {(i A-B) \sqrt {\cot (c+d x)}}{4 d (i a+a \cot (c+d x))^2}+\frac {\operatorname {Subst}\left (\int \frac {\frac {3}{2} a^2 (i A+3 B)+\frac {1}{2} a^2 (A+5 i B) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{4 a^4 d}\\ &=\frac {(A+5 i B) \sqrt {\cot (c+d x)}}{8 a^2 d (i+\cot (c+d x))}+\frac {(i A-B) \sqrt {\cot (c+d x)}}{4 d (i a+a \cot (c+d x))^2}+\frac {\left (\left (\frac {1}{16}+\frac {i}{16}\right ) ((1+2 i) A+(2-7 i) B)\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a^2 d}+\frac {((1+3 i) A+(9+5 i) B) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{16 a^2 d}\\ &=\frac {(A+5 i B) \sqrt {\cot (c+d x)}}{8 a^2 d (i+\cot (c+d x))}+\frac {(i A-B) \sqrt {\cot (c+d x)}}{4 d (i a+a \cot (c+d x))^2}+\frac {((1-3 i) A-(9-5 i) B) \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{32 \sqrt {2} a^2 d}+\frac {((1-3 i) A-(9-5 i) B) \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{32 \sqrt {2} a^2 d}+\frac {((1+3 i) A+(9+5 i) B) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{32 a^2 d}+\frac {((1+3 i) A+(9+5 i) B) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{32 a^2 d}\\ &=\frac {(A+5 i B) \sqrt {\cot (c+d x)}}{8 a^2 d (i+\cot (c+d x))}+\frac {(i A-B) \sqrt {\cot (c+d x)}}{4 d (i a+a \cot (c+d x))^2}+\frac {((1-3 i) A-(9-5 i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^2 d}-\frac {((1-3 i) A-(9-5 i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^2 d}+\frac {((1+3 i) A+(9+5 i) B) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^2 d}-\frac {((1+3 i) A+(9+5 i) B) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^2 d}\\ &=-\frac {((1+3 i) A+(9+5 i) B) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^2 d}+\frac {((1+3 i) A+(9+5 i) B) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^2 d}+\frac {(A+5 i B) \sqrt {\cot (c+d x)}}{8 a^2 d (i+\cot (c+d x))}+\frac {(i A-B) \sqrt {\cot (c+d x)}}{4 d (i a+a \cot (c+d x))^2}+\frac {((1-3 i) A-(9-5 i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^2 d}-\frac {((1-3 i) A-(9-5 i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^2 d}\\ \end {align*}
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Mathematica [A] time = 2.82, size = 241, normalized size = 0.85 \[ \frac {\sec (c+d x) (\cos (d x)+i \sin (d x))^2 (A+B \tan (c+d x)) \left (4 \cos (c+d x) (\cos (2 d x)-i \sin (2 d x)) ((-7 B+3 i A) \sin (c+d x)+(A+5 i B) \cos (c+d x))-(1+i) (-\sin (2 c)+i \cos (2 c)) \sqrt {\sin (2 (c+d x))} \csc (c+d x) \left (((2+7 i) B-(1-2 i) A) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+((7+2 i) B-(2-i) A) \log \left (\sin (c+d x)+\sqrt {\sin (2 (c+d x))}+\cos (c+d x)\right )\right )\right )}{32 d \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^2 (A \cos (c+d x)+B \sin (c+d x))} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.61, size = 662, normalized size = 2.33 \[ -\frac {{\left (2 \, a^{2} d \sqrt {\frac {-i \, A^{2} - 2 \, A B + i \, B^{2}}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-\frac {2 \, {\left ({\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2} 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} - 2 \, A B + i \, B^{2}}{a^{4} d^{2}}} + {\left (A - i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{i \, A + B}\right ) - 2 \, a^{2} d \sqrt {\frac {-i \, A^{2} - 2 \, A B + i \, B^{2}}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (\frac {2 \, {\left ({\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2} 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} - 2 \, A B + i \, B^{2}}{a^{4} d^{2}}} - {\left (A - i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{i \, A + B}\right ) + a^{2} d \sqrt {\frac {i \, A^{2} + 14 \, A B - 49 i \, B^{2}}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-\frac {{\left ({\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2} 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} + 14 \, A B - 49 i \, B^{2}}{a^{4} d^{2}}} + A - 7 i \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{2} d}\right ) - a^{2} d \sqrt {\frac {i \, A^{2} + 14 \, A B - 49 i \, B^{2}}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (\frac {{\left ({\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2} 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} + 14 \, A B - 49 i \, B^{2}}{a^{4} d^{2}}} - A + 7 i \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{2} d}\right ) - 2 \, {\left ({\left (-2 i \, A + 6 \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (3 i \, A - 7 \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, A + B\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}}\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{32 \, 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 \[ \int \frac {B \tan \left (d x + c\right ) + A}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} \cot \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.09, size = 5042, normalized size = 17.75 \[ \text {output too large to display} \]
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 \[ \int \frac {A+B\,\mathrm {tan}\left (c+d\,x\right )}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]
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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