Integrand size = 29, antiderivative size = 185 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {2 a b \csc (c+d x)}{d}-\frac {a^2 \csc ^2(c+d x)}{2 d}-\frac {\left (12 a^2+15 a b+4 b^2\right ) \log (1-\sin (c+d x))}{8 d}+\frac {\left (3 a^2+b^2\right ) \log (\sin (c+d x))}{d}-\frac {\left (12 a^2-15 a b+4 b^2\right ) \log (1+\sin (c+d x))}{8 d}+\frac {\sec ^4(c+d x) \left (a^2+b^2+2 a b \sin (c+d x)\right )}{4 d}+\frac {\sec ^2(c+d x) \left (2 \left (2 a^2+b^2\right )+7 a b \sin (c+d x)\right )}{4 d} \]
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Time = 0.27 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2916, 12, 1819, 1816} \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {\left (12 a^2+15 a b+4 b^2\right ) \log (1-\sin (c+d x))}{8 d}+\frac {\left (3 a^2+b^2\right ) \log (\sin (c+d x))}{d}-\frac {\left (12 a^2-15 a b+4 b^2\right ) \log (\sin (c+d x)+1)}{8 d}+\frac {\sec ^4(c+d x) \left (a^2+2 a b \sin (c+d x)+b^2\right )}{4 d}+\frac {\sec ^2(c+d x) \left (2 \left (2 a^2+b^2\right )+7 a b \sin (c+d x)\right )}{4 d}-\frac {a^2 \csc ^2(c+d x)}{2 d}-\frac {2 a b \csc (c+d x)}{d} \]
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Rule 12
Rule 1816
Rule 1819
Rule 2916
Rubi steps \begin{align*} \text {integral}& = \frac {b^5 \text {Subst}\left (\int \frac {b^3 (a+x)^2}{x^3 \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d} \\ & = \frac {b^8 \text {Subst}\left (\int \frac {(a+x)^2}{x^3 \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d} \\ & = \frac {\sec ^4(c+d x) \left (a^2+b^2+2 a b \sin (c+d x)\right )}{4 d}-\frac {b^6 \text {Subst}\left (\int \frac {-4 a^2-8 a x-4 \left (1+\frac {a^2}{b^2}\right ) x^2-\frac {6 a x^3}{b^2}}{x^3 \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 d} \\ & = \frac {\sec ^4(c+d x) \left (a^2+b^2+2 a b \sin (c+d x)\right )}{4 d}+\frac {\sec ^2(c+d x) \left (2 \left (2 a^2+b^2\right )+7 a b \sin (c+d x)\right )}{4 d}+\frac {b^4 \text {Subst}\left (\int \frac {8 a^2+16 a x+8 \left (1+\frac {2 a^2}{b^2}\right ) x^2+\frac {14 a x^3}{b^2}}{x^3 \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{8 d} \\ & = \frac {\sec ^4(c+d x) \left (a^2+b^2+2 a b \sin (c+d x)\right )}{4 d}+\frac {\sec ^2(c+d x) \left (2 \left (2 a^2+b^2\right )+7 a b \sin (c+d x)\right )}{4 d}+\frac {b^4 \text {Subst}\left (\int \left (\frac {12 a^2+15 a b+4 b^2}{b^4 (b-x)}+\frac {8 a^2}{b^2 x^3}+\frac {16 a}{b^2 x^2}+\frac {8 \left (3 a^2+b^2\right )}{b^4 x}+\frac {-12 a^2+15 a b-4 b^2}{b^4 (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{8 d} \\ & = -\frac {2 a b \csc (c+d x)}{d}-\frac {a^2 \csc ^2(c+d x)}{2 d}-\frac {\left (12 a^2+15 a b+4 b^2\right ) \log (1-\sin (c+d x))}{8 d}+\frac {\left (3 a^2+b^2\right ) \log (\sin (c+d x))}{d}-\frac {\left (12 a^2-15 a b+4 b^2\right ) \log (1+\sin (c+d x))}{8 d}+\frac {\sec ^4(c+d x) \left (a^2+b^2+2 a b \sin (c+d x)\right )}{4 d}+\frac {\sec ^2(c+d x) \left (2 \left (2 a^2+b^2\right )+7 a b \sin (c+d x)\right )}{4 d} \\ \end{align*}
Time = 3.69 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.98 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {-32 a b \csc (c+d x)-8 a^2 \csc ^2(c+d x)-2 \left (12 a^2+15 a b+4 b^2\right ) \log (1-\sin (c+d x))+16 \left (3 a^2+b^2\right ) \log (\sin (c+d x))-2 \left (12 a^2-15 a b+4 b^2\right ) \log (1+\sin (c+d x))+\frac {(a+b)^2}{(-1+\sin (c+d x))^2}-\frac {(a+b) (9 a+5 b)}{-1+\sin (c+d x)}+\frac {(a-b)^2}{(1+\sin (c+d x))^2}+\frac {(9 a-5 b) (a-b)}{1+\sin (c+d x)}}{16 d} \]
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Time = 1.18 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {a^{2} \left (\frac {1}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}+\frac {3}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )^{2}}+3 \ln \left (\tan \left (d x +c \right )\right )\right )+2 a b \left (\frac {1}{4 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{4}}+\frac {5}{8 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {15}{8 \sin \left (d x +c \right )}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+b^{2} \left (\frac {1}{4 \cos \left (d x +c \right )^{4}}+\frac {1}{2 \cos \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(165\) |
default | \(\frac {a^{2} \left (\frac {1}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}+\frac {3}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )^{2}}+3 \ln \left (\tan \left (d x +c \right )\right )\right )+2 a b \left (\frac {1}{4 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{4}}+\frac {5}{8 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {15}{8 \sin \left (d x +c \right )}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+b^{2} \left (\frac {1}{4 \cos \left (d x +c \right )^{4}}+\frac {1}{2 \cos \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(165\) |
parallelrisch | \(\frac {-384 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a^{2}+\frac {5}{4} a b +\frac {1}{3} b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-384 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a^{2}-\frac {5}{4} a b +\frac {1}{3} b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+384 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a^{2}+\frac {b^{2}}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-27 \left (\cos \left (2 d x +2 c \right )+\frac {2 \cos \left (4 d x +4 c \right )}{9}-\frac {\cos \left (6 d x +6 c \right )}{9}+\frac {2}{27}\right ) \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-160 b \left (\cos \left (2 d x +2 c \right )+\frac {3 \cos \left (4 d x +4 c \right )}{8}+\frac {9}{40}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) a \csc \left (\frac {d x}{2}+\frac {c}{2}\right )-32 b^{2} \left (\frac {3 \cos \left (4 d x +4 c \right )}{4}-\frac {7}{4}+\cos \left (2 d x +2 c \right )\right )}{32 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(303\) |
risch | \(-\frac {i \left (12 i a^{2} {\mathrm e}^{10 i \left (d x +c \right )}+4 i b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+15 a b \,{\mathrm e}^{11 i \left (d x +c \right )}+24 i a^{2} {\mathrm e}^{8 i \left (d x +c \right )}+4 i b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+25 a b \,{\mathrm e}^{9 i \left (d x +c \right )}+8 i b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+24 i a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-22 a b \,{\mathrm e}^{7 i \left (d x +c \right )}+8 i b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-8 i a^{2} {\mathrm e}^{6 i \left (d x +c \right )}+22 a b \,{\mathrm e}^{5 i \left (d x +c \right )}+12 i a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-24 i b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-25 a b \,{\mathrm e}^{3 i \left (d x +c \right )}-15 a b \,{\mathrm e}^{i \left (d x +c \right )}\right )}{2 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a b}{4 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b^{2}}{d}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}+\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a b}{4 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b^{2}}{d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{2}}{d}\) | \(426\) |
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Time = 0.28 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.54 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {4 \, {\left (3 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (3 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - 2 \, b^{2} + 8 \, {\left ({\left (3 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{6} - {\left (3 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - {\left ({\left (12 \, a^{2} - 15 \, a b + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - {\left (12 \, a^{2} - 15 \, a b + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left ({\left (12 \, a^{2} + 15 \, a b + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - {\left (12 \, a^{2} + 15 \, a b + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (15 \, a b \cos \left (d x + c\right )^{4} - 5 \, a b \cos \left (d x + c\right )^{2} - 2 \, a b\right )} \sin \left (d x + c\right )}{8 \, {\left (d \cos \left (d x + c\right )^{6} - d \cos \left (d x + c\right )^{4}\right )}} \]
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Timed out. \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.99 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {{\left (12 \, a^{2} - 15 \, a b + 4 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (12 \, a^{2} + 15 \, a b + 4 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) - 8 \, {\left (3 \, a^{2} + b^{2}\right )} \log \left (\sin \left (d x + c\right )\right ) + \frac {2 \, {\left (15 \, a b \sin \left (d x + c\right )^{5} - 25 \, a b \sin \left (d x + c\right )^{3} + 2 \, {\left (3 \, a^{2} + b^{2}\right )} \sin \left (d x + c\right )^{4} + 8 \, a b \sin \left (d x + c\right ) - 3 \, {\left (3 \, a^{2} + b^{2}\right )} \sin \left (d x + c\right )^{2} + 2 \, a^{2}\right )}}{\sin \left (d x + c\right )^{6} - 2 \, \sin \left (d x + c\right )^{4} + \sin \left (d x + c\right )^{2}}}{8 \, d} \]
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Time = 0.40 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.03 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {{\left (12 \, a^{2} - 15 \, a b + 4 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + {\left (12 \, a^{2} + 15 \, a b + 4 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - 8 \, {\left (3 \, a^{2} + b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + \frac {2 \, {\left (15 \, a b \sin \left (d x + c\right )^{5} + 6 \, a^{2} \sin \left (d x + c\right )^{4} + 2 \, b^{2} \sin \left (d x + c\right )^{4} - 25 \, a b \sin \left (d x + c\right )^{3} - 9 \, a^{2} \sin \left (d x + c\right )^{2} - 3 \, b^{2} \sin \left (d x + c\right )^{2} + 8 \, a b \sin \left (d x + c\right ) + 2 \, a^{2}\right )}}{{\left (\sin \left (d x + c\right )^{3} - \sin \left (d x + c\right )\right )}^{2}}}{8 \, d} \]
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Time = 12.45 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.05 \[ \int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {\ln \left (\sin \left (c+d\,x\right )\right )\,\left (3\,a^2+b^2\right )}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (\frac {3\,a^2}{2}-\frac {15\,a\,b}{8}+\frac {b^2}{2}\right )}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (\frac {3\,a^2}{2}+\frac {15\,a\,b}{8}+\frac {b^2}{2}\right )}{d}-\frac {\frac {a^2}{2}+{\sin \left (c+d\,x\right )}^4\,\left (\frac {3\,a^2}{2}+\frac {b^2}{2}\right )-{\sin \left (c+d\,x\right )}^2\,\left (\frac {9\,a^2}{4}+\frac {3\,b^2}{4}\right )+2\,a\,b\,\sin \left (c+d\,x\right )-\frac {25\,a\,b\,{\sin \left (c+d\,x\right )}^3}{4}+\frac {15\,a\,b\,{\sin \left (c+d\,x\right )}^5}{4}}{d\,\left ({\sin \left (c+d\,x\right )}^6-2\,{\sin \left (c+d\,x\right )}^4+{\sin \left (c+d\,x\right )}^2\right )} \]
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