Integrand size = 21, antiderivative size = 123 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}-\frac {2 \sec ^3(c+d x)}{21 a d (a+a \sin (c+d x))^2}-\frac {2 \sec ^3(c+d x)}{21 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac {8 \tan (c+d x)}{21 a^3 d}+\frac {8 \tan ^3(c+d x)}{63 a^3 d} \]
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Time = 0.10 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2751, 3852} \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {8 \tan ^3(c+d x)}{63 a^3 d}+\frac {8 \tan (c+d x)}{21 a^3 d}-\frac {2 \sec ^3(c+d x)}{21 d \left (a^3 \sin (c+d x)+a^3\right )}-\frac {2 \sec ^3(c+d x)}{21 a d (a \sin (c+d x)+a)^2}-\frac {\sec ^3(c+d x)}{9 d (a \sin (c+d x)+a)^3} \]
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Rule 2751
Rule 3852
Rubi steps \begin{align*} \text {integral}& = -\frac {\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}+\frac {2 \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx}{3 a} \\ & = -\frac {\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}-\frac {2 \sec ^3(c+d x)}{21 a d (a+a \sin (c+d x))^2}+\frac {10 \int \frac {\sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx}{21 a^2} \\ & = -\frac {\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}-\frac {2 \sec ^3(c+d x)}{21 a d (a+a \sin (c+d x))^2}-\frac {2 \sec ^3(c+d x)}{21 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac {8 \int \sec ^4(c+d x) \, dx}{21 a^3} \\ & = -\frac {\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}-\frac {2 \sec ^3(c+d x)}{21 a d (a+a \sin (c+d x))^2}-\frac {2 \sec ^3(c+d x)}{21 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac {8 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{21 a^3 d} \\ & = -\frac {\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}-\frac {2 \sec ^3(c+d x)}{21 a d (a+a \sin (c+d x))^2}-\frac {2 \sec ^3(c+d x)}{21 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac {8 \tan (c+d x)}{21 a^3 d}+\frac {8 \tan ^3(c+d x)}{63 a^3 d} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.69 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\sec ^3(c+d x) (-27 \cos (2 (c+d x))-12 \cos (4 (c+d x))+\cos (6 (c+d x))+36 \sin (c+d x)+2 \sin (3 (c+d x))-6 \sin (5 (c+d x)))}{126 a^3 d (1+\sin (c+d x))^3} \]
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Result contains complex when optimal does not.
Time = 1.30 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.79
method | result | size |
risch | \(\frac {-\frac {32 i}{63}-\frac {64 \,{\mathrm e}^{i \left (d x +c \right )}}{21}+\frac {64 \,{\mathrm e}^{3 i \left (d x +c \right )}}{63}+\frac {128 \,{\mathrm e}^{5 i \left (d x +c \right )}}{7}+\frac {96 i {\mathrm e}^{4 i \left (d x +c \right )}}{7}+\frac {128 i {\mathrm e}^{2 i \left (d x +c \right )}}{21}}{\left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{9} \left (-i+{\mathrm e}^{i \left (d x +c \right )}\right )^{3} d \,a^{3}}\) | \(97\) |
parallelrisch | \(\frac {\frac {38}{63}+2 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {26 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-6 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {68 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {100 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {28 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+12 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {26 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21}-\frac {470 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63}+\frac {34 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{21}}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}\) | \(178\) |
derivativedivides | \(\frac {-\frac {8}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {68}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {46}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {35}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {59}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {19}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {9}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {57}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{24 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {7}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{a^{3} d}\) | \(190\) |
default | \(\frac {-\frac {8}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {68}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {46}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {35}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {59}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {19}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {9}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {57}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{24 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {7}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{a^{3} d}\) | \(190\) |
norman | \(\frac {\frac {28 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {38}{63 a d}-\frac {6 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {26 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {12 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {2 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {2 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {34 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{21 d a}-\frac {68 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d a}-\frac {470 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63 d a}+\frac {26 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21 d a}-\frac {100 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d a}}{a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}\) | \(247\) |
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Time = 0.29 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.06 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {16 \, \cos \left (d x + c\right )^{6} - 72 \, \cos \left (d x + c\right )^{4} + 30 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (24 \, \cos \left (d x + c\right )^{4} - 20 \, \cos \left (d x + c\right )^{2} - 7\right )} \sin \left (d x + c\right ) + 7}{63 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3} + {\left (a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \]
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\[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\sec ^{4}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 482 vs. \(2 (113) = 226\).
Time = 0.22 (sec) , antiderivative size = 482, normalized size of antiderivative = 3.92 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {2 \, {\left (\frac {51 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {39 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {235 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {450 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {306 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {294 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {378 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {63 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {273 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {189 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {63 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + 19\right )}}{63 \, {\left (a^{3} + \frac {6 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {12 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {27 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {36 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {36 \, a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {27 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {2 \, a^{3} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {12 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {6 \, a^{3} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}\right )} d} \]
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Time = 0.33 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.39 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {21 \, {\left (21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 36 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 19\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {3591 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 19656 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 56196 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 95760 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 107730 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 79464 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 38484 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 10944 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1615}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{9}}}{2016 \, d} \]
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Time = 7.09 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.36 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {63\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{8}-\frac {171\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{8}-\frac {145\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{16}+\frac {49\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{16}+\frac {\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{2}+\frac {617\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}-\frac {329\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{16}+\frac {145\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{32}-\frac {113\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{32}-\frac {115\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{32}+\frac {19\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{32}\right )}{2016\,a^3\,d\,{\cos \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\right )}^9\,{\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d\,x}{2}\right )}^3} \]
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