Optimal. Leaf size=31 \[ \frac {a^5 \sin ^2(c+d x)}{2 d (a-a \sin (c+d x))^2} \]
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Rubi [A] time = 0.06, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2836, 12, 37} \[ \frac {a^5 \sin ^2(c+d x)}{2 d (a-a \sin (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 37
Rule 2836
Rubi steps
\begin {align*} \int \sec ^4(c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx &=\frac {a^5 \operatorname {Subst}\left (\int \frac {x}{a (a-x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^4 \operatorname {Subst}\left (\int \frac {x}{(a-x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^5 \sin ^2(c+d x)}{2 d (a-a \sin (c+d x))^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 30, normalized size = 0.97 \[ \frac {a^3 \sin ^2(c+d x)}{2 d (1-\sin (c+d x))^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 44, normalized size = 1.42 \[ -\frac {2 \, a^{3} \sin \left (d x + c\right ) - a^{3}}{2 \, {\left (d \cos \left (d x + c\right )^{2} + 2 \, d \sin \left (d x + c\right ) - 2 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 32, normalized size = 1.03 \[ \frac {2 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.27, size = 154, normalized size = 4.97 \[ \frac {a^{3} \left (\sin ^{5}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}-\frac {a^{3} \left (\sin ^{5}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}-\frac {a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{8 d}+\frac {3 a^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}+\frac {3 a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}+\frac {3 a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}+\frac {a^{3}}{4 d \cos \left (d x +c \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 42, normalized size = 1.35 \[ \frac {2 \, a^{3} \sin \left (d x + c\right ) - a^{3}}{2 \, {\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.29, size = 30, normalized size = 0.97 \[ \frac {a^3\,{\sin \left (c+d\,x\right )}^2}{8\,d\,{\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d\,x}{2}\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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