Integrand size = 31, antiderivative size = 372 \[ \int \frac {\sec ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=-\frac {\left (3 a^2 A+2 a b (6 A+B)+b^2 (15 A+8 B)\right ) \log (1-\sin (c+d x))}{16 (a+b)^4 d}+\frac {\left (3 a^2 A+b^2 (15 A-8 B)-2 a b (6 A-B)\right ) \log (1+\sin (c+d x))}{16 (a-b)^4 d}-\frac {b^4 \left (6 a A b-5 a^2 B-b^2 B\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^4 d}-\frac {b \left (3 a^4 A-12 a^2 A b^2-15 A b^4+2 a^3 b B+22 a b^3 B\right )}{8 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac {\sec ^4(c+d x) (A b-a B-(a A-b B) \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\sec ^2(c+d x) \left (b \left (a^2 A+5 A b^2-6 a b B\right )+\left (3 a^3 A-9 a A b^2+2 a^2 b B+4 b^3 B\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))} \]
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Time = 0.41 (sec) , antiderivative size = 372, 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} \[ \int \frac {\sec ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=-\frac {\left (3 a^2 A+2 a b (6 A+B)+b^2 (15 A+8 B)\right ) \log (1-\sin (c+d x))}{16 d (a+b)^4}+\frac {\left (3 a^2 A-2 a b (6 A-B)+b^2 (15 A-8 B)\right ) \log (\sin (c+d x)+1)}{16 d (a-b)^4}-\frac {\sec ^4(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {b^4 \left (-5 a^2 B+6 a A b-b^2 B\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^4}+\frac {\sec ^2(c+d x) \left (b \left (a^2 A-6 a b B+5 A b^2\right )+\left (3 a^3 A+2 a^2 b B-9 a A b^2+4 b^3 B\right ) \sin (c+d x)\right )}{8 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))}-\frac {b \left (3 a^4 A+2 a^3 b B-12 a^2 A b^2+22 a b^3 B-15 A b^4\right )}{8 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))} \]
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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)^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 \text {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 \text {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 \text {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*}
Time = 2.72 (sec) , antiderivative size = 370, normalized size of antiderivative = 0.99 \[ \int \frac {\sec ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\frac {-\frac {\left (3 a^3 A-9 a A b^2+2 a^2 b B+4 b^3 B\right ) ((a-b) \log (1-\sin (c+d x))-(a+b) \log (1+\sin (c+d x))+2 b \log (a+b \sin (c+d x)))}{(a-b) (a+b)}+\frac {2 \left (-a^2+b^2\right ) \sec ^4(c+d x) (A b-a B+(-a A+b B) \sin (c+d x))}{a+b \sin (c+d x)}+\frac {\sec ^2(c+d x) \left (b \left (a^2 A+5 A b^2-6 a b B\right )+\left (3 a^3 A-9 a A b^2+2 a^2 b B+4 b^3 B\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}+b \left (-3 a^4 A+12 a^2 A b^2+15 A b^4-2 a^3 b B-22 a b^3 B\right ) \left (-\frac {\log (1-\sin (c+d x))}{2 b (a+b)^2}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^2 b}-\frac {2 a \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}+\frac {1}{\left (a^2-b^2\right ) (a+b \sin (c+d x))}\right )}{8 \left (a^2-b^2\right )^2 d} \]
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Time = 4.37 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\frac {-\frac {b^{4} \left (6 A a b -5 B \,a^{2}-B \,b^{2}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}+\frac {\left (A b -B a \right ) b^{4}}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )}-\frac {A -B}{16 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {3 a A -7 A b -B a +5 B b}{16 \left (a -b \right )^{3} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (3 A \,a^{2}-12 A a b +15 A \,b^{2}+2 B a b -8 B \,b^{2}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{4}}-\frac {-A -B}{16 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {3 a A +7 A b +B a +5 B b}{16 \left (a +b \right )^{3} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-3 A \,a^{2}-12 A a b -15 A \,b^{2}-2 B a b -8 B \,b^{2}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{4}}}{d}\) | \(297\) |
default | \(\frac {-\frac {b^{4} \left (6 A a b -5 B \,a^{2}-B \,b^{2}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}+\frac {\left (A b -B a \right ) b^{4}}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )}-\frac {A -B}{16 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {3 a A -7 A b -B a +5 B b}{16 \left (a -b \right )^{3} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (3 A \,a^{2}-12 A a b +15 A \,b^{2}+2 B a b -8 B \,b^{2}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{4}}-\frac {-A -B}{16 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {3 a A +7 A b +B a +5 B b}{16 \left (a +b \right )^{3} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-3 A \,a^{2}-12 A a b -15 A \,b^{2}-2 B a b -8 B \,b^{2}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{4}}}{d}\) | \(297\) |
parallelrisch | \(\frac {-6 b^{4} \left (A a b -\frac {5}{6} B \,a^{2}-\frac {1}{6} B \,b^{2}\right ) a \left (\frac {b \sin \left (5 d x +5 c \right )}{2}+\frac {3 b \sin \left (3 d x +3 c \right )}{2}+b \sin \left (d x +c \right )+\cos \left (4 d x +4 c \right ) a +4 a \cos \left (2 d x +2 c \right )+3 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 )-\frac {3 \left (A \,a^{2}+4 \left (A +\frac {B}{6}\right ) b a +5 b^{2} \left (A +\frac {8 B}{15}\right )\right ) a \left (\frac {b \sin \left (5 d x +5 c \right )}{2}+\frac {3 b \sin \left (3 d x +3 c \right )}{2}+b \sin \left (d x +c \right )+\cos \left (4 d x +4 c \right ) a +4 a \cos \left (2 d x +2 c \right )+3 a \right ) \left (a -b \right )^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8}+\frac {3 \left (\left (A \,a^{2}-4 b \left (A -\frac {B}{6}\right ) a +5 b^{2} \left (A -\frac {8 B}{15}\right )\right ) a \left (\frac {b \sin \left (5 d x +5 c \right )}{2}+\frac {3 b \sin \left (3 d x +3 c \right )}{2}+b \sin \left (d x +c \right )+\cos \left (4 d x +4 c \right ) a +4 a \cos \left (2 d x +2 c \right )+3 a \right ) \left (a +b \right )^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\frac {22 \left (\frac {4 a \left (a -b \right )^{2} \left (a +b \right )^{2} \left (A b -B a \right ) \cos \left (2 d x +2 c \right )}{11}+\frac {a \left (a -b \right ) \left (a +b \right ) \left (A \,a^{2} b -7 A \,b^{3}-2 B \,a^{3}+8 B a \,b^{2}\right ) \cos \left (4 d x +4 c \right )}{22}+\left (-\frac {9}{11} A \,a^{4} b^{2}-\frac {6}{11} A \,a^{2} b^{4}-\frac {6}{11} A \,b^{6}+\frac {13}{22} B a \,b^{5}+\frac {3}{11} A \,a^{6}+\frac {1}{22} B \,a^{5} b +B \,a^{3} b^{3}\right ) \sin \left (3 d x +3 c \right )+\frac {\left (A \,a^{4} b -5 A \,a^{2} b^{3}-2 A \,b^{5}-\frac {1}{2} B \,a^{5}+3 B \,a^{3} b^{2}+\frac {7}{2} B a \,b^{4}\right ) b \sin \left (5 d x +5 c \right )}{11}+\left (-\frac {5}{11} B a \,b^{5}-\frac {26}{11} A \,a^{4} b^{2}+\frac {7}{11} A \,a^{2} b^{4}+A \,a^{6}-\frac {7}{11} B \,a^{5} b +\frac {24}{11} B \,a^{3} b^{3}-\frac {4}{11} A \,b^{6}\right ) \sin \left (d x +c \right )-\frac {9 \left (A \,a^{2} b -\frac {5}{3} A \,b^{3}-\frac {10}{9} B \,a^{3}+\frac {16}{9} B a \,b^{2}\right ) a \left (a +b \right ) \left (a -b \right )}{22}\right ) \left (a -b \right )}{3}\right ) \left (a +b \right )}{8}}{d a \left (\frac {b \sin \left (5 d x +5 c \right )}{2}+\frac {3 b \sin \left (3 d x +3 c \right )}{2}+b \sin \left (d x +c \right )+\cos \left (4 d x +4 c \right ) a +4 a \cos \left (2 d x +2 c \right )+3 a \right ) \left (a +b \right )^{4} \left (a -b \right )^{4}}\) | \(708\) |
norman | \(\text {Expression too large to display}\) | \(1149\) |
risch | \(\text {Expression too large to display}\) | \(2657\) |
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Leaf count of result is larger than twice the leaf count of optimal. 881 vs. \(2 (359) = 718\).
Time = 1.82 (sec) , antiderivative size = 881, normalized size of antiderivative = 2.37 \[ \int \frac {\sec ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\frac {4 \, B a^{7} - 4 \, A a^{6} b - 12 \, B a^{5} b^{2} + 12 \, A a^{4} b^{3} + 12 \, B a^{3} b^{4} - 12 \, A a^{2} b^{5} - 4 \, B a b^{6} + 4 \, A b^{7} - 2 \, {\left (3 \, A a^{6} b + 2 \, B a^{5} b^{2} - 15 \, A a^{4} b^{3} + 20 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5} - 22 \, B a b^{6} + 15 \, A b^{7}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (A a^{6} b - 6 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3} + 12 \, B a^{3} b^{4} - 9 \, A a^{2} b^{5} - 6 \, B a b^{6} + 5 \, A b^{7}\right )} \cos \left (d x + c\right )^{2} + 16 \, {\left ({\left (5 \, B a^{2} b^{5} - 6 \, A a b^{6} + B b^{7}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + {\left (5 \, B a^{3} b^{4} - 6 \, A a^{2} b^{5} + B a b^{6}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + {\left ({\left (3 \, A a^{6} b + 2 \, B a^{5} b^{2} - 15 \, A a^{4} b^{3} - 20 \, B a^{3} b^{4} + 5 \, {\left (9 \, A - 8 \, B\right )} a^{2} b^{5} + 6 \, {\left (8 \, A - 5 \, B\right )} a b^{6} + {\left (15 \, A - 8 \, B\right )} b^{7}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + {\left (3 \, A a^{7} + 2 \, B a^{6} b - 15 \, A a^{5} b^{2} - 20 \, B a^{4} b^{3} + 5 \, {\left (9 \, A - 8 \, B\right )} a^{3} b^{4} + 6 \, {\left (8 \, A - 5 \, B\right )} a^{2} b^{5} + {\left (15 \, A - 8 \, B\right )} a b^{6}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left ({\left (3 \, A a^{6} b + 2 \, B a^{5} b^{2} - 15 \, A a^{4} b^{3} - 20 \, B a^{3} b^{4} + 5 \, {\left (9 \, A + 8 \, B\right )} a^{2} b^{5} - 6 \, {\left (8 \, A + 5 \, B\right )} a b^{6} + {\left (15 \, A + 8 \, B\right )} b^{7}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + {\left (3 \, A a^{7} + 2 \, B a^{6} b - 15 \, A a^{5} b^{2} - 20 \, B a^{4} b^{3} + 5 \, {\left (9 \, A + 8 \, B\right )} a^{3} b^{4} - 6 \, {\left (8 \, A + 5 \, B\right )} a^{2} b^{5} + {\left (15 \, A + 8 \, B\right )} a b^{6}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, A a^{7} - 2 \, B a^{6} b - 6 \, A a^{5} b^{2} + 6 \, B a^{4} b^{3} + 6 \, A a^{3} b^{4} - 6 \, B a^{2} b^{5} - 2 \, A a b^{6} + 2 \, B b^{7} + {\left (3 \, A a^{7} + 2 \, B a^{6} b - 15 \, A a^{5} b^{2} + 21 \, A a^{3} b^{4} - 6 \, B a^{2} b^{5} - 9 \, A a b^{6} + 4 \, B b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, {\left ({\left (a^{8} b - 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} - 4 \, a^{2} b^{7} + b^{9}\right )} d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + {\left (a^{9} - 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} - 4 \, a^{3} b^{6} + a b^{8}\right )} d \cos \left (d x + c\right )^{4}\right )}} \]
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\[ \int \frac {\sec ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\int \frac {\left (A + B \sin {\left (c + d x \right )}\right ) \sec ^{5}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{2}}\, dx \]
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Time = 0.24 (sec) , antiderivative size = 659, normalized size of antiderivative = 1.77 \[ \int \frac {\sec ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\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} \]
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Leaf count of result is larger than twice the leaf count of optimal. 761 vs. \(2 (359) = 718\).
Time = 0.41 (sec) , antiderivative size = 761, 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))^2} \, dx=\frac {\frac {16 \, {\left (5 \, B a^{2} b^{5} - 6 \, A a b^{6} + B b^{7}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{8} b - 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} - 4 \, a^{2} b^{7} + b^{9}} - \frac {{\left (3 \, A a^{2} + 12 \, A a b + 2 \, B a b + 15 \, A b^{2} + 8 \, B b^{2}\right )} \log \left ({\left | -\sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {{\left (3 \, A a^{2} - 12 \, A a b + 2 \, B a b + 15 \, A b^{2} - 8 \, B b^{2}\right )} \log \left ({\left | -\sin \left (d x + c\right ) - 1 \right |}\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} - \frac {16 \, {\left (5 \, B a^{2} b^{5} \sin \left (d x + c\right ) - 6 \, A a b^{6} \sin \left (d x + c\right ) + B b^{7} \sin \left (d x + c\right ) + 6 \, B a^{3} b^{4} - 7 \, A a^{2} b^{5} + A b^{7}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}} + \frac {2 \, {\left (30 \, B a^{2} b^{4} \sin \left (d x + c\right )^{4} - 36 \, A a b^{5} \sin \left (d x + c\right )^{4} + 6 \, B b^{6} \sin \left (d x + c\right )^{4} - 3 \, A a^{6} \sin \left (d x + c\right )^{3} - 2 \, B a^{5} b \sin \left (d x + c\right )^{3} + 15 \, A a^{4} b^{2} \sin \left (d x + c\right )^{3} - 12 \, B a^{3} b^{3} \sin \left (d x + c\right )^{3} - 5 \, A a^{2} b^{4} \sin \left (d x + c\right )^{3} + 14 \, B a b^{5} \sin \left (d x + c\right )^{3} - 7 \, A b^{6} \sin \left (d x + c\right )^{3} + 12 \, B a^{4} b^{2} \sin \left (d x + c\right )^{2} - 16 \, A a^{3} b^{3} \sin \left (d x + c\right )^{2} - 68 \, B a^{2} b^{4} \sin \left (d x + c\right )^{2} + 88 \, A a b^{5} \sin \left (d x + c\right )^{2} - 16 \, B b^{6} \sin \left (d x + c\right )^{2} + 5 \, A a^{6} \sin \left (d x + c\right ) - 2 \, B a^{5} b \sin \left (d x + c\right ) - 17 \, A a^{4} b^{2} \sin \left (d x + c\right ) + 20 \, B a^{3} b^{3} \sin \left (d x + c\right ) + 3 \, A a^{2} b^{4} \sin \left (d x + c\right ) - 18 \, B a b^{5} \sin \left (d x + c\right ) + 9 \, A b^{6} \sin \left (d x + c\right ) + 2 \, B a^{6} - 4 \, A a^{5} b - 14 \, B a^{4} b^{2} + 24 \, A a^{3} b^{3} + 36 \, B a^{2} b^{4} - 56 \, A a b^{5} + 12 \, B b^{6}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \]
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Time = 12.91 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.65 \[ \int \frac {\sec ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {B\,a^5-2\,A\,a^4\,b-6\,B\,a^3\,b^2+10\,A\,a^2\,b^3-7\,B\,a\,b^4+4\,A\,b^5}{4\,\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\sin \left (c+d\,x\right )}^4\,\left (3\,A\,a^4\,b+2\,B\,a^3\,b^2-12\,A\,a^2\,b^3+22\,B\,a\,b^4-15\,A\,b^5\right )}{8\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {\sin \left (c+d\,x\right )\,\left (5\,A\,a^3-11\,A\,a\,b^2+6\,B\,b^3\right )}{8\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\sin \left (c+d\,x\right )}^3\,\left (3\,A\,a^3+2\,B\,a^2\,b-9\,A\,a\,b^2+4\,B\,b^3\right )}{8\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {{\sin \left (c+d\,x\right )}^2\,\left (5\,A\,a^4\,b+10\,B\,a^3\,b^2-28\,A\,a^2\,b^3+38\,B\,a\,b^4-25\,A\,b^5\right )}{8\,\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left (b\,{\sin \left (c+d\,x\right )}^5+a\,{\sin \left (c+d\,x\right )}^4-2\,b\,{\sin \left (c+d\,x\right )}^3-2\,a\,{\sin \left (c+d\,x\right )}^2+b\,\sin \left (c+d\,x\right )+a\right )}+\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (5\,B\,a^2\,b^4-6\,A\,a\,b^5+B\,b^6\right )}{d\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (3\,A\,a^2+\left (12\,A+2\,B\right )\,a\,b+\left (15\,A+8\,B\right )\,b^2\right )}{d\,\left (16\,a^4+64\,a^3\,b+96\,a^2\,b^2+64\,a\,b^3+16\,b^4\right )}+\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (3\,A\,a^2+\left (2\,B-12\,A\right )\,a\,b+\left (15\,A-8\,B\right )\,b^2\right )}{d\,\left (16\,a^4-64\,a^3\,b+96\,a^2\,b^2-64\,a\,b^3+16\,b^4\right )} \]
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