Optimal. Leaf size=52 \[ \frac {\tan ^3(c+d x)}{3 a d}+\frac {\tan (c+d x)}{a d}-\frac {i \sec ^4(c+d x)}{4 a d} \]
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Rubi [A] time = 0.12, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3092, 3090, 3767, 2606, 30} \[ \frac {\tan ^3(c+d x)}{3 a d}+\frac {\tan (c+d x)}{a d}-\frac {i \sec ^4(c+d x)}{4 a d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2606
Rule 3090
Rule 3092
Rule 3767
Rubi steps
\begin {align*} \int \frac {\sec ^5(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx &=-\frac {i \int \sec ^5(c+d x) (i a \cos (c+d x)+a \sin (c+d x)) \, dx}{a^2}\\ &=-\frac {i \int \left (i a \sec ^4(c+d x)+a \sec ^4(c+d x) \tan (c+d x)\right ) \, dx}{a^2}\\ &=-\frac {i \int \sec ^4(c+d x) \tan (c+d x) \, dx}{a}+\frac {\int \sec ^4(c+d x) \, dx}{a}\\ &=-\frac {i \operatorname {Subst}\left (\int x^3 \, dx,x,\sec (c+d x)\right )}{a d}-\frac {\operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{a d}\\ &=-\frac {i \sec ^4(c+d x)}{4 a d}+\frac {\tan (c+d x)}{a d}+\frac {\tan ^3(c+d x)}{3 a d}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 53, normalized size = 1.02 \[ -\frac {i \sec ^4(c+d x) (i \sec (c) (4 \sin (c+2 d x)+\sin (3 c+4 d x))-3 i \tan (c)+3)}{12 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 72, normalized size = 1.38 \[ \frac {16 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 4 i}{3 \, {\left (a d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 47, normalized size = 0.90 \[ -\frac {3 i \, \tan \left (d x + c\right )^{4} - 4 \, \tan \left (d x + c\right )^{3} + 6 i \, \tan \left (d x + c\right )^{2} - 12 \, \tan \left (d x + c\right )}{12 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 47, normalized size = 0.90 \[ \frac {\tan \left (d x +c \right )-\frac {i \left (\tan ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\frac {i \left (\tan ^{2}\left (d x +c \right )\right )}{2}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 211, normalized size = 4.06 \[ \frac {2 \, {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {3 i \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {5 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {5 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {3 i \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{3 \, {\left (a - \frac {4 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {4 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.29, size = 99, normalized size = 1.90 \[ -\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,3{}\mathrm {i}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,3{}\mathrm {i}-3\right )}{3\,a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}^4} \]
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 \[ \frac {\int \frac {\sec ^{5}{\left (c + d x \right )}}{i \sin {\left (c + d x \right )} + \cos {\left (c + d x \right )}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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