Optimal. Leaf size=34 \[ \frac {i \tan ^4(c+d x) (-\cot (c+d x)+i)^4}{4 a^3 d} \]
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Rubi [A] time = 0.06, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3088, 848, 37} \[ \frac {i \tan ^4(c+d x) (-\cot (c+d x)+i)^4}{4 a^3 d} \]
Antiderivative was successfully verified.
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Rule 37
Rule 848
Rule 3088
Rubi steps
\begin {align*} \int \frac {\sec ^5(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^5 (i a+a x)^3} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\left (-\frac {i}{a}+\frac {x}{a}\right )^3}{x^5} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {i (i-\cot (c+d x))^4 \tan ^4(c+d x)}{4 a^3 d}\\ \end {align*}
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Mathematica [B] time = 0.47, size = 90, normalized size = 2.65 \[ -\frac {i \sec (c) \sec ^4(c+d x) (2 i \sin (c+2 d x)-2 i \sin (3 c+2 d x)+i \sin (3 c+4 d x)+2 \cos (c+2 d x)+2 \cos (3 c+2 d x)-3 i \sin (c)+3 \cos (c))}{4 a^3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.42, size = 69, normalized size = 2.03 \[ \frac {4 i}{a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 47, normalized size = 1.38 \[ -\frac {-i \, \tan \left (d x + c\right )^{4} + 4 \, \tan \left (d x + c\right )^{3} + 6 i \, \tan \left (d x + c\right )^{2} - 4 \, \tan \left (d x + c\right )}{4 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 47, normalized size = 1.38 \[ \frac {\tan \left (d x +c \right )+\frac {i \left (\tan ^{4}\left (d x +c \right )\right )}{4}-\left (\tan ^{3}\left (d x +c \right )\right )-\frac {3 i \left (\tan ^{2}\left (d x +c \right )\right )}{2}}{d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.38, size = 240, normalized size = 7.06 \[ \frac {2 \, {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {3 i \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {7 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {8 i \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {7 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {3 i \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {\sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{{\left (a^{3} - \frac {4 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {4 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.88, size = 55, normalized size = 1.62 \[ \frac {{\sin \left (c+d\,x\right )}^2\,1{}\mathrm {i}-\frac {{\sin \left (2\,c+2\,d\,x\right )}^2\,7{}\mathrm {i}}{4}+\sin \left (4\,c+4\,d\,x\right )}{4\,a^3\,d\,{\left ({\sin \left (c+d\,x\right )}^2-1\right )}^2} \]
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 \[ \frac {\int \frac {\sec ^{5}{\left (c + d x \right )}}{- i \sin ^{3}{\left (c + d x \right )} - 3 \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )} + 3 i \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + \cos ^{3}{\left (c + d x \right )}}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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