Optimal. Leaf size=138 \[ \frac{2 i \sec ^5(c+d x)}{1155 a^3 d (a+i a \tan (c+d x))^5}+\frac{2 i \sec ^5(c+d x)}{231 a^2 d (a+i a \tan (c+d x))^6}+\frac{i \sec ^5(c+d x)}{33 a d (a+i a \tan (c+d x))^7}+\frac{i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8} \]
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Rubi [A] time = 0.175295, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3502, 3488} \[ \frac{2 i \sec ^5(c+d x)}{1155 a^3 d (a+i a \tan (c+d x))^5}+\frac{2 i \sec ^5(c+d x)}{231 a^2 d (a+i a \tan (c+d x))^6}+\frac{i \sec ^5(c+d x)}{33 a d (a+i a \tan (c+d x))^7}+\frac{i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8} \]
Antiderivative was successfully verified.
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Rule 3502
Rule 3488
Rubi steps
\begin{align*} \int \frac{\sec ^5(c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=\frac{i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8}+\frac{3 \int \frac{\sec ^5(c+d x)}{(a+i a \tan (c+d x))^7} \, dx}{11 a}\\ &=\frac{i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8}+\frac{i \sec ^5(c+d x)}{33 a d (a+i a \tan (c+d x))^7}+\frac{2 \int \frac{\sec ^5(c+d x)}{(a+i a \tan (c+d x))^6} \, dx}{33 a^2}\\ &=\frac{i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8}+\frac{i \sec ^5(c+d x)}{33 a d (a+i a \tan (c+d x))^7}+\frac{2 i \sec ^5(c+d x)}{231 a^2 d (a+i a \tan (c+d x))^6}+\frac{2 \int \frac{\sec ^5(c+d x)}{(a+i a \tan (c+d x))^5} \, dx}{231 a^3}\\ &=\frac{i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8}+\frac{i \sec ^5(c+d x)}{33 a d (a+i a \tan (c+d x))^7}+\frac{2 i \sec ^5(c+d x)}{231 a^2 d (a+i a \tan (c+d x))^6}+\frac{2 i \sec ^5(c+d x)}{1155 a^3 d (a+i a \tan (c+d x))^5}\\ \end{align*}
Mathematica [A] time = 0.208569, size = 73, normalized size = 0.53 \[ \frac{i \sec ^8(c+d x) (55 i \sin (c+d x)+63 i \sin (3 (c+d x))+440 \cos (c+d x)+168 \cos (3 (c+d x)))}{4620 a^8 d (\tan (c+d x)-i)^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.109, size = 189, normalized size = 1.4 \begin{align*} 2\,{\frac{1}{d{a}^{8}} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-1}+{\frac{512}{3\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{9}}}-{\frac{88\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{4}}}+{\frac{932}{5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{5}}}-{\frac{128}{11\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{11}}}-{\frac{2376}{7\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{7}}}-{\frac{288\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{8}}}+{\frac{7\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{2}}}+{\frac{64\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{10}}}+{\frac{292\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{6}}}-30\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-3} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.23661, size = 131, normalized size = 0.95 \begin{align*} \frac{105 i \, \cos \left (11 \, d x + 11 \, c\right ) + 385 i \, \cos \left (9 \, d x + 9 \, c\right ) + 495 i \, \cos \left (7 \, d x + 7 \, c\right ) + 231 i \, \cos \left (5 \, d x + 5 \, c\right ) + 105 \, \sin \left (11 \, d x + 11 \, c\right ) + 385 \, \sin \left (9 \, d x + 9 \, c\right ) + 495 \, \sin \left (7 \, d x + 7 \, c\right ) + 231 \, \sin \left (5 \, d x + 5 \, c\right )}{9240 \, a^{8} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.4299, size = 177, normalized size = 1.28 \begin{align*} \frac{{\left (231 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 495 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 385 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 105 i\right )} e^{\left (-11 i \, d x - 11 i \, c\right )}}{9240 \, a^{8} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25228, size = 204, normalized size = 1.48 \begin{align*} \frac{2 \,{\left (1155 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} - 3465 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 13860 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 23100 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 37422 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 32802 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 27060 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 11220 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 4895 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 517 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 152\right )}}{1155 \, a^{8} d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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