Optimal. Leaf size=65 \[ \frac {\cos ^3(c+d x)}{3 a^2 d}-\frac {3 \cos (c+d x)}{a^2 d}+\frac {\sec ^3(c+d x)}{3 a^2 d}-\frac {3 \sec (c+d x)}{a^2 d} \]
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Rubi [A] time = 0.08, antiderivative size = 65, 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} \[ \frac {\cos ^3(c+d x)}{3 a^2 d}-\frac {3 \cos (c+d x)}{a^2 d}+\frac {\sec ^3(c+d x)}{3 a^2 d}-\frac {3 \sec (c+d x)}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2590
Rule 3175
Rubi steps
\begin {align*} \int \frac {\sin ^7(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx &=\frac {\int \sin ^3(c+d x) \tan ^4(c+d x) \, dx}{a^2}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^3}{x^4} \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (3+\frac {1}{x^4}-\frac {3}{x^2}-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac {3 \cos (c+d x)}{a^2 d}+\frac {\cos ^3(c+d x)}{3 a^2 d}-\frac {3 \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 = 0.05, size = 59, normalized size = 0.91 \[ \frac {-\frac {11 \cos (c+d x)}{4 d}+\frac {\cos (3 (c+d x))}{12 d}+\frac {\sec ^3(c+d x)}{3 d}-\frac {3 \sec (c+d x)}{d}}{a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 46, normalized size = 0.71 \[ \frac {\cos \left (d x + c\right )^{6} - 9 \, \cos \left (d x + c\right )^{4} - 9 \, \cos \left (d x + c\right )^{2} + 1}{3 \, a^{2} d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 57, normalized size = 0.88 \[ -\frac {32 \, {\left (\frac {3 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )}}{3 \, a^{2} d {\left (\frac {{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 47, normalized size = 0.72 \[ \frac {\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}-3 \cos \left (d x +c \right )-\frac {3}{\cos \left (d x +c \right )}+\frac {1}{3 \cos \left (d x +c \right )^{3}}}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 52, normalized size = 0.80 \[ \frac {\frac {\cos \left (d x + c\right )^{3} - 9 \, \cos \left (d x + c\right )}{a^{2}} - \frac {9 \, \cos \left (d x + c\right )^{2} - 1}{a^{2} \cos \left (d x + c\right )^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.64, size = 48, normalized size = 0.74 \[ -\frac {-{\cos \left (c+d\,x\right )}^6+9\,{\cos \left (c+d\,x\right )}^4+9\,{\cos \left (c+d\,x\right )}^2-1}{3\,a^2\,d\,{\cos \left (c+d\,x\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 106.84, size = 156, normalized size = 2.40 \[ \begin {cases} - \frac {96 \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{12}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 9 a^{2} d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} + \frac {32}{3 a^{2} d \tan ^{12}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 9 a^{2} d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{7}{\relax (c )}}{\left (- a \sin ^{2}{\relax (c )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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