Optimal. Leaf size=100 \[ \frac {a^3 (A+B)}{8 d (a-a \sin (c+d x))^2}-\frac {a^2 (A-B)}{8 d (a \sin (c+d x)+a)}+\frac {a^2 A}{4 d (a-a \sin (c+d x))}+\frac {a (3 A-B) \tanh ^{-1}(\sin (c+d x))}{8 d} \]
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Rubi [A] time = 0.12, antiderivative size = 100, 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} \[ \frac {a^3 (A+B)}{8 d (a-a \sin (c+d x))^2}-\frac {a^2 (A-B)}{8 d (a \sin (c+d x)+a)}+\frac {a^2 A}{4 d (a-a \sin (c+d x))}+\frac {a (3 A-B) \tanh ^{-1}(\sin (c+d x))}{8 d} \]
Antiderivative was successfully verified.
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Rule 77
Rule 206
Rule 2836
Rubi steps
\begin {align*} \int \sec ^5(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx &=\frac {a^5 \operatorname {Subst}\left (\int \frac {A+\frac {B x}{a}}{(a-x)^3 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^5 \operatorname {Subst}\left (\int \left (\frac {A+B}{4 a^2 (a-x)^3}+\frac {A}{4 a^3 (a-x)^2}+\frac {A-B}{8 a^3 (a+x)^2}+\frac {3 A-B}{8 a^3 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^3 (A+B)}{8 d (a-a \sin (c+d x))^2}+\frac {a^2 A}{4 d (a-a \sin (c+d x))}-\frac {a^2 (A-B)}{8 d (a+a \sin (c+d x))}+\frac {\left (a^2 (3 A-B)\right ) \operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{8 d}\\ &=\frac {a (3 A-B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {a^3 (A+B)}{8 d (a-a \sin (c+d x))^2}+\frac {a^2 A}{4 d (a-a \sin (c+d x))}-\frac {a^2 (A-B)}{8 d (a+a \sin (c+d x))}\\ \end {align*}
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Mathematica [C] time = 1.56, size = 357, normalized size = 3.57 \[ \frac {a (\sin (c+d x)+1) \left (i x (3 A-B) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2+\frac {2 (A+B) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^4}-\frac {2 (3 A-B) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {(3 A-B) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2 \log \left (\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2\right )}{d}-\frac {2 i (3 A-B) \tan ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}{d}+\frac {2 (B-A)}{d}+\frac {4 A \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}\right )}{16 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 182, normalized size = 1.82 \[ -\frac {2 \, {\left (3 \, A - B\right )} a \cos \left (d x + c\right )^{2} + 2 \, {\left (3 \, A - B\right )} a \sin \left (d x + c\right ) - 2 \, {\left (A - 3 \, B\right )} a - {\left ({\left (3 \, A - B\right )} a \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - {\left (3 \, A - B\right )} a \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left ({\left (3 \, A - B\right )} a \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - {\left (3 \, A - B\right )} a \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{16 \, {\left (d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - d \cos \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 152, normalized size = 1.52 \[ \frac {2 \, {\left (3 \, A a - B a\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 2 \, {\left (3 \, A a - B a\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (3 \, A a \sin \left (d x + c\right ) - B a \sin \left (d x + c\right ) + 5 \, A a - 3 \, B a\right )}}{\sin \left (d x + c\right ) + 1} + \frac {9 \, A a \sin \left (d x + c\right )^{2} - 3 \, B a \sin \left (d x + c\right )^{2} - 26 \, A a \sin \left (d x + c\right ) + 6 \, B a \sin \left (d x + c\right ) + 21 \, A a + B a}{{\left (\sin \left (d x + c\right ) - 1\right )}^{2}}}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.60, size = 173, normalized size = 1.73 \[ \frac {a A}{4 d \cos \left (d x +c \right )^{4}}+\frac {a B \left (\sin ^{3}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}+\frac {a B \left (\sin ^{3}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}+\frac {a B \sin \left (d x +c \right )}{8 d}-\frac {a B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {a A \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{4 d}+\frac {3 a A \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {3 a A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {a B}{4 d \cos \left (d x +c \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 115, normalized size = 1.15 \[ \frac {{\left (3 \, A - B\right )} a \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (3 \, A - B\right )} a \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left ({\left (3 \, A - B\right )} a \sin \left (d x + c\right )^{2} - {\left (3 \, A - B\right )} a \sin \left (d x + c\right ) - 2 \, {\left (A + B\right )} a\right )}}{\sin \left (d x + c\right )^{3} - \sin \left (d x + c\right )^{2} - \sin \left (d x + c\right ) + 1}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 98, normalized size = 0.98 \[ \frac {a\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )\,\left (3\,A-B\right )}{8\,d}-\frac {\left (\frac {B\,a}{8}-\frac {3\,A\,a}{8}\right )\,{\sin \left (c+d\,x\right )}^2+\left (\frac {3\,A\,a}{8}-\frac {B\,a}{8}\right )\,\sin \left (c+d\,x\right )+\frac {A\,a}{4}+\frac {B\,a}{4}}{d\,\left (-{\sin \left (c+d\,x\right )}^3+{\sin \left (c+d\,x\right )}^2+\sin \left (c+d\,x\right )-1\right )} \]
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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