Optimal. Leaf size=27 \[ \frac {i}{7 a d (a+i a \tan (c+d x))^7} \]
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Rubi [A] time = 0.04, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3487, 32} \[ \frac {i}{7 a d (a+i a \tan (c+d x))^7} \]
Antiderivative was successfully verified.
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Rule 32
Rule 3487
Rubi steps
\begin {align*} \int \frac {\sec ^2(c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=-\frac {i \operatorname {Subst}\left (\int \frac {1}{(a+x)^8} \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=\frac {i}{7 a d (a+i a \tan (c+d x))^7}\\ \end {align*}
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Mathematica [B] time = 0.31, size = 100, normalized size = 3.70 \[ \frac {i \sec ^8(c+d x) (14 i \sin (2 (c+d x))+14 i \sin (4 (c+d x))+6 i \sin (6 (c+d x))+56 \cos (2 (c+d x))+28 \cos (4 (c+d x))+8 \cos (6 (c+d x))+35)}{896 a^8 d (\tan (c+d x)-i)^8} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.80, size = 85, normalized size = 3.15 \[ \frac {{\left (7 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 21 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 35 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 35 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 21 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 7 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-14 i \, d x - 14 i \, c\right )}}{896 \, a^{8} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.58, size = 189, normalized size = 7.00 \[ -\frac {2 \, {\left (7 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 42 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 182 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 490 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 1001 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1484 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1716 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1484 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1001 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 490 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 182 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 42 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 7 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{7 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 24, normalized size = 0.89 \[ \frac {i}{7 a d \left (a +i a \tan \left (d x +c \right )\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 21, normalized size = 0.78 \[ \frac {i}{7 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{7} a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.56, size = 19, normalized size = 0.70 \[ -\frac {1}{7\,a^8\,d\,{\left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 36.83, size = 1081, normalized size = 40.04 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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