Optimal. Leaf size=47 \[ \frac {a^2 (A+B)}{2 d (a-a \sin (c+d x))}+\frac {a (A-B) \tanh ^{-1}(\sin (c+d x))}{2 d} \]
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Rubi [A] time = 0.09, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2836, 77, 206} \[ \frac {a^2 (A+B)}{2 d (a-a \sin (c+d x))}+\frac {a (A-B) \tanh ^{-1}(\sin (c+d x))}{2 d} \]
Antiderivative was successfully verified.
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Rule 77
Rule 206
Rule 2836
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx &=\frac {a^3 \operatorname {Subst}\left (\int \frac {A+\frac {B x}{a}}{(a-x)^2 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^3 \operatorname {Subst}\left (\int \left (\frac {A+B}{2 a (a-x)^2}+\frac {A-B}{2 a \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^2 (A+B)}{2 d (a-a \sin (c+d x))}+\frac {\left (a^2 (A-B)\right ) \operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{2 d}\\ &=\frac {a (A-B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a^2 (A+B)}{2 d (a-a \sin (c+d x))}\\ \end {align*}
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Mathematica [C] time = 0.63, size = 260, normalized size = 5.53 \[ \frac {a \left (2 i (A-B) (\sin (c+d x)-1) \tan ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )+(A-B) \sin (c+d x) \left (2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2\right )-i d x\right )-2 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+A \log \left (\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2\right )+i A d x+2 A+2 B \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-B \log \left (\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2\right )-i B d x+2 B\right )}{4 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.71, size = 90, normalized size = 1.91 \[ -\frac {2 \, {\left (A + B\right )} a - {\left ({\left (A - B\right )} a \sin \left (d x + c\right ) - {\left (A - B\right )} a\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left ({\left (A - B\right )} a \sin \left (d x + c\right ) - {\left (A - B\right )} a\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{4 \, {\left (d \sin \left (d x + c\right ) - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 84, normalized size = 1.79 \[ \frac {{\left (A a - B a\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - {\left (A a - B a\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + \frac {A a \sin \left (d x + c\right ) - B a \sin \left (d x + c\right ) - 3 \, A a - B a}{\sin \left (d x + c\right ) - 1}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.57, size = 129, normalized size = 2.74 \[ \frac {a A}{2 d \cos \left (d x +c \right )^{2}}+\frac {a B \left (\sin ^{3}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {a B \sin \left (d x +c \right )}{2 d}-\frac {a B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {a A \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {a A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {a B}{2 d \cos \left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 55, normalized size = 1.17 \[ \frac {{\left (A - B\right )} a \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (A - B\right )} a \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (A + B\right )} a}{\sin \left (d x + c\right ) - 1}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.12, size = 43, normalized size = 0.91 \[ \frac {a\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )\,\left (A-B\right )}{2\,d}-\frac {\frac {A\,a}{2}+\frac {B\,a}{2}}{d\,\left (\sin \left (c+d\,x\right )-1\right )} \]
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 \[ a \left (\int A \sec ^{3}{\left (c + d x \right )}\, dx + \int A \sin {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int B \sin {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int B \sin ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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