Optimal. Leaf size=47 \[ -\frac {\cos (c+d x)}{a^2 d}+\frac {\sec ^3(c+d x)}{3 a^2 d}-\frac {2 \sec (c+d x)}{a^2 d} \]
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Rubi [A] time = 0.07, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3175, 2590, 270} \[ -\frac {\cos (c+d x)}{a^2 d}+\frac {\sec ^3(c+d x)}{3 a^2 d}-\frac {2 \sec (c+d x)}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2590
Rule 3175
Rubi steps
\begin {align*} \int \frac {\sin ^5(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx &=\frac {\int \sin (c+d x) \tan ^4(c+d x) \, dx}{a^2}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^4} \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (1+\frac {1}{x^4}-\frac {2}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac {\cos (c+d x)}{a^2 d}-\frac {2 \sec (c+d x)}{a^2 d}+\frac {\sec ^3(c+d x)}{3 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 42, normalized size = 0.89 \[ \frac {-\frac {\cos (c+d x)}{d}+\frac {\sec ^3(c+d x)}{3 d}-\frac {2 \sec (c+d x)}{d}}{a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 38, normalized size = 0.81 \[ -\frac {3 \, \cos \left (d x + c\right )^{4} + 6 \, \cos \left (d x + c\right )^{2} - 1}{3 \, a^{2} d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 106, normalized size = 2.26 \[ \frac {2 \, {\left (\frac {3}{a^{2} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}} - \frac {\frac {12 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {3 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 5}{a^{2} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{3}}\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 37, normalized size = 0.79 \[ \frac {-\cos \left (d x +c \right )-\frac {2}{\cos \left (d x +c \right )}+\frac {1}{3 \cos \left (d x +c \right )^{3}}}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 41, normalized size = 0.87 \[ -\frac {\frac {3 \, \cos \left (d x + c\right )}{a^{2}} + \frac {6 \, \cos \left (d x + c\right )^{2} - 1}{a^{2} \cos \left (d x + c\right )^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 36, normalized size = 0.77 \[ -\frac {{\cos \left (c+d\,x\right )}^4+2\,{\cos \left (c+d\,x\right )}^2-\frac {1}{3}}{a^2\,d\,{\cos \left (c+d\,x\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 49.03, size = 156, normalized size = 3.32 \[ \begin {cases} - \frac {32 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} + \frac {16}{3 a^{2} d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{5}{\relax (c )}}{\left (- a \sin ^{2}{\relax (c )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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