Optimal. Leaf size=57 \[ \frac {4 i}{3 a^2 d (a+i a \tan (c+d x))^{3/2}}-\frac {2 i}{a^3 d \sqrt {a+i a \tan (c+d x)}} \]
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Rubi [A]
time = 0.05, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3568, 45}
\begin {gather*} \frac {4 i}{3 a^2 d (a+i a \tan (c+d x))^{3/2}}-\frac {2 i}{a^3 d \sqrt {a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3568
Rubi steps
\begin {align*} \int \frac {\sec ^4(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx &=-\frac {i \text {Subst}\left (\int \frac {a-x}{(a+x)^{5/2}} \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac {i \text {Subst}\left (\int \left (\frac {2 a}{(a+x)^{5/2}}-\frac {1}{(a+x)^{3/2}}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=\frac {4 i}{3 a^2 d (a+i a \tan (c+d x))^{3/2}}-\frac {2 i}{a^3 d \sqrt {a+i a \tan (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 80, normalized size = 1.40 \begin {gather*} \frac {2 \sec ^2(c+d x) (\cos (2 (c+d x))+i \sin (2 (c+d x))) (1+3 i \tan (c+d x))}{3 a^3 d (-i+\tan (c+d x))^3 \sqrt {a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.88, size = 88, normalized size = 1.54
method | result | size |
default | \(\frac {2 \cos \left (d x +c \right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (4 i \left (\cos ^{3}\left (d x +c \right )\right )+4 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-5 i \cos \left (d x +c \right )-3 \sin \left (d x +c \right )\right )}{3 d \,a^{4}}\) | \(88\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 32, normalized size = 0.56 \begin {gather*} -\frac {2 i \, {\left (3 i \, a \tan \left (d x + c\right ) + a\right )}}{3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 61, normalized size = 1.07 \begin {gather*} \frac {\sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-2 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{3 \, a^{4} 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*} \int \frac {\sec ^{4}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {7}{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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\cos \left (c+d\,x\right )}^4\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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