Optimal. Leaf size=46 \[ -\frac {\cos ^3(c+d x)}{3 a d}+\frac {2 \cos (c+d x)}{a d}+\frac {\sec (c+d x)}{a d} \]
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Rubi [A] time = 0.08, antiderivative size = 46, 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} \[ -\frac {\cos ^3(c+d x)}{3 a d}+\frac {2 \cos (c+d x)}{a d}+\frac {\sec (c+d x)}{a d} \]
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)}{a-a \sin ^2(c+d x)} \, dx &=\frac {\int \sin ^3(c+d x) \tan ^2(c+d x) \, dx}{a}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^2} \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (-2+\frac {1}{x^2}+x^2\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac {2 \cos (c+d x)}{a d}-\frac {\cos ^3(c+d x)}{3 a d}+\frac {\sec (c+d x)}{a d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 43, normalized size = 0.93 \[ \frac {\frac {7 \cos (c+d x)}{4 d}-\frac {\cos (3 (c+d x))}{12 d}+\frac {\sec (c+d x)}{d}}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 36, normalized size = 0.78 \[ -\frac {\cos \left (d x + c\right )^{4} - 6 \, \cos \left (d x + c\right )^{2} - 3}{3 \, a d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 105, normalized size = 2.28 \[ \frac {2 \, {\left (\frac {3}{a {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}} + \frac {\frac {12 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {3 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 5}{a {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{3}}\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 35, normalized size = 0.76 \[ \frac {-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+2 \cos \left (d x +c \right )+\frac {1}{\cos \left (d x +c \right )}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 40, normalized size = 0.87 \[ -\frac {\frac {\cos \left (d x + c\right )^{3} - 6 \, \cos \left (d x + c\right )}{a} - \frac {3}{a \cos \left (d x + c\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 38, normalized size = 0.83 \[ \frac {-{\cos \left (c+d\,x\right )}^4+6\,{\cos \left (c+d\,x\right )}^2+3}{3\,a\,d\,\cos \left (c+d\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 14.35, size = 143, normalized size = 3.11 \[ \begin {cases} - \frac {32 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a d} - \frac {16}{3 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{5}{\relax (c )}}{- a \sin ^{2}{\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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