Optimal. Leaf size=205 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {151 (a+i a \tan (c+d x))^{3/2}}{30 a^3 d}-\frac {39 \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{10 a^2 d}+\frac {78 \sqrt {a+i a \tan (c+d x)}}{5 a^2 d}-\frac {\tan ^4(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {19 i \tan ^3(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.49, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {3558, 3595, 3597, 3592, 3527, 3480, 206} \[ -\frac {39 \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{10 a^2 d}-\frac {151 (a+i a \tan (c+d x))^{3/2}}{30 a^3 d}+\frac {78 \sqrt {a+i a \tan (c+d x)}}{5 a^2 d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {\tan ^4(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {19 i \tan ^3(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3480
Rule 3527
Rule 3558
Rule 3592
Rule 3595
Rule 3597
Rubi steps
\begin {align*} \int \frac {\tan ^5(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx &=-\frac {\tan ^4(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}-\frac {\int \frac {\tan ^3(c+d x) \left (-4 a+\frac {11}{2} i a \tan (c+d x)\right )}{\sqrt {a+i a \tan (c+d x)}} \, dx}{3 a^2}\\ &=-\frac {\tan ^4(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {19 i \tan ^3(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {\int \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {57 i a^2}{2}-\frac {117}{4} a^2 \tan (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac {\tan ^4(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {19 i \tan ^3(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {39 \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{10 a^2 d}+\frac {2 \int \tan (c+d x) \sqrt {a+i a \tan (c+d x)} \left (\frac {117 a^3}{2}-\frac {453}{8} i a^3 \tan (c+d x)\right ) \, dx}{15 a^5}\\ &=-\frac {\tan ^4(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {19 i \tan ^3(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {39 \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{10 a^2 d}-\frac {151 (a+i a \tan (c+d x))^{3/2}}{30 a^3 d}+\frac {2 \int \sqrt {a+i a \tan (c+d x)} \left (\frac {453 i a^3}{8}+\frac {117}{2} a^3 \tan (c+d x)\right ) \, dx}{15 a^5}\\ &=-\frac {\tan ^4(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {19 i \tan ^3(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {78 \sqrt {a+i a \tan (c+d x)}}{5 a^2 d}-\frac {39 \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{10 a^2 d}-\frac {151 (a+i a \tan (c+d x))^{3/2}}{30 a^3 d}-\frac {i \int \sqrt {a+i a \tan (c+d x)} \, dx}{4 a^2}\\ &=-\frac {\tan ^4(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {19 i \tan ^3(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {78 \sqrt {a+i a \tan (c+d x)}}{5 a^2 d}-\frac {39 \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{10 a^2 d}-\frac {151 (a+i a \tan (c+d x))^{3/2}}{30 a^3 d}-\frac {\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 {\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 {\tan ^4(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {19 i \tan ^3(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {78 \sqrt {a+i a \tan (c+d x)}}{5 a^2 d}-\frac {39 \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{10 a^2 d}-\frac {151 (a+i a \tan (c+d x))^{3/2}}{30 a^3 d}\\ \end {align*}
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Mathematica [A] time = 1.83, size = 149, normalized size = 0.73 \[ \frac {e^{-2 i (c+d x)} \left (115 e^{2 i (c+d x)}+855 e^{4 i (c+d x)}+1105 e^{6 i (c+d x)}+466 e^{8 i (c+d x)}-15 e^{3 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{5/2} \sinh ^{-1}\left (e^{i (c+d x)}\right )-5\right )}{30 a d \left (1+e^{2 i (c+d x)}\right )^3 \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 388, normalized size = 1.89 \[ -\frac {15 \, \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (7 i \, d x + 7 i \, c\right )} + 2 \, a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} + a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \sqrt {\frac {1}{a^{3} d^{2}}} \log \left (4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{3} d^{2}}} + a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - 15 \, \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (7 i \, d x + 7 i \, c\right )} + 2 \, a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} + a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \sqrt {\frac {1}{a^{3} d^{2}}} \log \left (-4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{3} d^{2}}} - a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (466 \, e^{\left (8 i \, d x + 8 i \, c\right )} + 1105 \, e^{\left (6 i \, d x + 6 i \, c\right )} + 855 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 115 \, e^{\left (2 i \, d x + 2 i \, c\right )} - 5\right )}}{60 \, {\left (a^{2} d e^{\left (7 i \, d x + 7 i \, c\right )} + 2 \, a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} + a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )}} \]
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 \[ \int \frac {\tan \left (d x + c\right )^{5}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 131, normalized size = 0.64 \[ \frac {\frac {2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}} a +8 a^{2} \sqrt {a +i a \tan \left (d x +c \right )}+\frac {9 a^{3}}{2 \sqrt {a +i a \tan \left (d x +c \right )}}-\frac {a^{4}}{3 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {a^{\frac {5}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{4}}{d \,a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 157, normalized size = 0.77 \[ \frac {15 \, \sqrt {2} a^{\frac {9}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right ) + 48 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2} - 240 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{3} + 960 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} a^{4} + \frac {20 \, {\left (27 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{5} - 2 \, a^{6}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}}{120 \, a^{6} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.20, size = 138, normalized size = 0.67 \[ \frac {8\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{a^2\,d}-\frac {2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{a^3\,d}+\frac {2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}}{5\,a^4\,d}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,\sqrt {a}}\right )\,1{}\mathrm {i}}{4\,a^{3/2}\,d}+\frac {\frac {25\,a}{6}+\frac {a\,\mathrm {tan}\left (c+d\,x\right )\,9{}\mathrm {i}}{2}}{a\,d\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \]
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 \[ \int \frac {\tan ^{5}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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