Optimal. Leaf size=80 \[ \frac {4 i (e \sec (c+d x))^{3/2}}{21 a d (a+i a \tan (c+d x))^{3/2}}+\frac {2 i (e \sec (c+d x))^{3/2}}{7 d (a+i a \tan (c+d x))^{5/2}} \]
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Rubi [A] time = 0.17, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3502, 3488} \[ \frac {4 i (e \sec (c+d x))^{3/2}}{21 a d (a+i a \tan (c+d x))^{3/2}}+\frac {2 i (e \sec (c+d x))^{3/2}}{7 d (a+i a \tan (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3488
Rule 3502
Rubi steps
\begin {align*} \int \frac {(e \sec (c+d x))^{3/2}}{(a+i a \tan (c+d x))^{5/2}} \, dx &=\frac {2 i (e \sec (c+d x))^{3/2}}{7 d (a+i a \tan (c+d x))^{5/2}}+\frac {2 \int \frac {(e \sec (c+d x))^{3/2}}{(a+i a \tan (c+d x))^{3/2}} \, dx}{7 a}\\ &=\frac {2 i (e \sec (c+d x))^{3/2}}{7 d (a+i a \tan (c+d x))^{5/2}}+\frac {4 i (e \sec (c+d x))^{3/2}}{21 a d (a+i a \tan (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 63, normalized size = 0.79 \[ \frac {2 (2 \tan (c+d x)-5 i) (e \sec (c+d x))^{3/2}}{21 a^2 d (\tan (c+d x)-i)^2 \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 79, normalized size = 0.99 \[ \frac {{\left (7 i \, e e^{\left (4 i \, d x + 4 i \, c\right )} + 10 i \, e e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i \, e\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {7}{2} i \, d x - \frac {7}{2} i \, c\right )}}{21 \, a^{3} d} \]
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 \[ \int \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {3}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.11, size = 112, normalized size = 1.40 \[ -\frac {2 i \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right ) \left (12 i \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )-12 \left (\cos ^{4}\left (d x +c \right )\right )+i \cos \left (d x +c \right ) \sin \left (d x +c \right )+5 \left (\cos ^{2}\left (d x +c \right )\right )+2\right )}{21 d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 86, normalized size = 1.08 \[ \frac {{\left (3 i \, e \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 7 i \, e \cos \left (\frac {3}{7} \, \arctan \left (\sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ), \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )\right )\right ) + 3 \, e \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 7 \, e \sin \left (\frac {3}{7} \, \arctan \left (\sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ), \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )\right )\right )\right )} \sqrt {e}}{21 \, a^{\frac {5}{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.23, size = 102, normalized size = 1.28 \[ \frac {e\,\sqrt {\frac {e}{\cos \left (c+d\,x\right )}}\,\left (7\,\sin \left (c+d\,x\right )+3\,\sin \left (3\,c+3\,d\,x\right )+\cos \left (c+d\,x\right )\,7{}\mathrm {i}+\cos \left (3\,c+3\,d\,x\right )\,3{}\mathrm {i}\right )}{21\,a^2\,d\,\sqrt {\frac {a\,\left (\cos \left (2\,c+2\,d\,x\right )+1+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,c+2\,d\,x\right )+1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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