Optimal. Leaf size=192 \[ \frac {7 i a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{16 \sqrt {2} d}+\frac {7 i a^2 \cos (c+d x)}{24 d \sqrt {a+i a \tan (c+d x)}}-\frac {7 i a \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{16 d}-\frac {7 i a \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{30 d}-\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d} \]
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Rubi [A]
time = 0.21, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {7 i a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{16 \sqrt {2} d}+\frac {7 i a^2 \cos (c+d x)}{24 d \sqrt {a+i a \tan (c+d x)}}-\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}-\frac {7 i a \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{30 d}-\frac {7 i a \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{16 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 ^5(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx &=-\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}+\frac {1}{10} (7 a) \int \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx\\ &=-\frac {7 i a \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{30 d}-\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}+\frac {1}{12} \left (7 a^2\right ) \int \frac {\cos (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx\\ &=\frac {7 i a^2 \cos (c+d x)}{24 d \sqrt {a+i a \tan (c+d x)}}-\frac {7 i a \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{30 d}-\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}+\frac {1}{16} (7 a) \int \cos (c+d x) \sqrt {a+i a \tan (c+d x)} \, dx\\ &=\frac {7 i a^2 \cos (c+d x)}{24 d \sqrt {a+i a \tan (c+d x)}}-\frac {7 i a \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{16 d}-\frac {7 i a \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{30 d}-\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}+\frac {1}{32} \left (7 a^2\right ) \int \frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx\\ &=\frac {7 i a^2 \cos (c+d x)}{24 d \sqrt {a+i a \tan (c+d x)}}-\frac {7 i a \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{16 d}-\frac {7 i a \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{30 d}-\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}+\frac {\left (7 i a^2\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 )}{16 d}\\ &=\frac {7 i a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{16 \sqrt {2} d}+\frac {7 i a^2 \cos (c+d x)}{24 d \sqrt {a+i a \tan (c+d x)}}-\frac {7 i a \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{16 d}-\frac {7 i a \cos ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{30 d}-\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}\\ \end {align*}
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Mathematica [A]
time = 1.63, size = 160, normalized size = 0.83 \begin {gather*} -\frac {i a e^{-3 i (c+d x)} \sqrt {\frac {a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left (-15+101 e^{2 i (c+d x)}+148 e^{4 i (c+d x)}+38 e^{6 i (c+d x)}+6 e^{8 i (c+d x)}-105 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 )}{240 \sqrt {2} 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. 913 vs. \(2 (155 ) = 310\).
time = 0.82, size = 914, normalized size = 4.76
method | result | size |
default | \(\text {Expression too large to display}\) | \(914\) |
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 = 0.40, size = 274, normalized size = 1.43 \begin {gather*} \frac {{\left (105 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{3}}{d^{2}}} d e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (\frac {7 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {-\frac {a^{3}}{d^{2}}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} + i \, a^{2}\right )} e^{\left (-i \, d x - i \, c\right )}}{8 \, d}\right ) - 105 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{3}}{d^{2}}} d e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-\frac {7 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {-\frac {a^{3}}{d^{2}}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} - i \, a^{2}\right )} e^{\left (-i \, d x - i \, c\right )}}{8 \, d}\right ) + \sqrt {2} {\left (-6 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} - 38 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} - 148 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} - 101 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 15 i \, a\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 )}}{480 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = -0.01 \begin {gather*} \int {\cos \left (c+d\,x\right )}^5\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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