Optimal. Leaf size=81 \[ -\frac {a^3 (A+B) \sin ^2(c+d x)}{2 d}-\frac {3 a^3 (A+B) \sin (c+d x)}{d}-\frac {4 a^3 (A+B) \log (1-\sin (c+d x))}{d}-\frac {B (a \sin (c+d x)+a)^3}{3 d} \]
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Rubi [A] time = 0.09, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2836, 77} \[ -\frac {a^3 (A+B) \sin ^2(c+d x)}{2 d}-\frac {3 a^3 (A+B) \sin (c+d x)}{d}-\frac {4 a^3 (A+B) \log (1-\sin (c+d x))}{d}-\frac {B (a \sin (c+d x)+a)^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 77
Rule 2836
Rubi steps
\begin {align*} \int \sec (c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx &=\frac {a \operatorname {Subst}\left (\int \frac {(a+x)^2 \left (A+\frac {B x}{a}\right )}{a-x} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a \operatorname {Subst}\left (\int \left (-3 a (A+B)+\frac {4 a^2 (A+B)}{a-x}-(A+B) x-\frac {B (a+x)^2}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {4 a^3 (A+B) \log (1-\sin (c+d x))}{d}-\frac {3 a^3 (A+B) \sin (c+d x)}{d}-\frac {a^3 (A+B) \sin ^2(c+d x)}{2 d}-\frac {B (a+a \sin (c+d x))^3}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 68, normalized size = 0.84 \[ -\frac {a^3 \left (3 (A+3 B) \sin ^2(c+d x)+6 (3 A+4 B) \sin (c+d x)+24 (A+B) \log (1-\sin (c+d x))+2 B \sin ^3(c+d x)\right )}{6 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 77, normalized size = 0.95 \[ \frac {3 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} - 24 \, {\left (A + B\right )} a^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (B a^{3} \cos \left (d x + c\right )^{2} - {\left (9 \, A + 13 \, B\right )} a^{3}\right )} \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 289, normalized size = 3.57 \[ \frac {2 \, {\left (6 \, {\left (A a^{3} + B a^{3}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 12 \, {\left (A a^{3} + B a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {11 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 11 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 9 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 42 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 18 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 28 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 42 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 11 \, A a^{3} + 11 \, B a^{3}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.36, size = 161, normalized size = 1.99 \[ -\frac {a^{3} A \left (\sin ^{2}\left (d x +c \right )\right )}{2 d}-\frac {4 a^{3} A \ln \left (\cos \left (d x +c \right )\right )}{d}-\frac {B \,a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{3 d}-\frac {4 a^{3} B \sin \left (d x +c \right )}{d}+\frac {4 B \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}-\frac {3 a^{3} A \sin \left (d x +c \right )}{d}+\frac {4 a^{3} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}-\frac {3 B \,a^{3} \left (\sin ^{2}\left (d x +c \right )\right )}{2 d}-\frac {4 B \,a^{3} \ln \left (\cos \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 73, normalized size = 0.90 \[ -\frac {2 \, B a^{3} \sin \left (d x + c\right )^{3} + 3 \, {\left (A + 3 \, B\right )} a^{3} \sin \left (d x + c\right )^{2} + 24 \, {\left (A + B\right )} a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) + 6 \, {\left (3 \, A + 4 \, B\right )} a^{3} \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 100, normalized size = 1.23 \[ -\frac {{\sin \left (c+d\,x\right )}^2\,\left (\frac {a^3\,\left (A+2\,B\right )}{2}+\frac {B\,a^3}{2}\right )+\sin \left (c+d\,x\right )\,\left (a^3\,\left (A+2\,B\right )+a^3\,\left (2\,A+B\right )+B\,a^3\right )+\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (4\,A\,a^3+4\,B\,a^3\right )+\frac {B\,a^3\,{\sin \left (c+d\,x\right )}^3}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{3} \left (\int A \sec {\left (c + d x \right )}\, dx + \int 3 A \sin {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 A \sin ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int A \sin ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \sin {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 B \sin ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 B \sin ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \sin ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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