Optimal. Leaf size=65 \[ -\frac {8 i a^2 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}+\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^{3/2}}{d} \]
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Rubi [A]
time = 0.07, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3575, 3574}
\begin {gather*} \frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^{3/2}}{d}-\frac {8 i a^2 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3574
Rule 3575
Rubi steps
\begin {align*} \int \cos (c+d x) (a+i a \tan (c+d x))^{5/2} \, dx &=\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^{3/2}}{d}+(4 a) \int \cos (c+d x) (a+i a \tan (c+d x))^{3/2} \, dx\\ &=-\frac {8 i a^2 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}+\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^{3/2}}{d}\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 46, normalized size = 0.71 \begin {gather*} -\frac {2 i a^2 (3 \cos (c+d x)-i \sin (c+d x)) \sqrt {a+i a \tan (c+d x)}}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.90, size = 53, normalized size = 0.82
method | result | size |
default | \(-\frac {2 \left (3 i \cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a^{2}}{d}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 331 vs. \(2 (53) = 106\).
time = 0.61, size = 331, normalized size = 5.09 \begin {gather*} \frac {2 \, {\left (-3 i \, a^{\frac {5}{2}} - \frac {2 \, a^{\frac {5}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {9 i \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {4 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {9 i \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {2 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3 i \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} {\left (-\frac {2 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )}^{\frac {5}{2}}}{d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{2}} {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}^{\frac {5}{2}} {\left (\frac {4 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {5 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {5 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {4 i \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {\sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 45, normalized size = 0.69 \begin {gather*} -\frac {2 \, \sqrt {2} {\left (i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 2 i \, a^{2}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{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 [B]
time = 0.39, size = 64, normalized size = 0.98 \begin {gather*} -\frac {2\,a^2\,\left (\sin \left (c+d\,x\right )+\cos \left (c+d\,x\right )\,3{}\mathrm {i}\right )\,\sqrt {\frac {a\,\left (\cos \left (2\,c+2\,d\,x\right )+1+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,c+2\,d\,x\right )+1}}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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