Optimal. Leaf size=119 \[ \frac {10 e^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{3 a^2 d}+\frac {10 e^3 \sin (c+d x) (e \sec (c+d x))^{3/2}}{3 a^2 d}-\frac {4 i e^2 (e \sec (c+d x))^{5/2}}{d \left (a^2+i a^2 \tan (c+d x)\right )} \]
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Rubi [A] time = 0.09, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3500, 3768, 3771, 2641} \[ \frac {10 e^3 \sin (c+d x) (e \sec (c+d x))^{3/2}}{3 a^2 d}-\frac {4 i e^2 (e \sec (c+d x))^{5/2}}{d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {10 e^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{3 a^2 d} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3500
Rule 3768
Rule 3771
Rubi steps
\begin {align*} \int \frac {(e \sec (c+d x))^{9/2}}{(a+i a \tan (c+d x))^2} \, dx &=-\frac {4 i e^2 (e \sec (c+d x))^{5/2}}{d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (5 e^2\right ) \int (e \sec (c+d x))^{5/2} \, dx}{a^2}\\ &=\frac {10 e^3 (e \sec (c+d x))^{3/2} \sin (c+d x)}{3 a^2 d}-\frac {4 i e^2 (e \sec (c+d x))^{5/2}}{d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (5 e^4\right ) \int \sqrt {e \sec (c+d x)} \, dx}{3 a^2}\\ &=\frac {10 e^3 (e \sec (c+d x))^{3/2} \sin (c+d x)}{3 a^2 d}-\frac {4 i e^2 (e \sec (c+d x))^{5/2}}{d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (5 e^4 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 a^2}\\ &=\frac {10 e^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{3 a^2 d}+\frac {10 e^3 (e \sec (c+d x))^{3/2} \sin (c+d x)}{3 a^2 d}-\frac {4 i e^2 (e \sec (c+d x))^{5/2}}{d \left (a^2+i a^2 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 67, normalized size = 0.56 \[ \frac {2 e^3 (e \sec (c+d x))^{3/2} \left (-\sin (c+d x)-6 i \cos (c+d x)+5 \cos ^{\frac {3}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{3 a^2 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ \frac {\sqrt {2} {\left (-10 i \, e^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 14 i \, e^{4}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 3 \, {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} {\rm integral}\left (-\frac {5 i \, \sqrt {2} e^{4} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{3 \, a^{2} d}, x\right )}{3 \, {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )}} \]
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 \[ \int \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {9}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.18, size = 201, normalized size = 1.69 \[ \frac {2 \left (1+\cos \left (d x +c \right )\right )^{2} \left (-1+\cos \left (d x +c \right )\right )^{2} \left (5 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right ) \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+5 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \cos \left (d x +c \right )-6 i \cos \left (d x +c \right )-\sin \left (d x +c \right )\right ) \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {9}{2}} \left (\cos ^{3}\left (d x +c \right )\right )}{3 a^{2} d \sin \left (d x +c \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{9/2}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]
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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