Optimal. Leaf size=183 \[ \frac {78 i a^4 \sqrt {e \sec (c+d x)}}{7 d}+\frac {78 i \left (a^4+i a^4 \tan (c+d x)\right ) \sqrt {e \sec (c+d x)}}{35 d}+\frac {78 a^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{7 d}+\frac {26 i \left (a^2+i a^2 \tan (c+d x)\right )^2 \sqrt {e \sec (c+d x)}}{35 d}+\frac {2 i a (a+i a \tan (c+d x))^3 \sqrt {e \sec (c+d x)}}{7 d} \]
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Rubi [A] time = 0.20, antiderivative size = 183, 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 = {3498, 3486, 3771, 2641} \[ \frac {78 i a^4 \sqrt {e \sec (c+d x)}}{7 d}+\frac {26 i \left (a^2+i a^2 \tan (c+d x)\right )^2 \sqrt {e \sec (c+d x)}}{35 d}+\frac {78 i \left (a^4+i a^4 \tan (c+d x)\right ) \sqrt {e \sec (c+d x)}}{35 d}+\frac {78 a^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{7 d}+\frac {2 i a (a+i a \tan (c+d x))^3 \sqrt {e \sec (c+d x)}}{7 d} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3486
Rule 3498
Rule 3771
Rubi steps
\begin {align*} \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^4 \, dx &=\frac {2 i a \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3}{7 d}+\frac {1}{7} (13 a) \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3 \, dx\\ &=\frac {2 i a \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3}{7 d}+\frac {26 i \sqrt {e \sec (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )^2}{35 d}+\frac {1}{35} \left (117 a^2\right ) \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^2 \, dx\\ &=\frac {2 i a \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3}{7 d}+\frac {26 i \sqrt {e \sec (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )^2}{35 d}+\frac {78 i \sqrt {e \sec (c+d x)} \left (a^4+i a^4 \tan (c+d x)\right )}{35 d}+\frac {1}{7} \left (39 a^3\right ) \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x)) \, dx\\ &=\frac {78 i a^4 \sqrt {e \sec (c+d x)}}{7 d}+\frac {2 i a \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3}{7 d}+\frac {26 i \sqrt {e \sec (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )^2}{35 d}+\frac {78 i \sqrt {e \sec (c+d x)} \left (a^4+i a^4 \tan (c+d x)\right )}{35 d}+\frac {1}{7} \left (39 a^4\right ) \int \sqrt {e \sec (c+d x)} \, dx\\ &=\frac {78 i a^4 \sqrt {e \sec (c+d x)}}{7 d}+\frac {2 i a \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3}{7 d}+\frac {26 i \sqrt {e \sec (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )^2}{35 d}+\frac {78 i \sqrt {e \sec (c+d x)} \left (a^4+i a^4 \tan (c+d x)\right )}{35 d}+\frac {1}{7} \left (39 a^4 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {78 i a^4 \sqrt {e \sec (c+d x)}}{7 d}+\frac {78 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)}}{7 d}+\frac {2 i a \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3}{7 d}+\frac {26 i \sqrt {e \sec (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )^2}{35 d}+\frac {78 i \sqrt {e \sec (c+d x)} \left (a^4+i a^4 \tan (c+d x)\right )}{35 d}\\ \end {align*}
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Mathematica [A] time = 1.76, size = 101, normalized size = 0.55 \[ \frac {a^4 \sec ^4(c+d x) \sqrt {e \sec (c+d x)} \left (-150 \sin (2 (c+d x))-85 \sin (4 (c+d x))+1008 i \cos (2 (c+d x))+280 i \cos (4 (c+d x))+1560 \cos ^{\frac {9}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+728 i\right )}{140 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ \frac {\sqrt {2} {\left (730 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 1586 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 1326 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 390 i \, a^{4}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 35 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} {\rm integral}\left (-\frac {39 i \, \sqrt {2} a^{4} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{7 \, d}, x\right )}{35 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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 \[ \int \sqrt {e \sec \left (d x + c\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.00, size = 230, normalized size = 1.26 \[ \frac {2 a^{4} \left (1+\cos \left (d x +c \right )\right )^{2} \left (-1+\cos \left (d x +c \right )\right )^{2} \left (195 i \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+195 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \left (\cos ^{3}\left (d x +c \right )\right )+280 i \left (\cos ^{3}\left (d x +c \right )\right )-85 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-28 i \cos \left (d x +c \right )+5 \sin \left (d x +c \right )\right ) \sqrt {\frac {e}{\cos \left (d x +c \right )}}}{35 d \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )^{4}} \]
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 \[ \int \sqrt {e \sec \left (d x + c\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {\frac {e}{\cos \left (c+d\,x\right )}}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^4 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{4} \left (\int \sqrt {e \sec {\left (c + d x \right )}}\, dx + \int \left (- 6 \sqrt {e \sec {\left (c + d x \right )}} \tan ^{2}{\left (c + d x \right )}\right )\, dx + \int \sqrt {e \sec {\left (c + d x \right )}} \tan ^{4}{\left (c + d x \right )}\, dx + \int 4 i \sqrt {e \sec {\left (c + d x \right )}} \tan {\left (c + d x \right )}\, dx + \int \left (- 4 i \sqrt {e \sec {\left (c + d x \right )}} \tan ^{3}{\left (c + d x \right )}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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