Optimal. Leaf size=121 \[ -\frac {3 i \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{a^{7/2} d}-\frac {2 i \sec ^3(c+d x)}{a d (a+i a \tan (c+d x))^{5/2}}+\frac {6 i \sec (c+d x)}{a^2 d (a+i a \tan (c+d x))^{3/2}} \]
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Rubi [A]
time = 0.16, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3582, 3583,
3570, 212} \begin {gather*} -\frac {3 i \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{a^{7/2} d}+\frac {6 i \sec (c+d x)}{a^2 d (a+i a \tan (c+d x))^{3/2}}-\frac {2 i \sec ^3(c+d x)}{a d (a+i a \tan (c+d x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 3570
Rule 3582
Rule 3583
Rubi steps
\begin {align*} \int \frac {\sec ^5(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx &=-\frac {2 i \sec ^3(c+d x)}{a d (a+i a \tan (c+d x))^{5/2}}+\frac {6 \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx}{a}\\ &=-\frac {2 i \sec ^3(c+d x)}{a d (a+i a \tan (c+d x))^{5/2}}+\frac {12 i \sec (c+d x)}{a^2 d (a+i a \tan (c+d x))^{3/2}}-\frac {12 \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx}{a^2}\\ &=-\frac {2 i \sec ^3(c+d x)}{a d (a+i a \tan (c+d x))^{5/2}}+\frac {6 i \sec (c+d x)}{a^2 d (a+i a \tan (c+d x))^{3/2}}-\frac {3 \int \frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx}{a^3}\\ &=-\frac {2 i \sec ^3(c+d x)}{a d (a+i a \tan (c+d x))^{5/2}}+\frac {6 i \sec (c+d x)}{a^2 d (a+i a \tan (c+d x))^{3/2}}-\frac {(6 i) \text {Subst}\left (\int \frac {1}{2-a x^2} \, dx,x,\frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}}\right )}{a^3 d}\\ &=-\frac {3 i \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{a^{7/2} d}-\frac {2 i \sec ^3(c+d x)}{a d (a+i a \tan (c+d x))^{5/2}}+\frac {6 i \sec (c+d x)}{a^2 d (a+i a \tan (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 1.11, size = 126, normalized size = 1.04 \begin {gather*} \frac {16 e^{5 i (c+d x)} \left (-1-3 e^{2 i (c+d x)}+3 e^{2 i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \tanh ^{-1}\left (\sqrt {1+e^{2 i (c+d x)}}\right )\right )}{a^3 d \left (1+e^{2 i (c+d x)}\right )^4 (-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 [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 317 vs. \(2 (102 ) = 204\).
time = 0.82, size = 318, normalized size = 2.63
method | result | size |
default | \(-\frac {\left (3 i \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\left (i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )\right ) \sqrt {2}}{2 \sin \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right ) \cos \left (d x +c \right )+3 i \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\left (i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )\right ) \sqrt {2}}{2 \sin \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )+3 \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\left (i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )\right ) \sqrt {2}}{2 \sin \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right ) \sin \left (d x +c \right )-8 i \left (\cos ^{3}\left (d x +c \right )\right )-8 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-4 \sin \left (d x +c \right )\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{2 d \,a^{4}}\) | \(318\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 245 vs. \(2 (96) = 192\).
time = 0.40, size = 245, normalized size = 2.02 \begin {gather*} \frac {{\left (-3 i \, \sqrt {2} a^{4} d \sqrt {\frac {1}{a^{7} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-\frac {12 \, {\left ({\left (i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{3} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{7} d^{2}}} + i\right )} e^{\left (-i \, d x - i \, c\right )}}{a^{3} d}\right ) + 3 i \, \sqrt {2} a^{4} d \sqrt {\frac {1}{a^{7} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-\frac {12 \, {\left ({\left (-i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{3} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{7} d^{2}}} + i\right )} e^{\left (-i \, d x - i \, c\right )}}{a^{3} d}\right ) - 2 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-3 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - i\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{2 \, 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 ^{5}{\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 = -0.01 \begin {gather*} \int \frac {1}{{\cos \left (c+d\,x\right )}^5\,{\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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