Optimal. Leaf size=71 \[ \frac {8 i a^2 \cos ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac {2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^{5/2}}{d} \]
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Rubi [A] time = 0.12, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3494, 3493} \[ \frac {8 i a^2 \cos ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac {2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^{5/2}}{d} \]
Antiderivative was successfully verified.
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Rule 3493
Rule 3494
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx &=-\frac {2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^{5/2}}{d}-(4 a) \int \cos ^3(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx\\ &=\frac {8 i a^2 \cos ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac {2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^{5/2}}{d}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 86, normalized size = 1.21 \[ \frac {2 a^3 \cos (c+d x) \sqrt {a+i a \tan (c+d x)} (3 \sin (c+d x)+i \cos (c+d x)) (\cos (c+4 d x)+i \sin (c+4 d x))}{3 d (\cos (d x)+i \sin (d x))^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 59, normalized size = 0.83 \[ \frac {\sqrt {2} {\left (-i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 2 i \, a^{3}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{3 \, d} \]
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 \[ \int {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \cos \left (d x + c\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.13, size = 71, normalized size = 1.00 \[ \frac {2 \left (-2 i \left (\cos ^{2}\left (d x +c \right )\right )+2 \cos \left (d x +c \right ) \sin \left (d x +c \right )+3 i\right ) \cos \left (d x +c \right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a^{3}}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.72, size = 504, normalized size = 7.10 \[ -\frac {2 \, {\left (-i \, a^{\frac {7}{2}} - \frac {6 \, a^{\frac {7}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {5 i \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {24 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {10 i \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {36 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {10 i \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {24 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {5 i \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {6 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {i \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\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 {12 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {9 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {24 i \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {42 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {42 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {24 i \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {9 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {12 i \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {3 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.85, size = 85, normalized size = 1.20 \[ \frac {a^3\,\sqrt {\frac {a\,\left (\cos \left (2\,c+2\,d\,x\right )+1+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,c+2\,d\,x\right )+1}}\,\left (\sin \left (c+d\,x\right )+\sin \left (3\,c+3\,d\,x\right )+\cos \left (c+d\,x\right )\,3{}\mathrm {i}-\cos \left (3\,c+3\,d\,x\right )\,1{}\mathrm {i}\right )}{3\,d} \]
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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