Optimal. Leaf size=116 \[ \frac {2 a^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{7 d e^4}+\frac {2 a^2 \sin (c+d x)}{7 d e^3 \sqrt {e \sec (c+d x)}}-\frac {4 i \left (a^2+i a^2 \tan (c+d x)\right )}{7 d (e \sec (c+d x))^{7/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 116, 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 = {3577, 3854,
3856, 2720} \begin {gather*} \frac {2 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)}}{7 d e^4}+\frac {2 a^2 \sin (c+d x)}{7 d e^3 \sqrt {e \sec (c+d x)}}-\frac {4 i \left (a^2+i a^2 \tan (c+d x)\right )}{7 d (e \sec (c+d x))^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 3577
Rule 3854
Rule 3856
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (c+d x))^2}{(e \sec (c+d x))^{7/2}} \, dx &=-\frac {4 i \left (a^2+i a^2 \tan (c+d x)\right )}{7 d (e \sec (c+d x))^{7/2}}+\frac {\left (3 a^2\right ) \int \frac {1}{(e \sec (c+d x))^{3/2}} \, dx}{7 e^2}\\ &=\frac {2 a^2 \sin (c+d x)}{7 d e^3 \sqrt {e \sec (c+d x)}}-\frac {4 i \left (a^2+i a^2 \tan (c+d x)\right )}{7 d (e \sec (c+d x))^{7/2}}+\frac {a^2 \int \sqrt {e \sec (c+d x)} \, dx}{7 e^4}\\ &=\frac {2 a^2 \sin (c+d x)}{7 d e^3 \sqrt {e \sec (c+d x)}}-\frac {4 i \left (a^2+i a^2 \tan (c+d x)\right )}{7 d (e \sec (c+d x))^{7/2}}+\frac {\left (a^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{7 e^4}\\ &=\frac {2 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)}}{7 d e^4}+\frac {2 a^2 \sin (c+d x)}{7 d e^3 \sqrt {e \sec (c+d x)}}-\frac {4 i \left (a^2+i a^2 \tan (c+d x)\right )}{7 d (e \sec (c+d x))^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.81, size = 133, normalized size = 1.15 \begin {gather*} \frac {a^2 \sqrt {e \sec (c+d x)} \left (-2 i-2 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+2 d x))+i \sin (2 (c+2 d x)))}{7 d e^4 (\cos (d x)+i \sin (d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.90, size = 188, normalized size = 1.62
method | result | size |
default | \(\frac {2 a^{2} \left (i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )-2 i \left (\cos ^{4}\left (d x +c \right )\right )+i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+2 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )+\sin \left (d x +c \right ) \cos \left (d x +c \right )\right )}{7 d \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {7}{2}} \cos \left (d x +c \right )^{4}}\) | \(188\) |
risch | \(-\frac {i {\mathrm e}^{i \left (d x +c \right )} \left ({\mathrm e}^{2 i \left (d x +c \right )}+3\right ) a^{2} \sqrt {2}}{14 d \,e^{3} \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}+\frac {2 \sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (d x +c \right )}}\, \EllipticF \left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right ) a^{2} \sqrt {e \,{\mathrm e}^{i \left (d x +c \right )} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}}{7 d \sqrt {e \,{\mathrm e}^{3 i \left (d x +c \right )}+e \,{\mathrm e}^{i \left (d x +c \right )}}\, e^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}\) | \(245\) |
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 = 0.76 \begin {gather*} \frac {{\left (-4 i \, \sqrt {2} a^{2} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right ) + \frac {\sqrt {2} {\left (-i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 4 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 3 i \, a^{2}\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 )}}{14 \, 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*} - a^{2} \left (\int \left (- \frac {1}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {7}{2}}}\right )\, dx + \int \frac {\tan ^{2}{\left (c + d x \right )}}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx + \int \left (- \frac {2 i \tan {\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 )}^2}{{\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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