Optimal. Leaf size=156 \[ -\frac {2 a^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{77 d e^6}-\frac {2 a^4 \sin (c+d x)}{77 d e^5 \sqrt {e \sec (c+d x)}}+\frac {4 i \left (a^4+i a^4 \tan (c+d x)\right )}{77 d e^2 (e \sec (c+d x))^{7/2}}-\frac {4 i a (a+i a \tan (c+d x))^3}{11 d (e \sec (c+d x))^{11/2}} \]
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Rubi [A] time = 0.15, antiderivative size = 156, 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 = {3496, 3769, 3771, 2641} \[ -\frac {2 a^4 \sin (c+d x)}{77 d e^5 \sqrt {e \sec (c+d x)}}+\frac {4 i \left (a^4+i a^4 \tan (c+d x)\right )}{77 d e^2 (e \sec (c+d x))^{7/2}}-\frac {2 a^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{77 d e^6}-\frac {4 i a (a+i a \tan (c+d x))^3}{11 d (e \sec (c+d x))^{11/2}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3496
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{11/2}} \, dx &=-\frac {4 i a (a+i a \tan (c+d x))^3}{11 d (e \sec (c+d x))^{11/2}}-\frac {a^2 \int \frac {(a+i a \tan (c+d x))^2}{(e \sec (c+d x))^{7/2}} \, dx}{11 e^2}\\ &=-\frac {4 i a (a+i a \tan (c+d x))^3}{11 d (e \sec (c+d x))^{11/2}}+\frac {4 i \left (a^4+i a^4 \tan (c+d x)\right )}{77 d e^2 (e \sec (c+d x))^{7/2}}-\frac {\left (3 a^4\right ) \int \frac {1}{(e \sec (c+d x))^{3/2}} \, dx}{77 e^4}\\ &=-\frac {2 a^4 \sin (c+d x)}{77 d e^5 \sqrt {e \sec (c+d x)}}-\frac {4 i a (a+i a \tan (c+d x))^3}{11 d (e \sec (c+d x))^{11/2}}+\frac {4 i \left (a^4+i a^4 \tan (c+d x)\right )}{77 d e^2 (e \sec (c+d x))^{7/2}}-\frac {a^4 \int \sqrt {e \sec (c+d x)} \, dx}{77 e^6}\\ &=-\frac {2 a^4 \sin (c+d x)}{77 d e^5 \sqrt {e \sec (c+d x)}}-\frac {4 i a (a+i a \tan (c+d x))^3}{11 d (e \sec (c+d x))^{11/2}}+\frac {4 i \left (a^4+i a^4 \tan (c+d x)\right )}{77 d e^2 (e \sec (c+d x))^{7/2}}-\frac {\left (a^4 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{77 e^6}\\ &=-\frac {2 a^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{77 d e^6}-\frac {2 a^4 \sin (c+d x)}{77 d e^5 \sqrt {e \sec (c+d x)}}-\frac {4 i a (a+i a \tan (c+d x))^3}{11 d (e \sec (c+d x))^{11/2}}+\frac {4 i \left (a^4+i a^4 \tan (c+d x)\right )}{77 d e^2 (e \sec (c+d x))^{7/2}}\\ \end {align*}
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Mathematica [A] time = 1.65, size = 148, normalized size = 0.95 \[ -\frac {a^4 \sqrt {e \sec (c+d x)} (\cos (3 c+7 d x)+i \sin (3 c+7 d x)) \left (-3 \sin (c+d x)-3 \sin (3 (c+d x))+37 i \cos (c+d x)+11 i \cos (3 (c+d x))+4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (\cos (3 (c+d x))-i \sin (3 (c+d x)))\right )}{154 d e^6 (\cos (d x)+i \sin (d x))^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ \frac {154 \, d e^{6} {\rm integral}\left (\frac {i \, \sqrt {2} a^{4} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{77 \, d e^{6}}, x\right ) + \sqrt {2} {\left (-7 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 20 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 17 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 4 i \, a^{4}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{154 \, d e^{6}} \]
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 \[ \int \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4}}{\left (e \sec \left (d x + c\right )\right )^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.03, size = 215, normalized size = 1.38 \[ -\frac {2 a^{4} \left (56 i \left (\cos ^{6}\left (d x +c \right )\right )-56 \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )-44 i \left (\cos ^{4}\left (d x +c \right )\right )+i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \cos \left (d x +c \right )+i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+16 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\cos \left (d x +c \right ) \sin \left (d x +c \right )\right )}{77 d \cos \left (d x +c \right )^{6} \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4}}{\left (e \sec \left (d x + c\right )\right )^{\frac {11}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^4}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{11/2}} \,d x \]
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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