Optimal. Leaf size=84 \[ \frac {8 i}{a^3 d \sqrt {a+i a \tan (c+d x)}}+\frac {8 i \sqrt {a+i a \tan (c+d x)}}{a^4 d}-\frac {2 i (a+i a \tan (c+d x))^{3/2}}{3 a^5 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 84, 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 (a+i a \tan (c+d x))^{3/2}}{3 a^5 d}+\frac {8 i \sqrt {a+i a \tan (c+d x)}}{a^4 d}+\frac {8 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 ^6(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx &=-\frac {i \text {Subst}\left (\int \frac {(a-x)^2}{(a+x)^{3/2}} \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac {i \text {Subst}\left (\int \left (\frac {4 a^2}{(a+x)^{3/2}}-\frac {4 a}{\sqrt {a+x}}+\sqrt {a+x}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=\frac {8 i}{a^3 d \sqrt {a+i a \tan (c+d x)}}+\frac {8 i \sqrt {a+i a \tan (c+d x)}}{a^4 d}-\frac {2 i (a+i a \tan (c+d x))^{3/2}}{3 a^5 d}\\ \end {align*}
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Mathematica [A]
time = 0.53, size = 61, normalized size = 0.73 \begin {gather*} \frac {2 i \sec ^2(c+d x) (12+11 \cos (2 (c+d x))+5 i \sin (2 (c+d x)))}{3 a^3 d \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 = 1.05
method | result | size |
default | \(\frac {2 \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (12 i \left (\cos ^{3}\left (d x +c \right )\right )+12 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+11 i \cos \left (d x +c \right )+\sin \left (d x +c \right )\right )}{3 d \cos \left (d x +c \right ) a^{4}}\) | \(88\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 62, normalized size = 0.74 \begin {gather*} \frac {2 i \, {\left (\frac {12}{\sqrt {i \, a \tan \left (d x + c\right ) + a} a^{2}} - \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} - 12 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} a}{a^{4}}\right )}}{3 \, a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 77, normalized size = 0.92 \begin {gather*} -\frac {4 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-8 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 12 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 3 i\right )}}{3 \, {\left (a^{4} d e^{\left (3 i \, d x + 3 i \, c\right )} + a^{4} d e^{\left (i \, d x + i \, c\right )}\right )}} \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 ^{6}{\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 [B]
time = 0.74, size = 110, normalized size = 1.31 \begin {gather*} \frac {2\,\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}}\,\left (\cos \left (2\,c+2\,d\,x\right )\,23{}\mathrm {i}+\cos \left (4\,c+4\,d\,x\right )\,3{}\mathrm {i}+7\,\sin \left (2\,c+2\,d\,x\right )+3\,\sin \left (4\,c+4\,d\,x\right )+20{}\mathrm {i}\right )}{3\,a^4\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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