Optimal. Leaf size=157 \[ \frac {a^4 (A+B)}{24 d (a-a \sin (c+d x))^3}+\frac {a^3 (3 A+B)}{32 d (a-a \sin (c+d x))^2}-\frac {a^3 (A-B)}{32 d (a \sin (c+d x)+a)^2}-\frac {a^2 (2 A-B)}{16 d (a \sin (c+d x)+a)}+\frac {3 a^2 A}{16 d (a-a \sin (c+d x))}+\frac {a (5 A-B) \tanh ^{-1}(\sin (c+d x))}{16 d} \]
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Rubi [A] time = 0.17, antiderivative size = 157, 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} \[ \frac {a^4 (A+B)}{24 d (a-a \sin (c+d x))^3}+\frac {a^3 (3 A+B)}{32 d (a-a \sin (c+d x))^2}-\frac {a^3 (A-B)}{32 d (a \sin (c+d x)+a)^2}-\frac {a^2 (2 A-B)}{16 d (a \sin (c+d x)+a)}+\frac {3 a^2 A}{16 d (a-a \sin (c+d x))}+\frac {a (5 A-B) \tanh ^{-1}(\sin (c+d x))}{16 d} \]
Antiderivative was successfully verified.
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Rule 77
Rule 206
Rule 2836
Rubi steps
\begin {align*} \int \sec ^7(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx &=\frac {a^7 \operatorname {Subst}\left (\int \frac {A+\frac {B x}{a}}{(a-x)^4 (a+x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^7 \operatorname {Subst}\left (\int \left (\frac {A+B}{8 a^3 (a-x)^4}+\frac {3 A+B}{16 a^4 (a-x)^3}+\frac {3 A}{16 a^5 (a-x)^2}+\frac {A-B}{16 a^4 (a+x)^3}+\frac {2 A-B}{16 a^5 (a+x)^2}+\frac {5 A-B}{16 a^5 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^4 (A+B)}{24 d (a-a \sin (c+d x))^3}+\frac {a^3 (3 A+B)}{32 d (a-a \sin (c+d x))^2}+\frac {3 a^2 A}{16 d (a-a \sin (c+d x))}-\frac {a^3 (A-B)}{32 d (a+a \sin (c+d x))^2}-\frac {a^2 (2 A-B)}{16 d (a+a \sin (c+d x))}+\frac {\left (a^2 (5 A-B)\right ) \operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{16 d}\\ &=\frac {a (5 A-B) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {a^4 (A+B)}{24 d (a-a \sin (c+d x))^3}+\frac {a^3 (3 A+B)}{32 d (a-a \sin (c+d x))^2}+\frac {3 a^2 A}{16 d (a-a \sin (c+d x))}-\frac {a^3 (A-B)}{32 d (a+a \sin (c+d x))^2}-\frac {a^2 (2 A-B)}{16 d (a+a \sin (c+d x))}\\ \end {align*}
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Mathematica [C] time = 1.75, size = 451, normalized size = 2.87 \[ \frac {a (\sin (c+d x)+1) \left (3 i x (5 A-B) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^4+\frac {3 (3 A+B) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^4}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {4 (A+B) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^4}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^6}-\frac {6 (2 A-B) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}{d}-\frac {6 (5 A-B) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {3 (5 A-B) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^4 \log \left (\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2\right )}{d}-\frac {6 i (5 A-B) \tan ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^4}{d}+\frac {3 (B-A)}{d}+\frac {18 A \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^4}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}\right )}{96 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 222, normalized size = 1.41 \[ -\frac {6 \, {\left (5 \, A - B\right )} a \cos \left (d x + c\right )^{4} - 2 \, {\left (5 \, A - B\right )} a \cos \left (d x + c\right )^{2} - 4 \, {\left (A - 5 \, B\right )} a - 3 \, {\left ({\left (5 \, A - B\right )} a \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - {\left (5 \, A - B\right )} a \cos \left (d x + c\right )^{4}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left ({\left (5 \, A - B\right )} a \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - {\left (5 \, A - B\right )} a \cos \left (d x + c\right )^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (3 \, {\left (5 \, A - B\right )} a \cos \left (d x + c\right )^{2} + 2 \, {\left (5 \, A - B\right )} a\right )} \sin \left (d x + c\right )}{96 \, {\left (d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - d \cos \left (d x + c\right )^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 201, normalized size = 1.28 \[ \frac {6 \, {\left (5 \, A a - B a\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 6 \, {\left (5 \, A a - B a\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {3 \, {\left (15 \, A a \sin \left (d x + c\right )^{2} - 3 \, B a \sin \left (d x + c\right )^{2} + 38 \, A a \sin \left (d x + c\right ) - 10 \, B a \sin \left (d x + c\right ) + 25 \, A a - 9 \, B a\right )}}{{\left (\sin \left (d x + c\right ) + 1\right )}^{2}} + \frac {55 \, A a \sin \left (d x + c\right )^{3} - 11 \, B a \sin \left (d x + c\right )^{3} - 201 \, A a \sin \left (d x + c\right )^{2} + 33 \, B a \sin \left (d x + c\right )^{2} + 255 \, A a \sin \left (d x + c\right ) - 27 \, B a \sin \left (d x + c\right ) - 117 \, A a - 3 \, B a}{{\left (\sin \left (d x + c\right ) - 1\right )}^{3}}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.60, size = 217, normalized size = 1.38 \[ \frac {a A}{6 d \cos \left (d x +c \right )^{6}}+\frac {a B \left (\sin ^{3}\left (d x +c \right )\right )}{6 d \cos \left (d x +c \right )^{6}}+\frac {a B \left (\sin ^{3}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{4}}+\frac {a B \left (\sin ^{3}\left (d x +c \right )\right )}{16 d \cos \left (d x +c \right )^{2}}+\frac {a B \sin \left (d x +c \right )}{16 d}-\frac {a B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16 d}+\frac {a A \tan \left (d x +c \right ) \left (\sec ^{5}\left (d x +c \right )\right )}{6 d}+\frac {5 a A \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{24 d}+\frac {5 a A \sec \left (d x +c \right ) \tan \left (d x +c \right )}{16 d}+\frac {5 a A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16 d}+\frac {a B}{6 d \cos \left (d x +c \right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 171, normalized size = 1.09 \[ \frac {3 \, {\left (5 \, A - B\right )} a \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (5 \, A - B\right )} a \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (3 \, {\left (5 \, A - B\right )} a \sin \left (d x + c\right )^{4} - 3 \, {\left (5 \, A - B\right )} a \sin \left (d x + c\right )^{3} - 5 \, {\left (5 \, A - B\right )} a \sin \left (d x + c\right )^{2} + 5 \, {\left (5 \, A - B\right )} a \sin \left (d x + c\right ) + 8 \, {\left (A + B\right )} a\right )}}{\sin \left (d x + c\right )^{5} - \sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{3} + 2 \, \sin \left (d x + c\right )^{2} + \sin \left (d x + c\right ) - 1}}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.26, size = 155, normalized size = 0.99 \[ \frac {a\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )\,\left (5\,A-B\right )}{16\,d}-\frac {\left (\frac {5\,A\,a}{16}-\frac {B\,a}{16}\right )\,{\sin \left (c+d\,x\right )}^4+\left (\frac {B\,a}{16}-\frac {5\,A\,a}{16}\right )\,{\sin \left (c+d\,x\right )}^3+\left (\frac {5\,B\,a}{48}-\frac {25\,A\,a}{48}\right )\,{\sin \left (c+d\,x\right )}^2+\left (\frac {25\,A\,a}{48}-\frac {5\,B\,a}{48}\right )\,\sin \left (c+d\,x\right )+\frac {A\,a}{6}+\frac {B\,a}{6}}{d\,\left ({\sin \left (c+d\,x\right )}^5-{\sin \left (c+d\,x\right )}^4-2\,{\sin \left (c+d\,x\right )}^3+2\,{\sin \left (c+d\,x\right )}^2+\sin \left (c+d\,x\right )-1\right )} \]
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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