Optimal. Leaf size=263 \[ -\frac{b^4 (A b-a B) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^3}-\frac{\left (3 a^2 A+a b (9 A+B)+b^2 (8 A+3 B)\right ) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac{\left (3 a^2 A-a b (9 A-B)+b^2 (8 A-3 B)\right ) \log (\sin (c+d x)+1)}{16 d (a-b)^3}-\frac{\sec ^4(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{4 d \left (a^2-b^2\right )}+\frac{\sec ^2(c+d x) \left (\left (3 a^3 A+a^2 b B-7 a A b^2+3 b^3 B\right ) \sin (c+d x)+4 b^2 (A b-a B)\right )}{8 d \left (a^2-b^2\right )^2} \]
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Rubi [A] time = 0.447782, antiderivative size = 263, 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} \[ -\frac{b^4 (A b-a B) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^3}-\frac{\left (3 a^2 A+a b (9 A+B)+b^2 (8 A+3 B)\right ) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac{\left (3 a^2 A-a b (9 A-B)+b^2 (8 A-3 B)\right ) \log (\sin (c+d x)+1)}{16 d (a-b)^3}-\frac{\sec ^4(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{4 d \left (a^2-b^2\right )}+\frac{\sec ^2(c+d x) \left (\left (3 a^3 A+a^2 b B-7 a A b^2+3 b^3 B\right ) \sin (c+d x)+4 b^2 (A b-a B)\right )}{8 d \left (a^2-b^2\right )^2} \]
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)} \, dx &=\frac{b^5 \operatorname{Subst}\left (\int \frac{A+\frac{B x}{b}}{(a+x) \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}-\frac{b^3 \operatorname{Subst}\left (\int \frac{-3 a^2 A+4 A b^2-a b B-3 (a A-b B) x}{(a+x) \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}+\frac{\sec ^2(c+d x) \left (4 b^2 (A b-a B)+\left (3 a^3 A-7 a A b^2+a^2 b B+3 b^3 B\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d}+\frac{b \operatorname{Subst}\left (\int \frac{3 a^4 A-7 a^2 A b^2+8 A b^4+a^3 b B-5 a b^3 B+\left (3 a^3 A-7 a A b^2+a^2 b B+3 b^3 B\right ) x}{(a+x) \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}+\frac{\sec ^2(c+d x) \left (4 b^2 (A b-a B)+\left (3 a^3 A-7 a A b^2+a^2 b B+3 b^3 B\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d}+\frac{b \operatorname{Subst}\left (\int \left (\frac{(a-b)^2 \left (3 a^2 A+a b (9 A+B)+b^2 (8 A+3 B)\right )}{2 b (a+b) (b-x)}+\frac{8 b^3 (-A b+a B)}{(a-b) (a+b) (a+x)}+\frac{(a+b)^2 \left (3 a^2 A+b^2 (8 A-3 B)-a b (9 A-B)\right )}{2 (a-b) 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+a b (9 A+B)+b^2 (8 A+3 B)\right ) \log (1-\sin (c+d x))}{16 (a+b)^3 d}+\frac{\left (3 a^2 A+b^2 (8 A-3 B)-a b (9 A-B)\right ) \log (1+\sin (c+d x))}{16 (a-b)^3 d}-\frac{b^4 (A b-a B) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^3 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}+\frac{\sec ^2(c+d x) \left (4 b^2 (A b-a B)+\left (3 a^3 A-7 a A b^2+a^2 b B+3 b^3 B\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d}\\ \end{align*}
Mathematica [A] time = 1.27897, size = 321, normalized size = 1.22 \[ \frac{\frac{16 b^4 (A b-a B) \log (a+b \sin (c+d x))}{\left (b^2-a^2\right )^3}-\frac{2 \left (3 a^2 A+a b (9 A+B)+b^2 (8 A+3 B)\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{(a+b)^3}+\frac{2 \left (3 a^2 A+a b (B-9 A)+b^2 (8 A-3 B)\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{(a-b)^3}+\frac{3 a A+a B+5 A b+3 b B}{(a+b)^2 \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{-3 a A+a B+5 A b-3 b B}{(a-b)^2 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{A+B}{(a+b) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^4}+\frac{B-A}{(a-b) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4}}{16 d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.106, size = 586, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01621, size = 495, normalized size = 1.88 \begin{align*} \frac{\frac{16 \,{\left (B a b^{4} - A b^{5}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac{{\left (3 \, A a^{2} -{\left (9 \, A - B\right )} a b +{\left (8 \, A - 3 \, B\right )} b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac{{\left (3 \, A a^{2} +{\left (9 \, A + B\right )} a b +{\left (8 \, A + 3 \, B\right )} b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac{2 \,{\left (2 \, B a^{3} - 2 \, A a^{2} b - 6 \, B a b^{2} + 6 \, A b^{3} -{\left (3 \, A a^{3} + B a^{2} b - 7 \, A a b^{2} + 3 \, B b^{3}\right )} \sin \left (d x + c\right )^{3} + 4 \,{\left (B a b^{2} - A b^{3}\right )} \sin \left (d x + c\right )^{2} +{\left (5 \, A a^{3} - B a^{2} b - 9 \, A a b^{2} + 5 \, B b^{3}\right )} \sin \left (d x + c\right )\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{4} + a^{4} - 2 \, a^{2} b^{2} + b^{4} - 2 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{2}}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 9.24453, size = 929, normalized size = 3.53 \begin{align*} \frac{4 \, B a^{5} - 4 \, A a^{4} b - 8 \, B a^{3} b^{2} + 8 \, A a^{2} b^{3} + 4 \, B a b^{4} - 4 \, A b^{5} + 16 \,{\left (B a b^{4} - A b^{5}\right )} \cos \left (d x + c\right )^{4} \log \left (b \sin \left (d x + c\right ) + a\right ) +{\left (3 \, A a^{5} + B a^{4} b - 10 \, A a^{3} b^{2} - 6 \, B a^{2} b^{3} +{\left (15 \, A - 8 \, B\right )} a b^{4} +{\left (8 \, A - 3 \, B\right )} b^{5}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (3 \, A a^{5} + B a^{4} b - 10 \, A a^{3} b^{2} - 6 \, B a^{2} b^{3} +{\left (15 \, A + 8 \, B\right )} a b^{4} -{\left (8 \, A + 3 \, B\right )} b^{5}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 8 \,{\left (B a^{3} b^{2} - A a^{2} b^{3} - B a b^{4} + A b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (2 \, A a^{5} - 2 \, B a^{4} b - 4 \, A a^{3} b^{2} + 4 \, B a^{2} b^{3} + 2 \, A a b^{4} - 2 \, B b^{5} +{\left (3 \, A a^{5} + B a^{4} b - 10 \, A a^{3} b^{2} + 2 \, B a^{2} b^{3} + 7 \, A a b^{4} - 3 \, B b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \,{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} d \cos \left (d x + c\right )^{4}} \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 = 1.29397, size = 728, normalized size = 2.77 \begin{align*} \frac{\frac{16 \,{\left (B a b^{5} - A b^{6}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}} - \frac{{\left (3 \, A a^{2} + 9 \, A a b + B a b + 8 \, A b^{2} + 3 \, B b^{2}\right )} \log \left ({\left | -\sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac{{\left (3 \, A a^{2} - 9 \, A a b + B a b + 8 \, A b^{2} - 3 \, B b^{2}\right )} \log \left ({\left | -\sin \left (d x + c\right ) - 1 \right |}\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac{2 \,{\left (6 \, B a b^{4} \sin \left (d x + c\right )^{4} - 6 \, A b^{5} \sin \left (d x + c\right )^{4} - 3 \, A a^{5} \sin \left (d x + c\right )^{3} - B a^{4} b \sin \left (d x + c\right )^{3} + 10 \, A a^{3} b^{2} \sin \left (d x + c\right )^{3} - 2 \, B a^{2} b^{3} \sin \left (d x + c\right )^{3} - 7 \, A a b^{4} \sin \left (d x + c\right )^{3} + 3 \, B b^{5} \sin \left (d x + c\right )^{3} + 4 \, B a^{3} b^{2} \sin \left (d x + c\right )^{2} - 4 \, A a^{2} b^{3} \sin \left (d x + c\right )^{2} - 16 \, B a b^{4} \sin \left (d x + c\right )^{2} + 16 \, A b^{5} \sin \left (d x + c\right )^{2} + 5 \, A a^{5} \sin \left (d x + c\right ) - B a^{4} b \sin \left (d x + c\right ) - 14 \, A a^{3} b^{2} \sin \left (d x + c\right ) + 6 \, B a^{2} b^{3} \sin \left (d x + c\right ) + 9 \, A a b^{4} \sin \left (d x + c\right ) - 5 \, B b^{5} \sin \left (d x + c\right ) + 2 \, B a^{5} - 2 \, A a^{4} b - 8 \, B a^{3} b^{2} + 8 \, A a^{2} b^{3} + 12 \, B a b^{4} - 12 \, A b^{5}\right )}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )}{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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