Optimal. Leaf size=130 \[ \frac {4 i \sqrt {2} a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {4 i a^2 \sqrt {a+i a \tan (c+d x)}}{d}-\frac {2 i a (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac {2 i (a+i a \tan (c+d x))^{7/2}}{7 a d} \]
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Rubi [A]
time = 0.08, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3624, 3559,
3561, 212} \begin {gather*} \frac {4 i \sqrt {2} a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {4 i a^2 \sqrt {a+i a \tan (c+d x)}}{d}-\frac {2 i (a+i a \tan (c+d x))^{7/2}}{7 a d}-\frac {2 i a (a+i a \tan (c+d x))^{3/2}}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 3559
Rule 3561
Rule 3624
Rubi steps
\begin {align*} \int \tan ^2(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx &=-\frac {2 i (a+i a \tan (c+d x))^{7/2}}{7 a d}-\int (a+i a \tan (c+d x))^{5/2} \, dx\\ &=-\frac {2 i a (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac {2 i (a+i a \tan (c+d x))^{7/2}}{7 a d}-(2 a) \int (a+i a \tan (c+d x))^{3/2} \, dx\\ &=-\frac {4 i a^2 \sqrt {a+i a \tan (c+d x)}}{d}-\frac {2 i a (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac {2 i (a+i a \tan (c+d x))^{7/2}}{7 a d}-\left (4 a^2\right ) \int \sqrt {a+i a \tan (c+d x)} \, dx\\ &=-\frac {4 i a^2 \sqrt {a+i a \tan (c+d x)}}{d}-\frac {2 i a (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac {2 i (a+i a \tan (c+d x))^{7/2}}{7 a d}+\frac {\left (8 i a^3\right ) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{d}\\ &=\frac {4 i \sqrt {2} a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {4 i a^2 \sqrt {a+i a \tan (c+d x)}}{d}-\frac {2 i a (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac {2 i (a+i a \tan (c+d x))^{7/2}}{7 a d}\\ \end {align*}
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Mathematica [A]
time = 2.82, size = 170, normalized size = 1.31 \begin {gather*} \frac {a^2 e^{-i (c+2 d x)} \sqrt {\frac {a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} (-i \cos (d x)+\sin (d x)) \left (-336 \sinh ^{-1}\left (e^{i (c+d x)}\right )+\sqrt {1+e^{2 i (c+d x)}} \sec ^3(c+d x) (86+122 \cos (2 (c+d x))+19 i \sec (c+d x) \sin (3 (c+d x))+7 i \tan (c+d x))\right )}{42 \sqrt {2} d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 96, normalized size = 0.74
method | result | size |
derivativedivides | \(-\frac {2 i \left (\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7}+\frac {a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 a^{3} \sqrt {a +i a \tan \left (d x +c \right )}-2 a^{\frac {7}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )\right )}{d a}\) | \(96\) |
default | \(-\frac {2 i \left (\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7}+\frac {a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 a^{3} \sqrt {a +i a \tan \left (d x +c \right )}-2 a^{\frac {7}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )\right )}{d a}\) | \(96\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 120, normalized size = 0.92 \begin {gather*} -\frac {2 i \, {\left (21 \, \sqrt {2} a^{\frac {11}{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 ) + 3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{2} + 7 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{4} + 42 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} a^{5}\right )}}{21 \, a^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 371 vs. \(2 (95) = 190\).
time = 0.49, size = 371, normalized size = 2.85 \begin {gather*} -\frac {2 \, {\left (21 \, \sqrt {2} \sqrt {-\frac {a^{5}}{d^{2}}} {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {4 \, {\left (a^{3} e^{\left (i \, d x + i \, c\right )} + \sqrt {-\frac {a^{5}}{d^{2}}} {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{a^{2}}\right ) - 21 \, \sqrt {2} \sqrt {-\frac {a^{5}}{d^{2}}} {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {4 \, {\left (a^{3} e^{\left (i \, d x + i \, c\right )} + \sqrt {-\frac {a^{5}}{d^{2}}} {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{a^{2}}\right ) + 2 \, \sqrt {2} {\left (40 i \, a^{2} e^{\left (7 i \, d x + 7 i \, c\right )} + 77 i \, a^{2} e^{\left (5 i \, d x + 5 i \, c\right )} + 70 i \, a^{2} e^{\left (3 i \, d x + 3 i \, c\right )} + 21 i \, a^{2} e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )}}{21 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end {gather*}
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 \begin {gather*} \int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {5}{2}} \tan ^{2}{\left (c + d x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.21, size = 107, normalized size = 0.82 \begin {gather*} -\frac {a^2\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,4{}\mathrm {i}}{d}-\frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/2}\,2{}\mathrm {i}}{7\,a\,d}-\frac {a\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,2{}\mathrm {i}}{3\,d}+\frac {\sqrt {2}\,{\left (-a\right )}^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {-a}}\right )\,4{}\mathrm {i}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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