Optimal. Leaf size=43 \[ \frac {a^3 (a A+a B \sin (c+d x))^2}{2 d (A+B) (a-a \sin (c+d x))^2} \]
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Rubi [A] time = 0.08, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2836, 37} \[ \frac {a^3 (a A+a B \sin (c+d x))^2}{2 d (A+B) (a-a \sin (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 37
Rule 2836
Rubi steps
\begin {align*} \int \sec ^5(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx &=\frac {a^5 \operatorname {Subst}\left (\int \frac {A+\frac {B x}{a}}{(a-x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^3 (a A+a B \sin (c+d x))^2}{2 (A+B) d (a-a \sin (c+d x))^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 37, normalized size = 0.86 \[ \frac {a^3 (A+B \sin (c+d x))^2}{2 d (A+B) (\sin (c+d x)-1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 49, normalized size = 1.14 \[ -\frac {2 \, B a^{3} \sin \left (d x + c\right ) + {\left (A - B\right )} a^{3}}{2 \, {\left (d \cos \left (d x + c\right )^{2} + 2 \, d \sin \left (d x + c\right ) - 2 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 82, normalized size = 1.91 \[ \frac {2 \, {\left (A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.58, size = 312, normalized size = 7.26 \[ \frac {a^{3} A \left (\sin ^{4}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}+\frac {B \,a^{3} \left (\sin ^{5}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}-\frac {B \,a^{3} \left (\sin ^{5}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}-\frac {B \,a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{8 d}+\frac {3 a^{3} A \left (\sin ^{3}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}+\frac {3 a^{3} A \left (\sin ^{3}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}+\frac {3 a^{3} A \sin \left (d x +c \right )}{8 d}+\frac {3 B \,a^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}+\frac {3 a^{3} A}{4 d \cos \left (d x +c \right )^{4}}+\frac {3 B \,a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}+\frac {3 B \,a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}+\frac {a^{3} A \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{4 d}+\frac {3 a^{3} A \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {B \,a^{3}}{4 d \cos \left (d x +c \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 47, normalized size = 1.09 \[ \frac {2 \, B a^{3} \sin \left (d x + c\right ) + {\left (A - B\right )} a^{3}}{2 \, {\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.15, size = 36, normalized size = 0.84 \[ \frac {\frac {a^3\,\left (A-B\right )}{2}+B\,a^3\,\sin \left (c+d\,x\right )}{d\,{\left (\sin \left (c+d\,x\right )-1\right )}^2} \]
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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