Optimal. Leaf size=35 \[ \frac {2 i a \sec ^9(c+d x)}{9 d (a+i a \tan (c+d x))^{9/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {3493} \[ \frac {2 i a \sec ^9(c+d x)}{9 d (a+i a \tan (c+d x))^{9/2}} \]
Antiderivative was successfully verified.
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Rule 3493
Rubi steps
\begin {align*} \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx &=\frac {2 i a \sec ^9(c+d x)}{9 d (a+i a \tan (c+d x))^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 59, normalized size = 1.69 \[ \frac {2 i (\tan (c+d x)+i) \sec ^7(c+d x)}{9 a^3 d (\tan (c+d x)-i)^3 \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.77, size = 89, normalized size = 2.54 \[ \frac {32 i \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{9 \, {\left (a^{4} d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a^{4} d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{4} d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a^{4} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4} d\right )}} \]
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 \[ \int \frac {\sec \left (d x + c\right )^{9}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.20, size = 115, normalized size = 3.29 \[ \frac {2 \left (16 i \left (\cos ^{5}\left (d x +c \right )\right )+16 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )-20 i \left (\cos ^{3}\left (d x +c \right )\right )-12 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+5 i \cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{9 d \cos \left (d x +c \right )^{4} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.02, size = 626, normalized size = 17.89 \[ -\frac {2 \, {\left (-i \, \sqrt {a} - \frac {2 \, \sqrt {a} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {6 i \, \sqrt {a} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {14 \, \sqrt {a} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {14 i \, \sqrt {a} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {42 \, \sqrt {a} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {14 i \, \sqrt {a} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {70 \, \sqrt {a} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {70 \, \sqrt {a} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {14 i \, \sqrt {a} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {42 \, \sqrt {a} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + \frac {14 i \, \sqrt {a} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac {14 \, \sqrt {a} \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} + \frac {6 i \, \sqrt {a} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} - \frac {2 \, \sqrt {a} \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}} + \frac {i \, \sqrt {a} \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}}\right )} {\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}}}{9 \, {\left (a^{4} - \frac {8 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {28 \, a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {56 \, a^{4} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {70 \, a^{4} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {56 \, a^{4} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {28 \, a^{4} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac {8 \, a^{4} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {a^{4} \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}}\right )} d {\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}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.49, size = 50, normalized size = 1.43 \[ \frac {{\mathrm {e}}^{-c\,5{}\mathrm {i}-d\,x\,5{}\mathrm {i}}\,\sqrt {a+\frac {a\,\sin \left (c+d\,x\right )\,1{}\mathrm {i}}{\cos \left (c+d\,x\right )}}\,2{}\mathrm {i}}{9\,a^4\,d\,{\cos \left (c+d\,x\right )}^4} \]
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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