Optimal. Leaf size=104 \[ -\frac {64 i a^3 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}+\frac {16 i a^2 \cos (c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^{5/2}}{3 d} \]
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Rubi [A] time = 0.15, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3494, 3493} \[ -\frac {64 i a^3 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}+\frac {16 i a^2 \cos (c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^{5/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 3493
Rule 3494
Rubi steps
\begin {align*} \int \cos (c+d x) (a+i a \tan (c+d x))^{7/2} \, dx &=\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^{5/2}}{3 d}+\frac {1}{3} (8 a) \int \cos (c+d x) (a+i a \tan (c+d x))^{5/2} \, dx\\ &=\frac {16 i a^2 \cos (c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^{5/2}}{3 d}+\frac {1}{3} \left (32 a^2\right ) \int \cos (c+d x) (a+i a \tan (c+d x))^{3/2} \, dx\\ &=-\frac {64 i a^3 \cos (c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}+\frac {16 i a^2 \cos (c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^{5/2}}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 59, normalized size = 0.57 \[ -\frac {2 i a^3 \sec (c+d x) \sqrt {a+i a \tan (c+d x)} (-5 i \sin (2 (c+d x))+11 \cos (2 (c+d x))+12)}{3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 71, normalized size = 0.68 \[ \frac {\sqrt {2} {\left (-12 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 48 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - 32 i \, a^{3}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{3 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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 \[ \int {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \cos \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.10, size = 73, normalized size = 0.70 \[ -\frac {2 \left (22 i \left (\cos ^{2}\left (d x +c \right )\right )+10 \cos \left (d x +c \right ) \sin \left (d x +c \right )+i\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a^{3}}{3 d \cos \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.03, size = 418, normalized size = 4.02 \[ \frac {2 \, {\left (23 i \, a^{\frac {7}{2}} + \frac {20 \, a^{\frac {7}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {88 i \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {60 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {130 i \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {60 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {88 i \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {20 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {23 i \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\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 {7}{2}}}{d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {7}{2}} {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}^{\frac {7}{2}} {\left (-\frac {18 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {42 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {42 i \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {42 i \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {42 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {18 i \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {3 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.72, size = 102, normalized size = 0.98 \[ -\frac {2\,a^3\,\sqrt {\frac {a\,\left (\cos \left (2\,c+2\,d\,x\right )+1+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,c+2\,d\,x\right )+1}}\,\left (5\,\sin \left (c+d\,x\right )+5\,\sin \left (3\,c+3\,d\,x\right )+\cos \left (c+d\,x\right )\,35{}\mathrm {i}+\cos \left (3\,c+3\,d\,x\right )\,11{}\mathrm {i}\right )}{3\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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