Optimal. Leaf size=156 \[ \frac {(A-i B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {A+i B}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {3 A+i B}{2 a d \sqrt {a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.51, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {3596, 3600, 3480, 206, 3599, 63, 208} \[ \frac {(A-i B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {A+i B}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {3 A+i B}{2 a d \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 208
Rule 3480
Rule 3596
Rule 3599
Rule 3600
Rubi steps
\begin {align*} \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx &=\frac {A+i B}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {\int \frac {\cot (c+d x) \left (3 a A-\frac {3}{2} a (i A-B) \tan (c+d x)\right )}{\sqrt {a+i a \tan (c+d x)}} \, dx}{3 a^2}\\ &=\frac {A+i B}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {3 A+i B}{2 a d \sqrt {a+i a \tan (c+d x)}}+\frac {\int \cot (c+d x) \sqrt {a+i a \tan (c+d x)} \left (3 a^2 A-\frac {3}{4} a^2 (3 i A-B) \tan (c+d x)\right ) \, dx}{3 a^4}\\ &=\frac {A+i B}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {3 A+i B}{2 a d \sqrt {a+i a \tan (c+d x)}}+\frac {A \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)} \, dx}{a^3}+\frac {(i A+B) \int \sqrt {a+i a \tan (c+d x)} \, dx}{4 a^2}\\ &=\frac {A+i B}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {3 A+i B}{2 a d \sqrt {a+i a \tan (c+d x)}}+\frac {A \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+i a x}} \, dx,x,\tan (c+d x)\right )}{a d}+\frac {(A-i B) \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{2 a d}\\ &=\frac {(A-i B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {A+i B}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {3 A+i B}{2 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(2 i A) \operatorname {Subst}\left (\int \frac {1}{i-\frac {i x^2}{a}} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{a^2 d}\\ &=-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {(A-i B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {A+i B}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {3 A+i B}{2 a d \sqrt {a+i a \tan (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 4.37, size = 192, normalized size = 1.23 \[ \frac {-\frac {12 i (A-i B) e^{3 i (c+d x)} \sinh ^{-1}\left (e^{i (c+d x)}\right )}{\left (1+e^{2 i (c+d x)}\right )^{3/2}}+18 A \tan (c+d x)+\frac {48 i \sqrt {2} A e^{3 i (c+d x)} \tanh ^{-1}\left (\frac {\sqrt {2} e^{i (c+d x)}}{\sqrt {1+e^{2 i (c+d x)}}}\right )}{\left (1+e^{2 i (c+d x)}\right )^{3/2}}-22 i A+6 i B \tan (c+d x)+10 B}{12 a d (\tan (c+d x)-i) \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 621, normalized size = 3.98 \[ \frac {{\left (3 \, \sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {A^{2} - 2 i \, A B - 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 {A^{2} - 2 i \, A B - B^{2}}{a^{3} d^{2}}} + {\left (4 i \, A + 4 \, 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 {A^{2} - 2 i \, A B - 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 {A^{2} - 2 i \, A B - B^{2}}{a^{3} d^{2}}} + {\left (4 i \, A + 4 \, B\right )} a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{i \, A + B}\right ) - 6 \, a^{2} d \sqrt {\frac {A^{2}}{a^{3} d^{2}}} e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (\frac {16 \, {\left (3 \, A a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + A a^{2} + 2 \, \sqrt {2} {\left (a^{3} d e^{\left (3 i \, d x + 3 i \, c\right )} + a^{3} d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {A^{2}}{a^{3} d^{2}}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{A}\right ) + 6 \, a^{2} d \sqrt {\frac {A^{2}}{a^{3} d^{2}}} e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (\frac {16 \, {\left (3 \, A a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + A a^{2} - 2 \, \sqrt {2} {\left (a^{3} d e^{\left (3 i \, d x + 3 i \, c\right )} + a^{3} d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {A^{2}}{a^{3} d^{2}}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{A}\right ) + \sqrt {2} {\left (2 \, {\left (5 \, A + 2 i \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (11 \, A + 5 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}}\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 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \cot \left (d x + c\right )}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.40, size = 1026, normalized size = 6.58 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.72, size = 161, normalized size = 1.03 \[ -\frac {\frac {3 \, \sqrt {2} {\left (A - i \, B\right )} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{a^{\frac {3}{2}}} - \frac {24 \, A \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {4 \, {\left (3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} {\left (3 \, A + i \, B\right )} + 2 \, {\left (A + i \, B\right )} a\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.70, size = 563, normalized size = 3.61 \[ \frac {\frac {A+B\,1{}\mathrm {i}}{3\,d}+\frac {\left (3\,A+B\,1{}\mathrm {i}\right )\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}{2\,a\,d}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}-\frac {2\,A\,\mathrm {atanh}\left (\frac {31\,A^3\,d\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{\sqrt {a^3}\,\left (\frac {31\,A^3\,d}{a}+\frac {A\,B^2\,d}{a}+\frac {A^2\,B\,d\,2{}\mathrm {i}}{a}\right )}+\frac {A\,B^2\,d\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{\sqrt {a^3}\,\left (\frac {31\,A^3\,d}{a}+\frac {A\,B^2\,d}{a}+\frac {A^2\,B\,d\,2{}\mathrm {i}}{a}\right )}+\frac {A^2\,B\,d\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,2{}\mathrm {i}}{\sqrt {a^3}\,\left (\frac {31\,A^3\,d}{a}+\frac {A\,B^2\,d}{a}+\frac {A^2\,B\,d\,2{}\mathrm {i}}{a}\right )}\right )}{d\,\sqrt {a^3}}+\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,A^3\,d\,\sqrt {-a^3}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,31{}\mathrm {i}}{16\,\left (\frac {31\,d\,A^3\,a^2}{8}-\frac {29{}\mathrm {i}\,d\,A^2\,B\,a^2}{8}+\frac {3\,d\,A\,B^2\,a^2}{8}-\frac {1{}\mathrm {i}\,d\,B^3\,a^2}{8}\right )}+\frac {\sqrt {2}\,B^3\,d\,\sqrt {-a^3}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{16\,\left (\frac {31\,d\,A^3\,a^2}{8}-\frac {29{}\mathrm {i}\,d\,A^2\,B\,a^2}{8}+\frac {3\,d\,A\,B^2\,a^2}{8}-\frac {1{}\mathrm {i}\,d\,B^3\,a^2}{8}\right )}+\frac {\sqrt {2}\,A\,B^2\,d\,\sqrt {-a^3}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,3{}\mathrm {i}}{16\,\left (\frac {31\,d\,A^3\,a^2}{8}-\frac {29{}\mathrm {i}\,d\,A^2\,B\,a^2}{8}+\frac {3\,d\,A\,B^2\,a^2}{8}-\frac {1{}\mathrm {i}\,d\,B^3\,a^2}{8}\right )}+\frac {29\,\sqrt {2}\,A^2\,B\,d\,\sqrt {-a^3}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{16\,\left (\frac {31\,d\,A^3\,a^2}{8}-\frac {29{}\mathrm {i}\,d\,A^2\,B\,a^2}{8}+\frac {3\,d\,A\,B^2\,a^2}{8}-\frac {1{}\mathrm {i}\,d\,B^3\,a^2}{8}\right )}\right )\,\left (B+A\,1{}\mathrm {i}\right )\,\sqrt {-a^3}}{4\,a^3\,d} \]
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 \[ \int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \cot {\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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