Optimal. Leaf size=89 \[ \frac {2 i a}{3 d (e \cos (c+d x))^{3/2}}-\frac {2 a \cos ^{\frac {3}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d (e \cos (c+d x))^{3/2}}+\frac {2 a \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}} \]
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Rubi [A] time = 0.10, antiderivative size = 92, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3515, 3486, 3768, 3771, 2639} \[ \frac {2 i a}{3 d (e \cos (c+d x))^{3/2}}-\frac {2 a \cos ^{\frac {3}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d (e \cos (c+d x))^{3/2}}+\frac {2 a \sin (c+d x) \cos (c+d x)}{d (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3486
Rule 3515
Rule 3768
Rule 3771
Rubi steps
\begin {align*} \int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx &=\frac {\int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x)) \, dx}{(e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\\ &=\frac {2 i a}{3 d (e \cos (c+d x))^{3/2}}+\frac {a \int (e \sec (c+d x))^{3/2} \, dx}{(e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\\ &=\frac {2 i a}{3 d (e \cos (c+d x))^{3/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{d (e \cos (c+d x))^{3/2}}-\frac {\left (a e^2\right ) \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{(e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\\ &=\frac {2 i a}{3 d (e \cos (c+d x))^{3/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{d (e \cos (c+d x))^{3/2}}-\frac {\left (a \cos ^{\frac {3}{2}}(c+d x)\right ) \int \sqrt {\cos (c+d x)} \, dx}{(e \cos (c+d x))^{3/2}}\\ &=\frac {2 i a}{3 d (e \cos (c+d x))^{3/2}}-\frac {2 a \cos ^{\frac {3}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d (e \cos (c+d x))^{3/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{d (e \cos (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [C] time = 3.79, size = 369, normalized size = 4.15 \[ \frac {\cos ^{\frac {5}{2}}(c+d x) (\cos (d x)-i \sin (d x)) (a+i a \tan (c+d x)) \left (\frac {2 \sin (c) (\cot (c)-i) (3 \csc (c) \cos (c+2 d x)+3 \cot (c)+2 i)}{3 \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 \sqrt {2} (\cot (c)-i) e^{-i d x} \left (e^{2 i d x} \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )+3 e^{2 i (c+d x)}-3 \sqrt {1-i e^{i (c+d x)}} \sqrt {e^{i (c+d x)} \left (e^{i (c+d x)}-i\right )} F\left (\left .\sin ^{-1}\left (\sqrt {\sin (c+d x)-i \cos (c+d x)}\right )\right |-1\right )+3 \sqrt {1-i e^{i (c+d x)}} \sqrt {e^{i (c+d x)} \left (e^{i (c+d x)}-i\right )} E\left (\left .\sin ^{-1}\left (\sqrt {\sin (c+d x)-i \cos (c+d x)}\right )\right |-1\right )+3\right )}{3 \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}\right )}{2 d (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ \frac {\sqrt {\frac {1}{2}} {\left (-12 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} - 4 i \, a e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} + 3 \, {\left (d e^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{2}\right )} {\rm integral}\left (\frac {2 i \, \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} a e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{d e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{2}}, x\right )}{3 \, {\left (d e^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{2}\right )}} \]
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 \[ \int \frac {i \, a \tan \left (d x + c\right ) + a}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 7.90, size = 214, normalized size = 2.40 \[ -\frac {2 \left (6 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+6 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{3 \left (2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e d} \]
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 \[ \int \frac {i \, a \tan \left (d x + c\right ) + a}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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 \[ i a \left (\int \left (- \frac {i}{\left (e \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}}\right )\, dx + \int \frac {\tan {\left (c + d x \right )}}{\left (e \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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