Optimal. Leaf size=121 \[ \frac {2 i}{7 d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^{3/2}}+\frac {8 i}{21 a d \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{21 a^2 d \sqrt {e \sec (c+d x)}} \]
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Rubi [A]
time = 0.15, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3583, 3569}
\begin {gather*} -\frac {16 i \sqrt {a+i a \tan (c+d x)}}{21 a^2 d \sqrt {e \sec (c+d x)}}+\frac {8 i}{21 a d \sqrt {a+i a \tan (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 i}{7 d (a+i a \tan (c+d x))^{3/2} \sqrt {e \sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3569
Rule 3583
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^{3/2}} \, dx &=\frac {2 i}{7 d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^{3/2}}+\frac {4 \int \frac {1}{\sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)}} \, dx}{7 a}\\ &=\frac {2 i}{7 d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^{3/2}}+\frac {8 i}{21 a d \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {8 \int \frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}} \, dx}{21 a^2}\\ &=\frac {2 i}{7 d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^{3/2}}+\frac {8 i}{21 a d \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{21 a^2 d \sqrt {e \sec (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.37, size = 83, normalized size = 0.69 \begin {gather*} -\frac {\sec ^2(c+d x) (-7+9 \cos (2 (c+d x))+12 i \sin (2 (c+d x)))}{21 a d \sqrt {e \sec (c+d x)} (-i+\tan (c+d x)) \sqrt {a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.17, size = 106, normalized size = 0.88
method | result | size |
default | \(-\frac {2 \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (9 i \left (\cos ^{2}\left (d x +c \right )\right )-12 \sin \left (d x +c \right ) \cos \left (d x +c \right )-8 i\right )}{21 d \left (2 i \cos \left (d x +c \right ) \sin \left (d x +c \right )+2 \left (\cos ^{2}\left (d x +c \right )\right )-1\right ) \sqrt {\frac {e}{\cos \left (d x +c \right )}}\, a^{2}}\) | \(106\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 129, normalized size = 1.07 \begin {gather*} \frac {{\left (3 i \, \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 14 i \, \cos \left (\frac {3}{7} \, \arctan \left (\sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ), \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )\right )\right ) - 21 i \, \cos \left (\frac {1}{7} \, \arctan \left (\sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ), \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )\right )\right ) + 3 \, \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 14 \, \sin \left (\frac {3}{7} \, \arctan \left (\sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ), \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )\right )\right ) + 21 \, \sin \left (\frac {1}{7} \, \arctan \left (\sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ), \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )\right )\right )\right )} e^{\left (-\frac {1}{2}\right )}}{42 \, a^{\frac {3}{2}} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 83, normalized size = 0.69 \begin {gather*} \frac {\sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-21 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 7 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 17 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )} e^{\left (-\frac {7}{2} i \, d x - \frac {7}{2} i \, c - \frac {1}{2}\right )}}{42 \, a^{2} d \sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \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*} \int \frac {1}{\sqrt {e \sec {\left (c + d x \right )}} \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \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 [B]
time = 4.16, size = 104, normalized size = 0.86 \begin {gather*} \frac {\sqrt {\frac {e}{\cos \left (c+d\,x\right )}}\,\left (35\,\sin \left (c+d\,x\right )+3\,\sin \left (3\,c+3\,d\,x\right )-\cos \left (c+d\,x\right )\,7{}\mathrm {i}+\cos \left (3\,c+3\,d\,x\right )\,3{}\mathrm {i}\right )}{42\,a\,d\,e\,\sqrt {\frac {a\,\left (\cos \left (2\,c+2\,d\,x\right )+1+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,c+2\,d\,x\right )+1}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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