Optimal. Leaf size=231 \[ \frac {9 i a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{32 \sqrt {2} d}+\frac {3 i a^3 \cos (c+d x)}{16 d \sqrt {a+i a \tan (c+d x)}}-\frac {9 i a^2 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{32 d}-\frac {3 i a^2 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{20 d}-\frac {9 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{70 d}-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{7 d} \]
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Rubi [A]
time = 0.25, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3578, 3583,
3571, 3570, 212} \begin {gather*} \frac {9 i a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{32 \sqrt {2} d}+\frac {3 i a^3 \cos (c+d x)}{16 d \sqrt {a+i a \tan (c+d x)}}-\frac {3 i a^2 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{20 d}-\frac {9 i a^2 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{32 d}-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{7 d}-\frac {9 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{70 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 3570
Rule 3571
Rule 3578
Rule 3583
Rubi steps
\begin {align*} \int \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx &=-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{7 d}+\frac {1}{14} (9 a) \int \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx\\ &=-\frac {9 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{70 d}-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{7 d}+\frac {1}{20} \left (9 a^2\right ) \int \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx\\ &=-\frac {3 i a^2 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{20 d}-\frac {9 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{70 d}-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{7 d}+\frac {1}{8} \left (3 a^3\right ) \int \frac {\cos (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx\\ &=\frac {3 i a^3 \cos (c+d x)}{16 d \sqrt {a+i a \tan (c+d x)}}-\frac {3 i a^2 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{20 d}-\frac {9 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{70 d}-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{7 d}+\frac {1}{32} \left (9 a^2\right ) \int \cos (c+d x) \sqrt {a+i a \tan (c+d x)} \, dx\\ &=\frac {3 i a^3 \cos (c+d x)}{16 d \sqrt {a+i a \tan (c+d x)}}-\frac {9 i a^2 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{32 d}-\frac {3 i a^2 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{20 d}-\frac {9 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{70 d}-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{7 d}+\frac {1}{64} \left (9 a^3\right ) \int \frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx\\ &=\frac {3 i a^3 \cos (c+d x)}{16 d \sqrt {a+i a \tan (c+d x)}}-\frac {9 i a^2 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{32 d}-\frac {3 i a^2 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{20 d}-\frac {9 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{70 d}-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{7 d}+\frac {\left (9 i a^3\right ) \text {Subst}\left (\int \frac {1}{2-a x^2} \, dx,x,\frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}}\right )}{32 d}\\ &=\frac {9 i a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{32 \sqrt {2} d}+\frac {3 i a^3 \cos (c+d x)}{16 d \sqrt {a+i a \tan (c+d x)}}-\frac {9 i a^2 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{32 d}-\frac {3 i a^2 \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{20 d}-\frac {9 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{70 d}-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^{5/2}}{7 d}\\ \end {align*}
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Mathematica [A]
time = 1.28, size = 155, normalized size = 0.67 \begin {gather*} -\frac {i a^2 e^{-3 i (c+d x)} \left (-35+353 e^{2 i (c+d x)}+544 e^{4 i (c+d x)}+214 e^{6 i (c+d x)}+68 e^{8 i (c+d x)}+10 e^{10 i (c+d x)}-315 e^{2 i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \tanh ^{-1}\left (\sqrt {1+e^{2 i (c+d x)}}\right )\right ) \sqrt {a+i a \tan (c+d x)}}{2240 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1259 vs. \(2 (188 ) = 376\).
time = 1.17, size = 1260, normalized size = 5.45
method | result | size |
default | \(\text {Expression too large to display}\) | \(1260\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 300, normalized size = 1.30 \begin {gather*} -\frac {{\left (315 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{5}}{d^{2}}} d e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-\frac {9 \, {\left (-i \, a^{3} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{5}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + 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 )}}{16 \, d}\right ) - 315 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{5}}{d^{2}}} d e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-\frac {9 \, {\left (-i \, a^{3} - \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{5}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + 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 )}}{16 \, d}\right ) - \sqrt {2} {\left (-10 i \, a^{2} e^{\left (10 i \, d x + 10 i \, c\right )} - 68 i \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} - 214 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} - 544 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 353 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 35 i \, a^{2}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{2240 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\cos \left (c+d\,x\right )}^7\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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