Optimal. Leaf size=146 \[ -\frac {10 i a^4 \sqrt {e \sec (c+d x)}}{d e^2}-\frac {10 a^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{d e^2}-\frac {4 i a (a+i a \tan (c+d x))^3}{3 d (e \sec (c+d x))^{3/2}}-\frac {2 i \sqrt {e \sec (c+d x)} \left (a^4+i a^4 \tan (c+d x)\right )}{d e^2} \]
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Rubi [A]
time = 0.12, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3577, 3579,
3567, 3856, 2720} \begin {gather*} -\frac {10 i a^4 \sqrt {e \sec (c+d x)}}{d e^2}-\frac {2 i \left (a^4+i a^4 \tan (c+d x)\right ) \sqrt {e \sec (c+d x)}}{d e^2}-\frac {10 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)}}{d e^2}-\frac {4 i a (a+i a \tan (c+d x))^3}{3 d (e \sec (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 3567
Rule 3577
Rule 3579
Rule 3856
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{3/2}} \, dx &=-\frac {4 i a (a+i a \tan (c+d x))^3}{3 d (e \sec (c+d x))^{3/2}}-\frac {\left (3 a^2\right ) \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^2 \, dx}{e^2}\\ &=-\frac {4 i a (a+i a \tan (c+d x))^3}{3 d (e \sec (c+d x))^{3/2}}-\frac {2 i \sqrt {e \sec (c+d x)} \left (a^4+i a^4 \tan (c+d x)\right )}{d e^2}-\frac {\left (5 a^3\right ) \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x)) \, dx}{e^2}\\ &=-\frac {10 i a^4 \sqrt {e \sec (c+d x)}}{d e^2}-\frac {4 i a (a+i a \tan (c+d x))^3}{3 d (e \sec (c+d x))^{3/2}}-\frac {2 i \sqrt {e \sec (c+d x)} \left (a^4+i a^4 \tan (c+d x)\right )}{d e^2}-\frac {\left (5 a^4\right ) \int \sqrt {e \sec (c+d x)} \, dx}{e^2}\\ &=-\frac {10 i a^4 \sqrt {e \sec (c+d x)}}{d e^2}-\frac {4 i a (a+i a \tan (c+d x))^3}{3 d (e \sec (c+d x))^{3/2}}-\frac {2 i \sqrt {e \sec (c+d x)} \left (a^4+i a^4 \tan (c+d x)\right )}{d e^2}-\frac {\left (5 a^4 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{e^2}\\ &=-\frac {10 i a^4 \sqrt {e \sec (c+d x)}}{d e^2}-\frac {10 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)}}{d e^2}-\frac {4 i a (a+i a \tan (c+d x))^3}{3 d (e \sec (c+d x))^{3/2}}-\frac {2 i \sqrt {e \sec (c+d x)} \left (a^4+i a^4 \tan (c+d x)\right )}{d e^2}\\ \end {align*}
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Mathematica [A]
time = 0.96, size = 130, normalized size = 0.89 \begin {gather*} \frac {a^4 \sec ^3(c+d x) \left (21+19 \cos (2 (c+d x))-30 i \cos ^{\frac {3}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (\cos (c+d x)-i \sin (c+d x))-11 i \sin (2 (c+d x))\right ) (-i \cos (c+5 d x)+\sin (c+5 d x))}{3 d (e \sec (c+d x))^{3/2} (\cos (d x)+i \sin (d x))^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.55, size = 198, normalized size = 1.36
method | result | size |
default | \(\frac {2 a^{4} \left (-15 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \left (\cos ^{2}\left (d x +c \right )\right )-15 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )-8 i \left (\cos ^{3}\left (d x +c \right )\right )+8 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-12 i \cos \left (d x +c \right )+\sin \left (d x +c \right )\right )}{3 d \cos \left (d x +c \right )^{3} \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}}}\) | \(198\) |
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 = 0.09, size = 122, normalized size = 0.84 \begin {gather*} -\frac {2 \, {\left (\frac {\sqrt {2} {\left (4 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 21 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 15 i \, a^{4}\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}} + 15 \, {\left (-i \, \sqrt {2} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, \sqrt {2} a^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )}}{3 \, {\left (d e^{\frac {3}{2}} + d e^{\left (2 i \, d x + 2 i \, c + \frac {3}{2}\right )}\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^{4} \left (\int \frac {1}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \left (- \frac {6 \tan ^{2}{\left (c + d x \right )}}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}}\right )\, dx + \int \frac {\tan ^{4}{\left (c + d x \right )}}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {4 i \tan {\left (c + d x \right )}}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \left (- \frac {4 i \tan ^{3}{\left (c + d x \right )}}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}}\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 \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 )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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