Optimal. Leaf size=192 \[ \frac{7 i a^2 \cos (c+d x)}{24 d \sqrt{a+i a \tan (c+d x)}}+\frac{7 i a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{16 \sqrt{2} d}-\frac{i \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}-\frac{7 i a \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{30 d}-\frac{7 i a \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{16 d} \]
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Rubi [A] time = 0.270018, antiderivative size = 192, normalized size of antiderivative = 1., 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 = {3497, 3502, 3490, 3489, 206} \[ \frac{7 i a^2 \cos (c+d x)}{24 d \sqrt{a+i a \tan (c+d x)}}+\frac{7 i a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sec (c+d x)}{\sqrt{2} \sqrt{a+i a \tan (c+d x)}}\right )}{16 \sqrt{2} d}-\frac{i \cos ^5(c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}-\frac{7 i a \cos ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{30 d}-\frac{7 i a \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{16 d} \]
Antiderivative was successfully verified.
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Rule 3497
Rule 3502
Rule 3490
Rule 3489
Rule 206
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 ) \operatorname{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*}
Mathematica [A] time = 1.37671, size = 160, normalized size = 0.83 \[ -\frac{i a e^{-3 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left (101 e^{2 i (c+d x)}+148 e^{4 i (c+d x)}+38 e^{6 i (c+d x)}+6 e^{8 i (c+d x)}-105 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left (\sqrt{1+e^{2 i (c+d x)}}\right )-15\right )}{240 \sqrt{2} d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.313, size = 914, normalized size = 4.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.39696, size = 892, normalized size = 4.65 \begin{align*} -\frac{{\left (105 \, \sqrt{\frac{1}{2}} \sqrt{-\frac{a^{3}}{d^{2}}} d e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (\frac{{\left (14 i \, \sqrt{\frac{1}{2}} \sqrt{-\frac{a^{3}}{d^{2}}} d e^{\left (i \, d x + i \, c\right )} + 7 \, \sqrt{2}{\left (a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{7 \, a}\right ) - 105 \, \sqrt{\frac{1}{2}} \sqrt{-\frac{a^{3}}{d^{2}}} d e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (\frac{{\left (-14 i \, \sqrt{\frac{1}{2}} \sqrt{-\frac{a^{3}}{d^{2}}} d e^{\left (i \, d x + i \, c\right )} + 7 \, \sqrt{2}{\left (a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{7 \, a}\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}} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \cos \left (d x + c\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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