Optimal. Leaf size=62 \[ \frac {2 a^4 (A+B)}{d (a-a \sin (c+d x))}+\frac {a^3 (A+3 B) \log (1-\sin (c+d x))}{d}+\frac {a^3 B \sin (c+d x)}{d} \]
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Rubi [A] time = 0.11, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2836, 77} \[ \frac {2 a^4 (A+B)}{d (a-a \sin (c+d x))}+\frac {a^3 (A+3 B) \log (1-\sin (c+d x))}{d}+\frac {a^3 B \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 77
Rule 2836
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx &=\frac {a^3 \operatorname {Subst}\left (\int \frac {(a+x) \left (A+\frac {B x}{a}\right )}{(a-x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^3 \operatorname {Subst}\left (\int \left (\frac {B}{a}+\frac {2 a (A+B)}{(a-x)^2}+\frac {-A-3 B}{a-x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^3 (A+3 B) \log (1-\sin (c+d x))}{d}+\frac {a^3 B \sin (c+d x)}{d}+\frac {2 a^4 (A+B)}{d (a-a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 48, normalized size = 0.77 \[ \frac {a^3 \left (-\frac {2 (A+B)}{\sin (c+d x)-1}+(A+3 B) \log (1-\sin (c+d x))+B \sin (c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 89, normalized size = 1.44 \[ -\frac {B a^{3} \cos \left (d x + c\right )^{2} + B a^{3} \sin \left (d x + c\right ) + {\left (2 \, A + B\right )} a^{3} - {\left ({\left (A + 3 \, B\right )} a^{3} \sin \left (d x + c\right ) - {\left (A + 3 \, B\right )} a^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{d \sin \left (d x + c\right ) - d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 228, normalized size = 3.68 \[ -\frac {{\left (A a^{3} + 3 \, B a^{3}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 2 \, {\left (A a^{3} + 3 \, B a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A a^{3} + 3 \, B a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + \frac {3 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 10 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 22 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A a^{3} + 9 \, B a^{3}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.58, size = 290, normalized size = 4.68 \[ \frac {a^{3} A \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a^{3} A \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {B \,a^{3} \left (\sin ^{5}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {B \,a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{2 d}+\frac {3 a^{3} B \sin \left (d x +c \right )}{d}-\frac {3 B \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {3 a^{3} A \left (\sin ^{3}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {3 a^{3} A \sin \left (d x +c \right )}{2 d}-\frac {a^{3} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {3 B \,a^{3} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {3 B \,a^{3} \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {3 a^{3} A}{2 d \cos \left (d x +c \right )^{2}}+\frac {3 B \,a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {a^{3} A \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {B \,a^{3}}{2 d \cos \left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 52, normalized size = 0.84 \[ \frac {{\left (A + 3 \, B\right )} a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) + B a^{3} \sin \left (d x + c\right ) - \frac {2 \, {\left (A + B\right )} a^{3}}{\sin \left (d x + c\right ) - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 63, normalized size = 1.02 \[ \frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (A\,a^3+3\,B\,a^3\right )-\frac {2\,A\,a^3+2\,B\,a^3}{\sin \left (c+d\,x\right )-1}+B\,a^3\,\sin \left (c+d\,x\right )}{d} \]
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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