Optimal. Leaf size=124 \[ -\frac {2 a^3 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{21 d e^4}-\frac {2 i (a+i a \tan (c+d x))^3}{7 d (e \sec (c+d x))^{7/2}}-\frac {4 i \left (a^3+i a^3 \tan (c+d x)\right )}{21 d e^2 (e \sec (c+d x))^{3/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 124, 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 = {3578, 3577,
3856, 2720} \begin {gather*} -\frac {2 a^3 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{21 d e^4}-\frac {4 i \left (a^3+i a^3 \tan (c+d x)\right )}{21 d e^2 (e \sec (c+d x))^{3/2}}-\frac {2 i (a+i a \tan (c+d x))^3}{7 d (e \sec (c+d x))^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 3577
Rule 3578
Rule 3856
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{7/2}} \, dx &=-\frac {2 i (a+i a \tan (c+d x))^3}{7 d (e \sec (c+d x))^{7/2}}+\frac {a \int \frac {(a+i a \tan (c+d x))^2}{(e \sec (c+d x))^{3/2}} \, dx}{7 e^2}\\ &=-\frac {2 i (a+i a \tan (c+d x))^3}{7 d (e \sec (c+d x))^{7/2}}-\frac {4 i \left (a^3+i a^3 \tan (c+d x)\right )}{21 d e^2 (e \sec (c+d x))^{3/2}}-\frac {a^3 \int \sqrt {e \sec (c+d x)} \, dx}{21 e^4}\\ &=-\frac {2 i (a+i a \tan (c+d x))^3}{7 d (e \sec (c+d x))^{7/2}}-\frac {4 i \left (a^3+i a^3 \tan (c+d x)\right )}{21 d e^2 (e \sec (c+d x))^{3/2}}-\frac {\left (a^3 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 e^4}\\ &=-\frac {2 a^3 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{21 d e^4}-\frac {2 i (a+i a \tan (c+d x))^3}{7 d (e \sec (c+d x))^{7/2}}-\frac {4 i \left (a^3+i a^3 \tan (c+d x)\right )}{21 d e^2 (e \sec (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.82, size = 133, normalized size = 1.07 \begin {gather*} -\frac {a^3 \sqrt {e \sec (c+d x)} \left (5 i+5 i \cos (2 (c+d x))+2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (\cos (2 (c+d x))-i \sin (2 (c+d x)))-\sin (2 (c+d x))\right ) (\cos (2 c+5 d x)+i \sin (2 c+5 d x))}{21 d e^4 (\cos (d x)+i \sin (d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.50, size = 199, normalized size = 1.60
method | result | size |
default | \(-\frac {2 a^{3} \left (12 i \left (\cos ^{4}\left (d x +c \right )\right )+i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )-12 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )-7 i \left (\cos ^{2}\left (d x +c \right )\right )+\sin \left (d x +c \right ) \cos \left (d x +c \right )\right )}{21 d \cos \left (d x +c \right )^{4} \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {7}{2}}}\) | \(199\) |
risch | \(-\frac {i {\mathrm e}^{i \left (d x +c \right )} \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}+2\right ) a^{3} \sqrt {2}}{21 d \,e^{3} \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}-\frac {2 \sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (d x +c \right )}}\, \EllipticF \left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right ) a^{3} \sqrt {e \,{\mathrm e}^{i \left (d x +c \right )} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}}{21 d \sqrt {e \,{\mathrm e}^{3 i \left (d x +c \right )}+e \,{\mathrm e}^{i \left (d x +c \right )}}\, e^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}\) | \(247\) |
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 = 88, normalized size = 0.71 \begin {gather*} \frac {{\left (2 i \, \sqrt {2} a^{3} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right ) + \frac {\sqrt {2} {\left (-3 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 5 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - 2 i \, a^{3}\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}}\right )} e^{\left (-\frac {7}{2}\right )}}{21 \, d} \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*} - i a^{3} \left (\int \frac {i}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx + \int \left (- \frac {3 \tan {\left (c + d x \right )}}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {7}{2}}}\right )\, dx + \int \frac {\tan ^{3}{\left (c + d x \right )}}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx + \int \left (- \frac {3 i \tan ^{2}{\left (c + d x \right )}}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {7}{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 )}^3}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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