Optimal. Leaf size=163 \[ -\frac {42 e^6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^4 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {42 e^5 \sin (c+d x) \sqrt {e \sec (c+d x)}}{5 a^4 d}-\frac {28 i e^4 (e \sec (c+d x))^{3/2}}{5 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{5 a d (a+i a \tan (c+d x))^3} \]
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Rubi [A] time = 0.15, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3500, 3768, 3771, 2639} \[ \frac {42 e^5 \sin (c+d x) \sqrt {e \sec (c+d x)}}{5 a^4 d}-\frac {28 i e^4 (e \sec (c+d x))^{3/2}}{5 d \left (a^4+i a^4 \tan (c+d x)\right )}-\frac {42 e^6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^4 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{5 a d (a+i a \tan (c+d x))^3} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3500
Rule 3768
Rule 3771
Rubi steps
\begin {align*} \int \frac {(e \sec (c+d x))^{11/2}}{(a+i a \tan (c+d x))^4} \, dx &=\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{5 a d (a+i a \tan (c+d x))^3}-\frac {\left (7 e^2\right ) \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx}{5 a^2}\\ &=\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{5 a d (a+i a \tan (c+d x))^3}-\frac {28 i e^4 (e \sec (c+d x))^{3/2}}{5 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {\left (21 e^4\right ) \int (e \sec (c+d x))^{3/2} \, dx}{5 a^4}\\ &=\frac {42 e^5 \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 a^4 d}+\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{5 a d (a+i a \tan (c+d x))^3}-\frac {28 i e^4 (e \sec (c+d x))^{3/2}}{5 d \left (a^4+i a^4 \tan (c+d x)\right )}-\frac {\left (21 e^6\right ) \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{5 a^4}\\ &=\frac {42 e^5 \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 a^4 d}+\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{5 a d (a+i a \tan (c+d x))^3}-\frac {28 i e^4 (e \sec (c+d x))^{3/2}}{5 d \left (a^4+i a^4 \tan (c+d x)\right )}-\frac {\left (21 e^6\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 a^4 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}\\ &=-\frac {42 e^6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^4 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {42 e^5 \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 a^4 d}+\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{5 a d (a+i a \tan (c+d x))^3}-\frac {28 i e^4 (e \sec (c+d x))^{3/2}}{5 d \left (a^4+i a^4 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [C] time = 0.63, size = 106, normalized size = 0.65 \[ -\frac {2 i e^5 e^{-3 i (c+d x)} \left (21 e^{2 i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-e^{2 i (c+d x)}\right )-7 e^{2 i (c+d x)}-2\right ) \sqrt {e \sec (c+d x)}}{5 a^4 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.11, size = 0, normalized size = 0.00 \[ \frac {{\left (5 \, a^{4} d e^{\left (3 i \, d x + 3 i \, c\right )} {\rm integral}\left (\frac {21 i \, \sqrt {2} e^{5} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{5 \, a^{4} d}, x\right ) + \sqrt {2} {\left (-42 i \, e^{5} e^{\left (4 i \, d x + 4 i \, c\right )} - 28 i \, e^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + 4 i \, e^{5}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{5 \, a^{4} d} \]
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 \[ \int \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {11}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.44, size = 379, normalized size = 2.33 \[ -\frac {2 \left (21 i \cos \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )-21 i \cos \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )-8 i \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )+21 i \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )-21 i \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+8 \left (\cos ^{4}\left (d x +c \right )\right )+20 i \cos \left (d x +c \right ) \sin \left (d x +c \right )-24 \left (\cos ^{2}\left (d x +c \right )\right )+21 \cos \left (d x +c \right )-5\right ) \left (\cos ^{5}\left (d x +c \right )\right ) \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {11}{2}} \left (1+\cos \left (d x +c \right )\right )^{2} \left (-1+\cos \left (d x +c \right )\right )^{2}}{5 a^{4} d \sin \left (d x +c \right )^{5}} \]
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 \[ \int \frac {{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{11/2}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^4} \,d x \]
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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