Optimal. Leaf size=181 \[ \frac {2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{7 a^2 d e^4}+\frac {2 \sin (c+d x)}{7 a^2 d e^3 \sqrt {e \sec (c+d x)}}+\frac {4 i e^2}{15 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{11/2}}+\frac {2 e \sin (c+d x)}{15 a^2 d (e \sec (c+d x))^{9/2}}+\frac {6 \sin (c+d x)}{35 a^2 d e (e \sec (c+d x))^{5/2}} \]
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Rubi [A] time = 0.13, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3500, 3769, 3771, 2641} \[ \frac {2 \sin (c+d x)}{7 a^2 d e^3 \sqrt {e \sec (c+d x)}}+\frac {4 i e^2}{15 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{11/2}}+\frac {2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{7 a^2 d e^4}+\frac {2 e \sin (c+d x)}{15 a^2 d (e \sec (c+d x))^{9/2}}+\frac {6 \sin (c+d x)}{35 a^2 d e (e \sec (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3500
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))^2} \, dx &=\frac {4 i e^2}{15 d (e \sec (c+d x))^{11/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (11 e^2\right ) \int \frac {1}{(e \sec (c+d x))^{11/2}} \, dx}{15 a^2}\\ &=\frac {2 e \sin (c+d x)}{15 a^2 d (e \sec (c+d x))^{9/2}}+\frac {4 i e^2}{15 d (e \sec (c+d x))^{11/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {3 \int \frac {1}{(e \sec (c+d x))^{7/2}} \, dx}{5 a^2}\\ &=\frac {2 e \sin (c+d x)}{15 a^2 d (e \sec (c+d x))^{9/2}}+\frac {6 \sin (c+d x)}{35 a^2 d e (e \sec (c+d x))^{5/2}}+\frac {4 i e^2}{15 d (e \sec (c+d x))^{11/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {3 \int \frac {1}{(e \sec (c+d x))^{3/2}} \, dx}{7 a^2 e^2}\\ &=\frac {2 e \sin (c+d x)}{15 a^2 d (e \sec (c+d x))^{9/2}}+\frac {6 \sin (c+d x)}{35 a^2 d e (e \sec (c+d x))^{5/2}}+\frac {2 \sin (c+d x)}{7 a^2 d e^3 \sqrt {e \sec (c+d x)}}+\frac {4 i e^2}{15 d (e \sec (c+d x))^{11/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\int \sqrt {e \sec (c+d x)} \, dx}{7 a^2 e^4}\\ &=\frac {2 e \sin (c+d x)}{15 a^2 d (e \sec (c+d x))^{9/2}}+\frac {6 \sin (c+d x)}{35 a^2 d e (e \sec (c+d x))^{5/2}}+\frac {2 \sin (c+d x)}{7 a^2 d e^3 \sqrt {e \sec (c+d x)}}+\frac {4 i e^2}{15 d (e \sec (c+d x))^{11/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{7 a^2 e^4}\\ &=\frac {2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{7 a^2 d e^4}+\frac {2 e \sin (c+d x)}{15 a^2 d (e \sec (c+d x))^{9/2}}+\frac {6 \sin (c+d x)}{35 a^2 d e (e \sec (c+d x))^{5/2}}+\frac {2 \sin (c+d x)}{7 a^2 d e^3 \sqrt {e \sec (c+d x)}}+\frac {4 i e^2}{15 d (e \sec (c+d x))^{11/2} \left (a^2+i a^2 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 1.00, size = 151, normalized size = 0.83 \[ -\frac {(e \sec (c+d x))^{5/2} \left (-17 \sin (2 (c+d x))+128 \sin (4 (c+d x))+11 \sin (6 (c+d x))+228 i \cos (2 (c+d x))-72 i \cos (4 (c+d x))-4 i \cos (6 (c+d x))+480 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+296 i\right )}{1680 a^2 d e^6 (\tan (c+d x)-i)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ \frac {{\left (3360 \, a^{2} d e^{4} e^{\left (8 i \, d x + 8 i \, c\right )} {\rm integral}\left (-\frac {i \, \sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{7 \, a^{2} d e^{4}}, x\right ) + \sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-15 i \, e^{\left (12 i \, d x + 12 i \, c\right )} - 200 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 245 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 592 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 211 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 56 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}\right )} e^{\left (-8 i \, d x - 8 i \, c\right )}}{3360 \, a^{2} d e^{4}} \]
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 \[ \int \frac {1}{\left (e \sec \left (d x + c\right )\right )^{\frac {7}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.55, size = 252, normalized size = 1.39 \[ \frac {2 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {7}{2}} \left (-1+\cos \left (d x +c \right )\right )^{2} \left (1+\cos \left (d x +c \right )\right )^{2} \left (14 i \left (\cos ^{8}\left (d x +c \right )\right )+14 \sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )+7 \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )+15 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \cos \left (d x +c \right )+9 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+15 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+15 \cos \left (d x +c \right ) \sin \left (d x +c \right )\right )}{105 a^{2} d \,e^{7} \sin \left (d x +c \right )^{4}} \]
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 \[ \int \frac {1}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{7/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]
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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