Optimal. Leaf size=383 \[ \frac{b^6 (A b-a B) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^4}-\frac{\left (a^2 b (20 A+B)+5 a^3 A+a b^2 (29 A+4 B)+b^3 (16 A+5 B)\right ) \log (1-\sin (c+d x))}{32 d (a+b)^4}+\frac{\left (-a^2 b (20 A-B)+5 a^3 A+a b^2 (29 A-4 B)-b^3 (16 A-5 B)\right ) \log (\sin (c+d x)+1)}{32 d (a-b)^4}-\frac{\sec ^6(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{6 d \left (a^2-b^2\right )}+\frac{\sec ^4(c+d x) \left (\left (5 a^3 A+a^2 b B-11 a A b^2+5 b^3 B\right ) \sin (c+d x)+6 b^2 (A b-a B)\right )}{24 d \left (a^2-b^2\right )^2}-\frac{\sec ^2(c+d x) \left (8 b^4 (A b-a B)-\left (-16 a^3 A b^2+5 a^5 A-4 a^2 b^3 B+a^4 b B+19 a A b^4-5 b^5 B\right ) \sin (c+d x)\right )}{16 d \left (a^2-b^2\right )^3} \]
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Rubi [A] time = 0.680853, antiderivative size = 383, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {2837, 823, 801} \[ \frac{b^6 (A b-a B) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^4}-\frac{\left (a^2 b (20 A+B)+5 a^3 A+a b^2 (29 A+4 B)+b^3 (16 A+5 B)\right ) \log (1-\sin (c+d x))}{32 d (a+b)^4}+\frac{\left (-a^2 b (20 A-B)+5 a^3 A+a b^2 (29 A-4 B)-b^3 (16 A-5 B)\right ) \log (\sin (c+d x)+1)}{32 d (a-b)^4}-\frac{\sec ^6(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{6 d \left (a^2-b^2\right )}+\frac{\sec ^4(c+d x) \left (\left (5 a^3 A+a^2 b B-11 a A b^2+5 b^3 B\right ) \sin (c+d x)+6 b^2 (A b-a B)\right )}{24 d \left (a^2-b^2\right )^2}-\frac{\sec ^2(c+d x) \left (8 b^4 (A b-a B)-\left (-16 a^3 A b^2+5 a^5 A-4 a^2 b^3 B+a^4 b B+19 a A b^4-5 b^5 B\right ) \sin (c+d x)\right )}{16 d \left (a^2-b^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 823
Rule 801
Rubi steps
\begin{align*} \int \frac{\sec ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx &=\frac{b^7 \operatorname{Subst}\left (\int \frac{A+\frac{B x}{b}}{(a+x) \left (b^2-x^2\right )^4} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac{\sec ^6(c+d x) (A b-a B-(a A-b B) \sin (c+d x))}{6 \left (a^2-b^2\right ) d}-\frac{b^5 \operatorname{Subst}\left (\int \frac{-5 a^2 A+6 A b^2-a b B-5 (a A-b B) x}{(a+x) \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{6 \left (a^2-b^2\right ) d}\\ &=-\frac{\sec ^6(c+d x) (A b-a B-(a A-b B) \sin (c+d x))}{6 \left (a^2-b^2\right ) d}+\frac{\sec ^4(c+d x) \left (6 b^2 (A b-a B)+\left (5 a^3 A-11 a A b^2+a^2 b B+5 b^3 B\right ) \sin (c+d x)\right )}{24 \left (a^2-b^2\right )^2 d}+\frac{b^3 \operatorname{Subst}\left (\int \frac{3 \left (5 a^4 A-11 a^2 A b^2+8 A b^4+a^3 b B-3 a b^3 B\right )+3 \left (5 a^3 A-11 a A b^2+a^2 b B+5 b^3 B\right ) x}{(a+x) \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{24 \left (a^2-b^2\right )^2 d}\\ &=-\frac{\sec ^6(c+d x) (A b-a B-(a A-b B) \sin (c+d x))}{6 \left (a^2-b^2\right ) d}+\frac{\sec ^4(c+d x) \left (6 b^2 (A b-a B)+\left (5 a^3 A-11 a A b^2+a^2 b B+5 b^3 B\right ) \sin (c+d x)\right )}{24 \left (a^2-b^2\right )^2 d}-\frac{\sec ^2(c+d x) \left (8 b^4 (A b-a B)-\left (5 a^5 A-16 a^3 A b^2+19 a A b^4+a^4 b B-4 a^2 b^3 B-5 b^5 B\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^3 d}-\frac{b \operatorname{Subst}\left (\int \frac{-3 \left (5 a^6 A-16 a^4 A b^2+19 a^2 A b^4-16 A b^6+a^5 b B-4 a^3 b^3 B+11 a b^5 B\right )-3 \left (5 a^5 A-16 a^3 A b^2+19 a A b^4+a^4 b B-4 a^2 b^3 B-5 b^5 B\right ) x}{(a+x) \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{48 \left (a^2-b^2\right )^3 d}\\ &=-\frac{\sec ^6(c+d x) (A b-a B-(a A-b B) \sin (c+d x))}{6 \left (a^2-b^2\right ) d}+\frac{\sec ^4(c+d x) \left (6 b^2 (A b-a B)+\left (5 a^3 A-11 a A b^2+a^2 b B+5 b^3 B\right ) \sin (c+d x)\right )}{24 \left (a^2-b^2\right )^2 d}-\frac{\sec ^2(c+d x) \left (8 b^4 (A b-a B)-\left (5 a^5 A-16 a^3 A b^2+19 a A b^4+a^4 b B-4 a^2 b^3 B-5 b^5 B\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^3 d}-\frac{b \operatorname{Subst}\left (\int \left (\frac{3 (a-b)^3 \left (-5 a^3 A-a^2 b (20 A+B)-a b^2 (29 A+4 B)-b^3 (16 A+5 B)\right )}{2 b (a+b) (b-x)}+\frac{48 b^5 (-A b+a B)}{(a-b) (a+b) (a+x)}+\frac{3 (a+b)^3 \left (-5 a^3 A+b^3 (16 A-5 B)-a b^2 (29 A-4 B)+a^2 b (20 A-B)\right )}{2 (a-b) b (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{48 \left (a^2-b^2\right )^3 d}\\ &=-\frac{\left (5 a^3 A+a^2 b (20 A+B)+a b^2 (29 A+4 B)+b^3 (16 A+5 B)\right ) \log (1-\sin (c+d x))}{32 (a+b)^4 d}+\frac{\left (5 a^3 A-b^3 (16 A-5 B)+a b^2 (29 A-4 B)-a^2 b (20 A-B)\right ) \log (1+\sin (c+d x))}{32 (a-b)^4 d}+\frac{b^6 (A b-a B) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^4 d}-\frac{\sec ^6(c+d x) (A b-a B-(a A-b B) \sin (c+d x))}{6 \left (a^2-b^2\right ) d}+\frac{\sec ^4(c+d x) \left (6 b^2 (A b-a B)+\left (5 a^3 A-11 a A b^2+a^2 b B+5 b^3 B\right ) \sin (c+d x)\right )}{24 \left (a^2-b^2\right )^2 d}-\frac{\sec ^2(c+d x) \left (8 b^4 (A b-a B)-\left (5 a^5 A-16 a^3 A b^2+19 a A b^4+a^4 b B-4 a^2 b^3 B-5 b^5 B\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^3 d}\\ \end{align*}
Mathematica [A] time = 2.59314, size = 565, normalized size = 1.48 \[ \frac{\frac{768 b^6 (A b-a B) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^4}-\frac{48 \left (a^2 b (20 A+B)+5 a^3 A+a b^2 (29 A+4 B)+b^3 (16 A+5 B)\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{(a+b)^4}+\frac{48 \left (a^2 b (B-20 A)+5 a^3 A+a b^2 (29 A-4 B)+b^3 (5 B-16 A)\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{(a-b)^4}+\frac{\sec ^6(c+d x) \left (-96 b^2 \left (a^2-3 b^2\right ) (a B-A b) \cos (2 (c+d x))-480 a^3 A b^2 \sin (c+d x)-272 a^3 A b^2 \sin (3 (c+d x))-48 a^3 A b^2 \sin (5 (c+d x))+352 a^2 A b^3-128 a^4 A b+198 a^5 A \sin (c+d x)+85 a^5 A \sin (3 (c+d x))+15 a^5 A \sin (5 (c+d x))+264 a^2 b^3 B \sin (c+d x)-4 a^2 b^3 B \sin (3 (c+d x))-12 a^2 b^3 B \sin (5 (c+d x))-352 a^3 b^2 B-114 a^4 b B \sin (c+d x)+17 a^4 b B \sin (3 (c+d x))+3 a^4 b B \sin (5 (c+d x))+128 a^5 B-48 b^4 (A b-a B) \cos (4 (c+d x))+330 a A b^4 \sin (c+d x)+259 a A b^4 \sin (3 (c+d x))+57 a A b^4 \sin (5 (c+d x))+368 a b^4 B-368 A b^5-198 b^5 B \sin (c+d x)-85 b^5 B \sin (3 (c+d x))-15 b^5 B \sin (5 (c+d x))\right )}{\left (a^2-b^2\right )^3}}{768 d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.109, size = 990, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11085, size = 853, normalized size = 2.23 \begin{align*} -\frac{\frac{96 \,{\left (B a b^{6} - A b^{7}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} - \frac{3 \,{\left (5 \, A a^{3} -{\left (20 \, A - B\right )} a^{2} b +{\left (29 \, A - 4 \, B\right )} a b^{2} -{\left (16 \, A - 5 \, B\right )} b^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} + \frac{3 \,{\left (5 \, A a^{3} +{\left (20 \, A + B\right )} a^{2} b +{\left (29 \, A + 4 \, B\right )} a b^{2} +{\left (16 \, A + 5 \, B\right )} b^{3}\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac{2 \,{\left (8 \, B a^{5} - 8 \, A a^{4} b - 28 \, B a^{3} b^{2} + 28 \, A a^{2} b^{3} + 44 \, B a b^{4} - 44 \, A b^{5} + 3 \,{\left (5 \, A a^{5} + B a^{4} b - 16 \, A a^{3} b^{2} - 4 \, B a^{2} b^{3} + 19 \, A a b^{4} - 5 \, B b^{5}\right )} \sin \left (d x + c\right )^{5} + 24 \,{\left (B a b^{4} - A b^{5}\right )} \sin \left (d x + c\right )^{4} - 8 \,{\left (5 \, A a^{5} + B a^{4} b - 16 \, A a^{3} b^{2} - 2 \, B a^{2} b^{3} + 17 \, A a b^{4} - 5 \, B b^{5}\right )} \sin \left (d x + c\right )^{3} + 12 \,{\left (B a^{3} b^{2} - A a^{2} b^{3} - 5 \, B a b^{4} + 5 \, A b^{5}\right )} \sin \left (d x + c\right )^{2} + 3 \,{\left (11 \, A a^{5} - B a^{4} b - 32 \, A a^{3} b^{2} + 4 \, B a^{2} b^{3} + 29 \, A a b^{4} - 11 \, B b^{5}\right )} \sin \left (d x + c\right )\right )}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sin \left (d x + c\right )^{6} - a^{6} + 3 \, a^{4} b^{2} - 3 \, a^{2} b^{4} + b^{6} - 3 \,{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sin \left (d x + c\right )^{4} + 3 \,{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sin \left (d x + c\right )^{2}}}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 23.3608, size = 1454, normalized size = 3.8 \begin{align*} \frac{16 \, B a^{7} - 16 \, A a^{6} b - 48 \, B a^{5} b^{2} + 48 \, A a^{4} b^{3} + 48 \, B a^{3} b^{4} - 48 \, A a^{2} b^{5} - 16 \, B a b^{6} + 16 \, A b^{7} - 96 \,{\left (B a b^{6} - A b^{7}\right )} \cos \left (d x + c\right )^{6} \log \left (b \sin \left (d x + c\right ) + a\right ) + 3 \,{\left (5 \, A a^{7} + B a^{6} b - 21 \, A a^{5} b^{2} - 5 \, B a^{4} b^{3} + 35 \, A a^{3} b^{4} + 15 \, B a^{2} b^{5} -{\left (35 \, A - 16 \, B\right )} a b^{6} -{\left (16 \, A - 5 \, B\right )} b^{7}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (5 \, A a^{7} + B a^{6} b - 21 \, A a^{5} b^{2} - 5 \, B a^{4} b^{3} + 35 \, A a^{3} b^{4} + 15 \, B a^{2} b^{5} -{\left (35 \, A + 16 \, B\right )} a b^{6} +{\left (16 \, A + 5 \, B\right )} b^{7}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 48 \,{\left (B a^{3} b^{4} - A a^{2} b^{5} - B a b^{6} + A b^{7}\right )} \cos \left (d x + c\right )^{4} - 24 \,{\left (B a^{5} b^{2} - A a^{4} b^{3} - 2 \, B a^{3} b^{4} + 2 \, A a^{2} b^{5} + B a b^{6} - A b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (8 \, A a^{7} - 8 \, B a^{6} b - 24 \, A a^{5} b^{2} + 24 \, B a^{4} b^{3} + 24 \, A a^{3} b^{4} - 24 \, B a^{2} b^{5} - 8 \, A a b^{6} + 8 \, B b^{7} + 3 \,{\left (5 \, A a^{7} + B a^{6} b - 21 \, A a^{5} b^{2} - 5 \, B a^{4} b^{3} + 35 \, A a^{3} b^{4} - B a^{2} b^{5} - 19 \, A a b^{6} + 5 \, B b^{7}\right )} \cos \left (d x + c\right )^{4} + 2 \,{\left (5 \, A a^{7} + B a^{6} b - 21 \, A a^{5} b^{2} + 3 \, B a^{4} b^{3} + 27 \, A a^{3} b^{4} - 9 \, B a^{2} b^{5} - 11 \, A a b^{6} + 5 \, B b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{96 \,{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} d \cos \left (d x + c\right )^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.45496, size = 1224, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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