Optimal. Leaf size=106 \[ \frac {10 i a^2 \sqrt {e \sec (c+d x)}}{3 d}+\frac {10 a^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{3 d}+\frac {2 i \sqrt {e \sec (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}{3 d} \]
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Rubi [A]
time = 0.07, antiderivative size = 106, 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 = {3579, 3567,
3856, 2720} \begin {gather*} \frac {10 i a^2 \sqrt {e \sec (c+d x)}}{3 d}+\frac {2 i \left (a^2+i a^2 \tan (c+d x)\right ) \sqrt {e \sec (c+d x)}}{3 d}+\frac {10 a^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 3567
Rule 3579
Rule 3856
Rubi steps
\begin {align*} \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^2 \, dx &=\frac {2 i \sqrt {e \sec (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}{3 d}+\frac {1}{3} (5 a) \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x)) \, dx\\ &=\frac {10 i a^2 \sqrt {e \sec (c+d x)}}{3 d}+\frac {2 i \sqrt {e \sec (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}{3 d}+\frac {1}{3} \left (5 a^2\right ) \int \sqrt {e \sec (c+d x)} \, dx\\ &=\frac {10 i a^2 \sqrt {e \sec (c+d x)}}{3 d}+\frac {2 i \sqrt {e \sec (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}{3 d}+\frac {1}{3} \left (5 a^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {10 i a^2 \sqrt {e \sec (c+d x)}}{3 d}+\frac {10 a^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{3 d}+\frac {2 i \sqrt {e \sec (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.56, size = 67, normalized size = 0.63 \begin {gather*} \frac {2 a^2 (e \sec (c+d x))^{3/2} \left (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 )-\sin (c+d x)\right )}{3 d e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.49, size = 201, normalized size = 1.90
method | result | size |
default | \(\frac {2 a^{2} \sqrt {\frac {e}{\cos \left (d x +c \right )}}\, \left (-1+\cos \left (d x +c \right )\right )^{2} \left (5 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \left (\cos ^{2}\left (d x +c \right )\right )+5 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+6 i \cos \left (d x +c \right )-\sin \left (d x +c \right )\right ) \left (1+\cos \left (d x +c \right )\right )^{2}}{3 d \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}}\) | \(201\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.11, size = 110, normalized size = 1.04 \begin {gather*} -\frac {2 \, {\left (\frac {\sqrt {2} {\left (-5 i \, a^{2} e^{\frac {1}{2}} - 7 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c + \frac {1}{2}\right )}\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} + 5 \, {\left (i \, \sqrt {2} a^{2} e^{\frac {1}{2}} + i \, \sqrt {2} a^{2} e^{\left (2 i \, d x + 2 i \, c + \frac {1}{2}\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )}}{3 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end {gather*}
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 \begin {gather*} - a^{2} \left (\int \left (- \sqrt {e \sec {\left (c + d x \right )}}\right )\, dx + \int \sqrt {e \sec {\left (c + d x \right )}} \tan ^{2}{\left (c + d x \right )}\, dx + \int \left (- 2 i \sqrt {e \sec {\left (c + d x \right )}} \tan {\left (c + d x \right )}\right )\, dx\right ) \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \sqrt {\frac {e}{\cos \left (c+d\,x\right )}}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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