Optimal. Leaf size=372 \[ -\frac{b \left (-12 a^2 A b^2+3 a^4 A+2 a^3 b B+22 a b^3 B-15 A b^4\right )}{8 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}-\frac{b^4 \left (-5 a^2 B+6 a A b-b^2 B\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^4}-\frac{\left (3 a^2 A+2 a b (6 A+B)+b^2 (15 A+8 B)\right ) \log (1-\sin (c+d x))}{16 d (a+b)^4}+\frac{\left (3 a^2 A-2 a b (6 A-B)+b^2 (15 A-8 B)\right ) \log (\sin (c+d x)+1)}{16 d (a-b)^4}-\frac{\sec ^4(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))}+\frac{\sec ^2(c+d x) \left (\left (3 a^3 A+2 a^2 b B-9 a A b^2+4 b^3 B\right ) \sin (c+d x)+b \left (a^2 A-6 a b B+5 A b^2\right )\right )}{8 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))} \]
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Rubi [A] time = 0.564163, antiderivative size = 372, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {2837, 823, 801} \[ -\frac{b \left (-12 a^2 A b^2+3 a^4 A+2 a^3 b B+22 a b^3 B-15 A b^4\right )}{8 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}-\frac{b^4 \left (-5 a^2 B+6 a A b-b^2 B\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^4}-\frac{\left (3 a^2 A+2 a b (6 A+B)+b^2 (15 A+8 B)\right ) \log (1-\sin (c+d x))}{16 d (a+b)^4}+\frac{\left (3 a^2 A-2 a b (6 A-B)+b^2 (15 A-8 B)\right ) \log (\sin (c+d x)+1)}{16 d (a-b)^4}-\frac{\sec ^4(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))}+\frac{\sec ^2(c+d x) \left (\left (3 a^3 A+2 a^2 b B-9 a A b^2+4 b^3 B\right ) \sin (c+d x)+b \left (a^2 A-6 a b B+5 A b^2\right )\right )}{8 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 823
Rule 801
Rubi steps
\begin{align*} \int \frac{\sec ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx &=\frac{b^5 \operatorname{Subst}\left (\int \frac{A+\frac{B x}{b}}{(a+x)^2 \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac{\sec ^4(c+d x) (A b-a B-(a A-b B) \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac{b^3 \operatorname{Subst}\left (\int \frac{-3 a^2 A+5 A b^2-2 a b B-4 (a A-b B) x}{(a+x)^2 \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 \left (a^2-b^2\right ) d}\\ &=-\frac{\sec ^4(c+d x) (A b-a B-(a A-b B) \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac{\sec ^2(c+d x) \left (b \left (a^2 A+5 A b^2-6 a b B\right )+\left (3 a^3 A-9 a A b^2+2 a^2 b B+4 b^3 B\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac{b \operatorname{Subst}\left (\int \frac{3 a^4 A-6 a^2 A b^2+15 A b^4+2 a^3 b B-14 a b^3 B+2 \left (3 a^3 A-9 a A b^2+2 a^2 b B+4 b^3 B\right ) x}{(a+x)^2 \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d}\\ &=-\frac{\sec ^4(c+d x) (A b-a B-(a A-b B) \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac{\sec ^2(c+d x) \left (b \left (a^2 A+5 A b^2-6 a b B\right )+\left (3 a^3 A-9 a A b^2+2 a^2 b B+4 b^3 B\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac{b \operatorname{Subst}\left (\int \left (\frac{(a-b)^2 \left (3 a^2 A+2 a b (6 A+B)+b^2 (15 A+8 B)\right )}{2 b (a+b)^2 (b-x)}+\frac{3 a^4 A-12 a^2 A b^2-15 A b^4+2 a^3 b B+22 a b^3 B}{\left (a^2-b^2\right ) (a+x)^2}+\frac{8 b^3 \left (-6 a A b+5 a^2 B+b^2 B\right )}{\left (-a^2+b^2\right )^2 (a+x)}+\frac{(a+b)^2 \left (3 a^2 A+b^2 (15 A-8 B)-2 a b (6 A-B)\right )}{2 (a-b)^2 b (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d}\\ &=-\frac{\left (3 a^2 A+2 a b (6 A+B)+b^2 (15 A+8 B)\right ) \log (1-\sin (c+d x))}{16 (a+b)^4 d}+\frac{\left (3 a^2 A+b^2 (15 A-8 B)-2 a b (6 A-B)\right ) \log (1+\sin (c+d x))}{16 (a-b)^4 d}-\frac{b^4 \left (6 a A b-5 a^2 B-b^2 B\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^4 d}-\frac{b \left (3 a^4 A-12 a^2 A b^2-15 A b^4+2 a^3 b B+22 a b^3 B\right )}{8 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac{\sec ^4(c+d x) (A b-a B-(a A-b B) \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac{\sec ^2(c+d x) \left (b \left (a^2 A+5 A b^2-6 a b B\right )+\left (3 a^3 A-9 a A b^2+2 a^2 b B+4 b^3 B\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 4.12523, size = 370, normalized size = 0.99 \[ \frac{-\frac{\left (3 a^3 A+2 a^2 b B-9 a A b^2+4 b^3 B\right ) ((a-b) \log (1-\sin (c+d x))-(a+b) \log (\sin (c+d x)+1)+2 b \log (a+b \sin (c+d x)))}{(a-b) (a+b)}+b \left (12 a^2 A b^2-3 a^4 A-2 a^3 b B-22 a b^3 B+15 A b^4\right ) \left (\frac{1}{\left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac{\log (1-\sin (c+d x))}{2 b (a+b)^2}+\frac{\log (\sin (c+d x)+1)}{2 b (a-b)^2}-\frac{2 a \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}\right )+\frac{2 \left (b^2-a^2\right ) \sec ^4(c+d x) ((b B-a A) \sin (c+d x)-a B+A b)}{a+b \sin (c+d x)}+\frac{\sec ^2(c+d x) \left (\left (3 a^3 A+2 a^2 b B-9 a A b^2+4 b^3 B\right ) \sin (c+d x)+b \left (a^2 A-6 a b B+5 A b^2\right )\right )}{a+b \sin (c+d x)}}{8 d \left (a^2-b^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.175, size = 675, normalized size = 1.8 \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 = 1.07899, size = 890, normalized size = 2.39 \begin{align*} \frac{\frac{16 \,{\left (5 \, B a^{2} b^{4} - 6 \, A a b^{5} + B b^{6}\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{{\left (3 \, A a^{2} - 2 \,{\left (6 \, A - B\right )} a b +{\left (15 \, A - 8 \, B\right )} b^{2}\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{{\left (3 \, A a^{2} + 2 \,{\left (6 \, A + B\right )} a b +{\left (15 \, A + 8 \, B\right )} b^{2}\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 (2 \, B a^{5} - 4 \, A a^{4} b - 12 \, B a^{3} b^{2} + 20 \, A a^{2} b^{3} - 14 \, B a b^{4} + 8 \, A b^{5} -{\left (3 \, A a^{4} b + 2 \, B a^{3} b^{2} - 12 \, A a^{2} b^{3} + 22 \, B a b^{4} - 15 \, A b^{5}\right )} \sin \left (d x + c\right )^{4} -{\left (3 \, A a^{5} + 2 \, B a^{4} b - 12 \, A a^{3} b^{2} + 2 \, B a^{2} b^{3} + 9 \, A a b^{4} - 4 \, B b^{5}\right )} \sin \left (d x + c\right )^{3} +{\left (5 \, A a^{4} b + 10 \, B a^{3} b^{2} - 28 \, A a^{2} b^{3} + 38 \, B a b^{4} - 25 \, A b^{5}\right )} \sin \left (d x + c\right )^{2} +{\left (5 \, A a^{5} - 16 \, A a^{3} b^{2} + 6 \, B a^{2} b^{3} + 11 \, A a b^{4} - 6 \, B b^{5}\right )} \sin \left (d x + c\right )\right )}}{a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6} +{\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \sin \left (d x + c\right )^{5} +{\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \sin \left (d x + c\right )^{4} - 2 \,{\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \sin \left (d x + c\right )^{3} - 2 \,{\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \sin \left (d x + c\right )^{2} +{\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \sin \left (d x + c\right )}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 21.0228, size = 1997, normalized size = 5.37 \begin{align*} \text{result too large to display} \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 = 1.35944, size = 1027, normalized size = 2.76 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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