Optimal. Leaf size=367 \[ \frac {7 (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{24 d \left (a^3 \cot (c+d x)+i a^3\right )}-\frac {5 (6 A+i B) \sqrt {\cot (c+d x)}}{8 a^3 d}-\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((29+i) A+(1+6 i) B) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a^3 d}+\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((29+i) A+(1+6 i) B) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a^3 d}-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) ((1+29 i) A-(6+i) B) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^3 d}+\frac {((30+28 i) A-(7-5 i) B) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{16 \sqrt {2} a^3 d}+\frac {(A+i B) \cot ^{\frac {7}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}+\frac {(5 A+2 i B) \cot ^{\frac {5}{2}}(c+d x)}{12 a d (a \cot (c+d x)+i a)^2} \]
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Rubi [A] time = 0.92, antiderivative size = 367, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3581, 3595, 3528, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac {7 (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{24 d \left (a^3 \cot (c+d x)+i a^3\right )}-\frac {5 (6 A+i B) \sqrt {\cot (c+d x)}}{8 a^3 d}-\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((29+i) A+(1+6 i) B) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a^3 d}+\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((29+i) A+(1+6 i) B) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a^3 d}-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) ((1+29 i) A-(6+i) B) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^3 d}+\frac {((30+28 i) A-(7-5 i) B) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{16 \sqrt {2} a^3 d}+\frac {(A+i B) \cot ^{\frac {7}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}+\frac {(5 A+2 i B) \cot ^{\frac {5}{2}}(c+d x)}{12 a d (a \cot (c+d x)+i a)^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rule 3528
Rule 3534
Rule 3581
Rule 3595
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} \, dx &=\int \frac {\cot ^{\frac {7}{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 {7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {\int \frac {\cot ^{\frac {5}{2}}(c+d x) \left (-\frac {7}{2} a (i A-B)+\frac {1}{2} a (13 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 {7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {(5 A+2 i B) \cot ^{\frac {5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac {\int \frac {\cot ^{\frac {3}{2}}(c+d x) \left (-5 a^2 (5 i A-2 B)+a^2 (31 A+4 i B) \cot (c+d x)\right )}{i a+a \cot (c+d x)} \, dx}{24 a^4}\\ &=\frac {(A+i B) \cot ^{\frac {7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {(5 A+2 i B) \cot ^{\frac {5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac {7 (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{24 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac {\int \sqrt {\cot (c+d x)} \left (-21 a^3 (4 i A-B)+15 a^3 (6 A+i B) \cot (c+d x)\right ) \, dx}{48 a^6}\\ &=-\frac {5 (6 A+i B) \sqrt {\cot (c+d x)}}{8 a^3 d}+\frac {(A+i B) \cot ^{\frac {7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {(5 A+2 i B) \cot ^{\frac {5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac {7 (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{24 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac {\int \frac {-15 a^3 (6 A+i B)-21 a^3 (4 i A-B) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{48 a^6}\\ &=-\frac {5 (6 A+i B) \sqrt {\cot (c+d x)}}{8 a^3 d}+\frac {(A+i B) \cot ^{\frac {7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {(5 A+2 i B) \cot ^{\frac {5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac {7 (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{24 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {15 a^3 (6 A+i B)+21 a^3 (4 i A-B) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{24 a^6 d}\\ &=-\frac {5 (6 A+i B) \sqrt {\cot (c+d x)}}{8 a^3 d}+\frac {(A+i B) \cot ^{\frac {7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {(5 A+2 i B) \cot ^{\frac {5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac {7 (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{24 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac {((30+28 i) A-(7-5 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 {((30-28 i) A+(7+5 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 {5 (6 A+i B) \sqrt {\cot (c+d x)}}{8 a^3 d}+\frac {(A+i B) \cot ^{\frac {7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {(5 A+2 i B) \cot ^{\frac {5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac {7 (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{24 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac {((30+28 i) A-(7-5 i) B) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{32 a^3 d}+\frac {((30+28 i) A-(7-5 i) B) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{32 a^3 d}-\frac {((30-28 i) A+(7+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^3 d}-\frac {((30-28 i) A+(7+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^3 d}\\ &=-\frac {5 (6 A+i B) \sqrt {\cot (c+d x)}}{8 a^3 d}+\frac {(A+i B) \cot ^{\frac {7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {(5 A+2 i B) \cot ^{\frac {5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac {7 (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{24 d \left (i a^3+a^3 \cot (c+d x)\right )}-\frac {((30-28 i) A+(7+5 i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^3 d}+\frac {((30-28 i) A+(7+5 i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^3 d}+\frac {((30+28 i) A-(7-5 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^3 d}-\frac {((30+28 i) A-(7-5 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^3 d}\\ &=-\frac {((30+28 i) A-(7-5 i) B) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^3 d}+\frac {((30+28 i) A-(7-5 i) B) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^3 d}-\frac {5 (6 A+i B) \sqrt {\cot (c+d x)}}{8 a^3 d}+\frac {(A+i B) \cot ^{\frac {7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac {(5 A+2 i B) \cot ^{\frac {5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac {7 (4 A+i B) \cot ^{\frac {3}{2}}(c+d x)}{24 d \left (i a^3+a^3 \cot (c+d x)\right )}-\frac {((30-28 i) A+(7+5 i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^3 d}+\frac {((30-28 i) A+(7+5 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 = 4.07, size = 284, normalized size = 0.77 \[ \frac {\sec ^2(c+d x) (\cos (d x)+i \sin (d x))^3 (A+B \tan (c+d x)) \left (\frac {2}{3} \cot (c+d x) (\cos (3 d x)-i \sin (3 d x)) ((49 A+19 i B) \cos (c+d x)-(145 A+19 i B) \cos (3 (c+d x))+6 \sin (c+d x) (7 (B-7 i A) \cos (2 (c+d x))-19 i A+2 B))+(-\sin (3 c)+i \cos (3 c)) \sqrt {\sin (2 (c+d x))} \csc (c+d x) \left (((28-30 i) A+(5+7 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))-(1+i) ((29+i) A+(1+6 i) B) \log \left (\sin (c+d x)+\sqrt {\sin (2 (c+d x))}+\cos (c+d x)\right )\right )\right )}{32 d \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 (A \cos (c+d x)+B \sin (c+d x))} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 690, normalized size = 1.88 \[ \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 ) + 3 \, a^{3} d \sqrt {\frac {-841 i \, A^{2} + 348 \, A B + 36 i \, B^{2}}{a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (\frac {{\left ({\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 {-841 i \, A^{2} + 348 \, A B + 36 i \, B^{2}}{a^{6} d^{2}}} + 29 i \, A - 6 \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{3} d}\right ) - 3 \, a^{3} d \sqrt {\frac {-841 i \, A^{2} + 348 \, A B + 36 i \, B^{2}}{a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-\frac {{\left ({\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 {-841 i \, A^{2} + 348 \, A B + 36 i \, B^{2}}{a^{6} d^{2}}} - 29 i \, A + 6 \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{3} d}\right ) - 2 \, {\left (2 \, {\left (73 \, A + 10 i \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} - {\left (41 \, A + 14 i \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - {\left (8 \, A + 5 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 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \cot \left (d x + c\right )^{\frac {3}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 4.95, size = 2577, normalized size = 7.02 \[ \text {Expression too large to display} \]
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 \[ \int \frac {{\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3} \,d x \]
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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