Optimal. Leaf size=178 \[ \frac {154 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {154 i a^4 (e \sec (c+d x))^{3/2}}{15 d e^2}-\frac {154 a^4 \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 d e}-\frac {4 i a (a+i a \tan (c+d x))^3}{d \sqrt {e \sec (c+d x)}}-\frac {22 i (e \sec (c+d x))^{3/2} \left (a^4+i a^4 \tan (c+d x)\right )}{5 d e^2} \]
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Rubi [A]
time = 0.13, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3577, 3579,
3567, 3853, 3856, 2719} \begin {gather*} -\frac {154 i a^4 (e \sec (c+d x))^{3/2}}{15 d e^2}-\frac {22 i \left (a^4+i a^4 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}{5 d e^2}-\frac {154 a^4 \sin (c+d x) \sqrt {e \sec (c+d x)}}{5 d e}+\frac {154 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {4 i a (a+i a \tan (c+d x))^3}{d \sqrt {e \sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 3567
Rule 3577
Rule 3579
Rule 3853
Rule 3856
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (c+d x))^4}{\sqrt {e \sec (c+d x)}} \, dx &=-\frac {4 i a (a+i a \tan (c+d x))^3}{d \sqrt {e \sec (c+d x)}}-\frac {\left (11 a^2\right ) \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2 \, dx}{e^2}\\ &=-\frac {4 i a (a+i a \tan (c+d x))^3}{d \sqrt {e \sec (c+d x)}}-\frac {22 i (e \sec (c+d x))^{3/2} \left (a^4+i a^4 \tan (c+d x)\right )}{5 d e^2}-\frac {\left (77 a^3\right ) \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x)) \, dx}{5 e^2}\\ &=-\frac {154 i a^4 (e \sec (c+d x))^{3/2}}{15 d e^2}-\frac {4 i a (a+i a \tan (c+d x))^3}{d \sqrt {e \sec (c+d x)}}-\frac {22 i (e \sec (c+d x))^{3/2} \left (a^4+i a^4 \tan (c+d x)\right )}{5 d e^2}-\frac {\left (77 a^4\right ) \int (e \sec (c+d x))^{3/2} \, dx}{5 e^2}\\ &=-\frac {154 i a^4 (e \sec (c+d x))^{3/2}}{15 d e^2}-\frac {154 a^4 \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 d e}-\frac {4 i a (a+i a \tan (c+d x))^3}{d \sqrt {e \sec (c+d x)}}-\frac {22 i (e \sec (c+d x))^{3/2} \left (a^4+i a^4 \tan (c+d x)\right )}{5 d e^2}+\frac {1}{5} \left (77 a^4\right ) \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx\\ &=-\frac {154 i a^4 (e \sec (c+d x))^{3/2}}{15 d e^2}-\frac {154 a^4 \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 d e}-\frac {4 i a (a+i a \tan (c+d x))^3}{d \sqrt {e \sec (c+d x)}}-\frac {22 i (e \sec (c+d x))^{3/2} \left (a^4+i a^4 \tan (c+d x)\right )}{5 d e^2}+\frac {\left (77 a^4\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}\\ &=\frac {154 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {154 i a^4 (e \sec (c+d x))^{3/2}}{15 d e^2}-\frac {154 a^4 \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 d e}-\frac {4 i a (a+i a \tan (c+d x))^3}{d \sqrt {e \sec (c+d x)}}-\frac {22 i (e \sec (c+d x))^{3/2} \left (a^4+i a^4 \tan (c+d x)\right )}{5 d e^2}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 3.99, size = 123, normalized size = 0.69 \begin {gather*} -\frac {2 i a^4 e^{i (c+d x)} \left (-77-176 e^{2 i (c+d x)}-111 e^{4 i (c+d x)}+77 \left (1+e^{2 i (c+d x)}\right )^{5/2} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )\right ) \sqrt {e \sec (c+d x)}}{15 d e \left (1+e^{2 i (c+d x)}\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1617 vs. \(2 (179 ) = 358\).
time = 0.59, size = 1618, normalized size = 9.09
method | result | size |
default | \(\text {Expression too large to display}\) | \(1618\) |
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 = 0.10, size = 164, normalized size = 0.92 \begin {gather*} -\frac {2 \, {\left (\frac {\sqrt {2} {\left (-111 i \, a^{4} e^{\left (5 i \, d x + 5 i \, c\right )} - 176 i \, a^{4} e^{\left (3 i \, d x + 3 i \, c\right )} - 77 i \, a^{4} e^{\left (i \, d x + i \, c\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}} + 231 \, {\left (-i \, \sqrt {2} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 2 i \, \sqrt {2} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, \sqrt {2} a^{4}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{15 \, {\left (d e^{\frac {1}{2}} + d e^{\left (4 i \, d x + 4 i \, c + \frac {1}{2}\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c + \frac {1}{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}{\sqrt {e \sec {\left (c + d x \right )}}}\, dx + \int \left (- \frac {6 \tan ^{2}{\left (c + d x \right )}}{\sqrt {e \sec {\left (c + d x \right )}}}\right )\, dx + \int \frac {\tan ^{4}{\left (c + d x \right )}}{\sqrt {e \sec {\left (c + d x \right )}}}\, dx + \int \frac {4 i \tan {\left (c + d x \right )}}{\sqrt {e \sec {\left (c + d x \right )}}}\, dx + \int \left (- \frac {4 i \tan ^{3}{\left (c + d x \right )}}{\sqrt {e \sec {\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 \frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^4}{\sqrt {\frac {e}{\cos \left (c+d\,x\right )}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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