Integrand size = 31, antiderivative size = 263 \[ \int \frac {\sec ^5(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx=-\frac {\left (3 a^2 A+a b (9 A+B)+b^2 (8 A+3 B)\right ) \log (1-\sin (c+d x))}{16 (a+b)^3 d}+\frac {\left (3 a^2 A+b^2 (8 A-3 B)-a b (9 A-B)\right ) \log (1+\sin (c+d x))}{16 (a-b)^3 d}-\frac {b^4 (A b-a B) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^3 d}-\frac {\sec ^4(c+d x) (A b-a B-(a A-b B) \sin (c+d x))}{4 \left (a^2-b^2\right ) d}+\frac {\sec ^2(c+d x) \left (4 b^2 (A b-a B)+\left (3 a^3 A-7 a A b^2+a^2 b B+3 b^3 B\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d} \]
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Time = 0.33 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {2916, 837, 815} \[ \int \frac {\sec ^5(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx=-\frac {\left (3 a^2 A+a b (9 A+B)+b^2 (8 A+3 B)\right ) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac {\left (3 a^2 A-a b (9 A-B)+b^2 (8 A-3 B)\right ) \log (\sin (c+d x)+1)}{16 d (a-b)^3}-\frac {\sec ^4(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{4 d \left (a^2-b^2\right )}-\frac {b^4 (A b-a B) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^3}+\frac {\sec ^2(c+d x) \left (\left (3 a^3 A+a^2 b B-7 a A b^2+3 b^3 B\right ) \sin (c+d x)+4 b^2 (A b-a B)\right )}{8 d \left (a^2-b^2\right )^2} \]
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Rule 815
Rule 837
Rule 2916
Rubi steps \begin{align*} \text {integral}& = \frac {b^5 \text {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 \text {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 \text {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 \text {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*}
Time = 6.16 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.36 \[ \int \frac {\sec ^5(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx=\frac {b^5 \left (-\frac {\sec ^4(c+d x) \left (-A b^2+a b B-b (-a A+b B) \sin (c+d x)\right )}{4 b^6 \left (-a^2+b^2\right )}+\frac {\frac {-\frac {(a-b)^2 \left (3 a^2 A+a b (9 A+B)+b^2 (8 A+3 B)\right ) \log (1-\sin (c+d x))}{2 b (a+b)}+\frac {(a+b)^2 \left (3 a^2 A+b^2 (8 A-3 B)-a b (9 A-B)\right ) \log (1+\sin (c+d x))}{2 (a-b) b}-\frac {8 b^3 (A b-a B) \log (a+b \sin (c+d x))}{(a-b) (a+b)}}{2 b^2 \left (-a^2+b^2\right )}-\frac {\sec ^2(c+d x) \left (-3 a b^2 (a A-b B)-b^2 \left (-3 a^2 A+4 A b^2-a b B\right )-b \left (-3 b^2 (a A-b B)-a \left (-3 a^2 A+4 A b^2-a b B\right )\right ) \sin (c+d x)\right )}{2 b^4 \left (-a^2+b^2\right )}}{4 b^2 \left (-a^2+b^2\right )}\right )}{d} \]
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Time = 1.42 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(\frac {-\frac {A -B}{2 \left (8 a -8 b \right ) \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {3 a A -5 A b -B a +3 B b}{16 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (3 A \,a^{2}-9 A a b +8 A \,b^{2}+B a b -3 B \,b^{2}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{3}}-\frac {\left (A b -B a \right ) b^{4} \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3}}-\frac {-A -B}{2 \left (8 a +8 b \right ) \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {3 a A +5 A b +B a +3 B b}{16 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-3 A \,a^{2}-9 A a b -8 A \,b^{2}-B a b -3 B \,b^{2}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{3}}}{d}\) | \(256\) |
default | \(\frac {-\frac {A -B}{2 \left (8 a -8 b \right ) \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {3 a A -5 A b -B a +3 B b}{16 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (3 A \,a^{2}-9 A a b +8 A \,b^{2}+B a b -3 B \,b^{2}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{3}}-\frac {\left (A b -B a \right ) b^{4} \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3}}-\frac {-A -B}{2 \left (8 a +8 b \right ) \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {3 a A +5 A b +B a +3 B b}{16 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-3 A \,a^{2}-9 A a b -8 A \,b^{2}-B a b -3 B \,b^{2}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{3}}}{d}\) | \(256\) |
parallelrisch | \(\frac {-16 b^{4} \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (A b -B a \right ) \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )-6 \left (\left (\frac {8 A}{3}+B \right ) b^{2}+3 \left (A +\frac {B}{9}\right ) a b +A \,a^{2}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a -b \right )^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+11 \left (a +b \right ) \left (\frac {6 \left (\left (\frac {8 A}{3}-B \right ) b^{2}-3 \left (A -\frac {B}{9}\right ) a b +A \,a^{2}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a +b \right )^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{11}+\left (\frac {4 \left (a -b \right ) \left (a +b \right ) \left (A b -B a \right ) \cos \left (2 d x +2 c \right )}{11}+\frac {\left (a^{2}-3 b^{2}\right ) \left (A b -B a \right ) \cos \left (4 d x +4 c \right )}{11}+\frac {\left (3 A \,a^{3}-7 A a \,b^{2}+B \,a^{2} b +3 B \,b^{3}\right ) \sin \left (3 d x +3 c \right )}{11}+\left (A \,a^{3}-\frac {15}{11} A a \,b^{2}-\frac {7}{11} B \,a^{2} b +B \,b^{3}\right ) \sin \left (d x +c \right )-\frac {5 \left (A b -B a \right ) \left (a^{2}-\frac {7 b^{2}}{5}\right )}{11}\right ) \left (a -b \right )\right )}{4 \left (a -b \right )^{3} \left (a +b \right )^{3} d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(395\) |
norman | \(\frac {\frac {2 \left (a A -B b \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (a^{2}-b^{2}\right )}+\frac {2 \left (a A -B b \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (a^{2}-b^{2}\right )}+\frac {\left (5 A \,a^{3}-9 A a \,b^{2}-B \,a^{2} b +5 B \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {\left (5 A \,a^{3}-9 A a \,b^{2}-B \,a^{2} b +5 B \,b^{3}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (2 A \,a^{2} b -4 A \,b^{3}-2 B \,a^{3}+4 B a \,b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (2 A \,a^{2} b -4 A \,b^{3}-2 B \,a^{3}+4 B a \,b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {\left (3 A \,a^{3}+A a \,b^{2}-7 B \,a^{2} b +3 B \,b^{3}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {2 \left (A \,a^{2} b -B \,a^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {2 \left (A \,a^{2} b -B \,a^{3}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\left (3 A \,a^{2}-9 A a b +8 A \,b^{2}+B a b -3 B \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}-\frac {\left (3 A \,a^{2}+9 A a b +8 A \,b^{2}+B a b +3 B \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}-\frac {b^{4} \left (A b -B a \right ) \ln \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}{d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}\) | \(698\) |
risch | \(\text {Expression too large to display}\) | \(1751\) |
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Time = 1.20 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.57 \[ \int \frac {\sec ^5(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx=\frac {4 \, B a^{5} - 4 \, A a^{4} b - 8 \, B a^{3} b^{2} + 8 \, A a^{2} b^{3} + 4 \, B a b^{4} - 4 \, A b^{5} + 16 \, {\left (B a b^{4} - A b^{5}\right )} \cos \left (d x + c\right )^{4} \log \left (b \sin \left (d x + c\right ) + a\right ) + {\left (3 \, A a^{5} + B a^{4} b - 10 \, A a^{3} b^{2} - 6 \, B a^{2} b^{3} + {\left (15 \, A - 8 \, B\right )} a b^{4} + {\left (8 \, A - 3 \, B\right )} b^{5}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (3 \, A a^{5} + B a^{4} b - 10 \, A a^{3} b^{2} - 6 \, B a^{2} b^{3} + {\left (15 \, A + 8 \, B\right )} a b^{4} - {\left (8 \, A + 3 \, B\right )} b^{5}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 8 \, {\left (B a^{3} b^{2} - A a^{2} b^{3} - B a b^{4} + A b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (2 \, A a^{5} - 2 \, B a^{4} b - 4 \, A a^{3} b^{2} + 4 \, B a^{2} b^{3} + 2 \, A a b^{4} - 2 \, B b^{5} + {\left (3 \, A a^{5} + B a^{4} b - 10 \, A a^{3} b^{2} + 2 \, B a^{2} b^{3} + 7 \, A a b^{4} - 3 \, B b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} d \cos \left (d x + c\right )^{4}} \]
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\[ \int \frac {\sec ^5(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx=\int \frac {\left (A + B \sin {\left (c + d x \right )}\right ) \sec ^{5}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
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Time = 0.32 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.40 \[ \int \frac {\sec ^5(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx=\frac {\frac {16 \, {\left (B a b^{4} - A b^{5}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac {{\left (3 \, A a^{2} - {\left (9 \, A - B\right )} a b + {\left (8 \, A - 3 \, B\right )} b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {{\left (3 \, A a^{2} + {\left (9 \, A + B\right )} a b + {\left (8 \, A + 3 \, B\right )} b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {2 \, {\left (2 \, B a^{3} - 2 \, A a^{2} b - 6 \, B a b^{2} + 6 \, A b^{3} - {\left (3 \, A a^{3} + B a^{2} b - 7 \, A a b^{2} + 3 \, B b^{3}\right )} \sin \left (d x + c\right )^{3} + 4 \, {\left (B a b^{2} - A b^{3}\right )} \sin \left (d x + c\right )^{2} + {\left (5 \, A a^{3} - B a^{2} b - 9 \, A a b^{2} + 5 \, B b^{3}\right )} \sin \left (d x + c\right )\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{4} + a^{4} - 2 \, a^{2} b^{2} + b^{4} - 2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{2}}}{16 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 539 vs. \(2 (254) = 508\).
Time = 0.37 (sec) , antiderivative size = 539, normalized size of antiderivative = 2.05 \[ \int \frac {\sec ^5(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx=\frac {\frac {16 \, {\left (B a b^{5} - A b^{6}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}} - \frac {{\left (3 \, A a^{2} + 9 \, A a b + B a b + 8 \, A b^{2} + 3 \, B b^{2}\right )} \log \left ({\left | -\sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {{\left (3 \, A a^{2} - 9 \, A a b + B a b + 8 \, A b^{2} - 3 \, B b^{2}\right )} \log \left ({\left | -\sin \left (d x + c\right ) - 1 \right |}\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac {2 \, {\left (6 \, B a b^{4} \sin \left (d x + c\right )^{4} - 6 \, A b^{5} \sin \left (d x + c\right )^{4} - 3 \, A a^{5} \sin \left (d x + c\right )^{3} - B a^{4} b \sin \left (d x + c\right )^{3} + 10 \, A a^{3} b^{2} \sin \left (d x + c\right )^{3} - 2 \, B a^{2} b^{3} \sin \left (d x + c\right )^{3} - 7 \, A a b^{4} \sin \left (d x + c\right )^{3} + 3 \, B b^{5} \sin \left (d x + c\right )^{3} + 4 \, B a^{3} b^{2} \sin \left (d x + c\right )^{2} - 4 \, A a^{2} b^{3} \sin \left (d x + c\right )^{2} - 16 \, B a b^{4} \sin \left (d x + c\right )^{2} + 16 \, A b^{5} \sin \left (d x + c\right )^{2} + 5 \, A a^{5} \sin \left (d x + c\right ) - B a^{4} b \sin \left (d x + c\right ) - 14 \, A a^{3} b^{2} \sin \left (d x + c\right ) + 6 \, B a^{2} b^{3} \sin \left (d x + c\right ) + 9 \, A a b^{4} \sin \left (d x + c\right ) - 5 \, B b^{5} \sin \left (d x + c\right ) + 2 \, B a^{5} - 2 \, A a^{4} b - 8 \, B a^{3} b^{2} + 8 \, A a^{2} b^{3} + 12 \, B a b^{4} - 12 \, A b^{5}\right )}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \]
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Time = 12.79 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.62 \[ \int \frac {\sec ^5(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx=\frac {\frac {B\,a^3-A\,a^2\,b-3\,B\,a\,b^2+3\,A\,b^3}{4\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {\sin \left (c+d\,x\right )\,\left (5\,A\,a^3-B\,a^2\,b-9\,A\,a\,b^2+5\,B\,b^3\right )}{8\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (A\,b^3-B\,a\,b^2\right )}{2\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\sin \left (c+d\,x\right )}^3\,\left (3\,A\,a^3+B\,a^2\,b-7\,A\,a\,b^2+3\,B\,b^3\right )}{8\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left ({\cos \left (c+d\,x\right )}^2+{\sin \left (c+d\,x\right )}^4-{\sin \left (c+d\,x\right )}^2\right )}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (3\,A\,a^2+\left (9\,A+B\right )\,a\,b+\left (8\,A+3\,B\right )\,b^2\right )}{d\,\left (16\,a^3+48\,a^2\,b+48\,a\,b^2+16\,b^3\right )}-\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (A\,b^5-B\,a\,b^4\right )}{d\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (3\,A\,a^2+\left (B-9\,A\right )\,a\,b+\left (8\,A-3\,B\right )\,b^2\right )}{d\,\left (16\,a^3-48\,a^2\,b+48\,a\,b^2-16\,b^3\right )} \]
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