Optimal. Leaf size=55 \[ \frac {4 i}{a^2 d \sqrt {a+i a \tan (c+d x)}}+\frac {2 i \sqrt {a+i a \tan (c+d x)}}{a^3 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 55, 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 {2 i \sqrt {a+i a \tan (c+d x)}}{a^3 d}+\frac {4 i}{a^2 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))^{5/2}} \, dx &=-\frac {i \text {Subst}\left (\int \frac {a-x}{(a+x)^{3/2}} \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac {i \text {Subst}\left (\int \left (\frac {2 a}{(a+x)^{3/2}}-\frac {1}{\sqrt {a+x}}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=\frac {4 i}{a^2 d \sqrt {a+i a \tan (c+d x)}}+\frac {2 i \sqrt {a+i a \tan (c+d x)}}{a^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 36, normalized size = 0.65 \begin {gather*} \frac {6 i-2 \tan (c+d x)}{a^2 d \sqrt {a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.81, size = 65, normalized size = 1.18
method | result | size |
default | \(\frac {2 \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (2 \sin \left (d x +c \right ) \cos \left (d x +c \right )+i+2 i \left (\cos ^{2}\left (d x +c \right )\right )\right )}{d \,a^{3}}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 44, normalized size = 0.80 \begin {gather*} \frac {2 i \, {\left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a}}{a^{2}} + \frac {2}{\sqrt {i \, a \tan \left (d x + c\right ) + a} a}\right )}}{a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 50, normalized size = 0.91 \begin {gather*} -\frac {2 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-2 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - i\right )} e^{\left (-i \, d x - i \, c\right )}}{a^{3} 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 {5}{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 = 0.27, size = 72, normalized size = 1.31 \begin {gather*} \frac {2\,\left (\cos \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}+\sin \left (2\,c+2\,d\,x\right )+2{}\mathrm {i}\right )\,\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}}}{a^3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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