Optimal. Leaf size=316 \[ \frac {(B+2 i A) \sqrt {\cot (c+d x)}}{8 d \left (a^3 \cot (c+d x)+i a^3\right )}+\frac {(2 i A+(1-i) B) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{32 \sqrt {2} a^3 d}-\frac {(2 i A+(1-i) B) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{32 \sqrt {2} a^3 d}-\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) (B+(1+i) A) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^3 d}+\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) (B+(1+i) A) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a^3 d}+\frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}+\frac {A \sqrt {\cot (c+d x)}}{4 a d (a \cot (c+d x)+i a)^2} \]
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Rubi [A] time = 0.76, antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3581, 3595, 3596, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac {(B+2 i A) \sqrt {\cot (c+d x)}}{8 d \left (a^3 \cot (c+d x)+i a^3\right )}+\frac {(2 i A+(1-i) B) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{32 \sqrt {2} a^3 d}-\frac {(2 i A+(1-i) B) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{32 \sqrt {2} a^3 d}-\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) (B+(1+i) A) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^3 d}+\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) (B+(1+i) A) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a^3 d}+\frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}+\frac {A \sqrt {\cot (c+d x)}}{4 a d (a \cot (c+d x)+i a)^2} \]
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 3595
Rule 3596
Rubi steps
\begin {align*} \int \frac {A+B \tan (c+d x)}{\sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3} \, dx &=\int \frac {\cot ^{\frac {3}{2}}(c+d x) (B+A \cot (c+d x))}{(i a+a \cot (c+d x))^3} \, dx\\ &=\frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {\int \frac {\sqrt {\cot (c+d x)} \left (-\frac {3}{2} a (i A-B)+\frac {3}{2} a (3 A-i B) \cot (c+d x)\right )}{(i a+a \cot (c+d x))^2} \, dx}{6 a^2}\\ &=\frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {A \sqrt {\cot (c+d x)}}{4 a d (i a+a \cot (c+d x))^2}+\frac {\int \frac {-3 i a^2 A+3 a^2 (3 A-2 i B) \cot (c+d x)}{\sqrt {\cot (c+d x)} (i a+a \cot (c+d x))} \, dx}{24 a^4}\\ &=\frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {A \sqrt {\cot (c+d x)}}{4 a d (i a+a \cot (c+d x))^2}+\frac {(2 i A+B) \sqrt {\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac {\int \frac {-3 i a^3 B-3 a^3 (2 i A+B) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{48 a^6}\\ &=\frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {A \sqrt {\cot (c+d x)}}{4 a d (i a+a \cot (c+d x))^2}+\frac {(2 i A+B) \sqrt {\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {3 i a^3 B+3 a^3 (2 i A+B) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{24 a^6 d}\\ &=\frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {A \sqrt {\cot (c+d x)}}{4 a d (i a+a \cot (c+d x))^2}+\frac {(2 i A+B) \sqrt {\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac {\left (\left (\frac {1}{16}+\frac {i}{16}\right ) ((1+i) A+B)\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a^3 d}-\frac {(2 i A+(1-i) B) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{16 a^3 d}\\ &=\frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {A \sqrt {\cot (c+d x)}}{4 a d (i a+a \cot (c+d x))^2}+\frac {(2 i A+B) \sqrt {\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac {\left (\left (\frac {1}{32}+\frac {i}{32}\right ) ((1+i) A+B)\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a^3 d}+\frac {\left (\left (\frac {1}{32}+\frac {i}{32}\right ) ((1+i) A+B)\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a^3 d}+\frac {(2 i A+(1-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^3 d}+\frac {(2 i A+(1-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^3 d}\\ &=\frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {A \sqrt {\cot (c+d x)}}{4 a d (i a+a \cot (c+d x))^2}+\frac {(2 i A+B) \sqrt {\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac {(2 i A+(1-i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^3 d}-\frac {(2 i A+(1-i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^3 d}+-\frac {\left (\left (\frac {1}{16}+\frac {i}{16}\right ) ((1+i) A+B)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^3 d}+\frac {\left (\left (\frac {1}{16}+\frac {i}{16}\right ) ((1+i) A+B)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^3 d}\\ &=-\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) ((1+i) A+B) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^3 d}+\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) ((1+i) A+B) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^3 d}+\frac {(A+i B) \cot ^{\frac {3}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {A \sqrt {\cot (c+d x)}}{4 a d (i a+a \cot (c+d x))^2}+\frac {(2 i A+B) \sqrt {\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac {(2 i A+(1-i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^3 d}-\frac {(2 i A+(1-i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^3 d}\\ \end {align*}
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Mathematica [A] time = 4.26, size = 272, normalized size = 0.86 \[ \frac {e^{-4 i (c+d x)} \sqrt {\cot (c+d x)} \sec (c+d x) (\cos (3 (c+d x))-i \sin (3 (c+d x))) \left (\left (-2 e^{2 i (c+d x)}+e^{4 i (c+d x)}+2 e^{6 i (c+d x)}-1\right ) \left (A e^{2 i (c+d x)}+A-2 i B e^{2 i (c+d x)}+i B\right )-6 (A-i B) e^{6 i (c+d x)} \sqrt {-1+e^{2 i (c+d x)}} \sqrt {1+e^{2 i (c+d x)}} \tanh ^{-1}\left (\sqrt {\frac {-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right )-3 A e^{6 i (c+d x)} \sqrt {-1+e^{4 i (c+d x)}} \tan ^{-1}\left (\sqrt {-1+e^{4 i (c+d x)}}\right )\right )}{96 a^3 d} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.54, size = 637, normalized size = 2.02 \[ -\frac {{\left (3 \, a^{3} d \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (\frac {{\left ({\left (16 i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - 16 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 \, A^{2} + 2 \, A B - i \, B^{2}}{a^{6} d^{2}}} - 16 \, {\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 )}}{8 \, {\left (i \, A + B\right )}}\right ) - 3 \, a^{3} d \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (\frac {{\left ({\left (-16 i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + 16 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 \, A^{2} + 2 \, A B - i \, B^{2}}{a^{6} d^{2}}} - 16 \, {\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 )}}{8 \, {\left (i \, A + B\right )}}\right ) - 24 \, a^{3} d \sqrt {-\frac {i \, A^{2}}{64 \, a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (\frac {{\left (8 \, {\left (a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - 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 \, A^{2}}{64 \, a^{6} d^{2}}} + i \, A\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{3} d}\right ) + 24 \, a^{3} d \sqrt {-\frac {i \, A^{2}}{64 \, a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-\frac {{\left (8 \, {\left (a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - 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 \, A^{2}}{64 \, a^{6} d^{2}}} - i \, A\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{3} d}\right ) - 2 \, {\left (2 \, {\left (A - 2 i \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (A + 4 i \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - {\left (2 \, A - i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - A - i \, 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 (-6 i \, d x - 6 i \, c\right )}}{96 \, a^{3} 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 \[ \int \frac {B \tan \left (d x + c\right ) + A}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} \sqrt {\cot \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 4.10, size = 5075, normalized size = 16.06 \[ \text {output too large to display} \]
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 \[ \int \frac {A+B\,\mathrm {tan}\left (c+d\,x\right )}{\sqrt {\mathrm {cot}\left (c+d\,x\right )}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3} \,d x \]
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 \[ \frac {i \left (\int \frac {A}{\tan ^{3}{\left (c + d x \right )} \sqrt {\cot {\left (c + d x \right )}} - 3 i \tan ^{2}{\left (c + d x \right )} \sqrt {\cot {\left (c + d x \right )}} - 3 \tan {\left (c + d x \right )} \sqrt {\cot {\left (c + d x \right )}} + i \sqrt {\cot {\left (c + d x \right )}}}\, dx + \int \frac {B \tan {\left (c + d x \right )}}{\tan ^{3}{\left (c + d x \right )} \sqrt {\cot {\left (c + d x \right )}} - 3 i \tan ^{2}{\left (c + d x \right )} \sqrt {\cot {\left (c + d x \right )}} - 3 \tan {\left (c + d x \right )} \sqrt {\cot {\left (c + d x \right )}} + i \sqrt {\cot {\left (c + d x \right )}}}\, dx\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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