Integrand size = 23, antiderivative size = 106 \[ \int \frac {\sin ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {a^3 \text {arctanh}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{b^{7/2} \sqrt {a+b} d}-\frac {\left (a^2-a b+b^2\right ) \cos (c+d x)}{b^3 d}-\frac {(a-2 b) \cos ^3(c+d x)}{3 b^2 d}-\frac {\cos ^5(c+d x)}{5 b d} \]
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Time = 0.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3265, 398, 214} \[ \int \frac {\sin ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {a^3 \text {arctanh}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{b^{7/2} d \sqrt {a+b}}-\frac {\left (a^2-a b+b^2\right ) \cos (c+d x)}{b^3 d}-\frac {(a-2 b) \cos ^3(c+d x)}{3 b^2 d}-\frac {\cos ^5(c+d x)}{5 b d} \]
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Rule 214
Rule 398
Rule 3265
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{a+b-b x^2} \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {a^2-a b+b^2}{b^3}+\frac {(a-2 b) x^2}{b^2}+\frac {x^4}{b}-\frac {a^3}{b^3 \left (a+b-b x^2\right )}\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {\left (a^2-a b+b^2\right ) \cos (c+d x)}{b^3 d}-\frac {(a-2 b) \cos ^3(c+d x)}{3 b^2 d}-\frac {\cos ^5(c+d x)}{5 b d}+\frac {a^3 \text {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\cos (c+d x)\right )}{b^3 d} \\ & = \frac {a^3 \text {arctanh}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{b^{7/2} \sqrt {a+b} d}-\frac {\left (a^2-a b+b^2\right ) \cos (c+d x)}{b^3 d}-\frac {(a-2 b) \cos ^3(c+d x)}{3 b^2 d}-\frac {\cos ^5(c+d x)}{5 b d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.27 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.70 \[ \int \frac {\sin ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {-240 a^3 \arctan \left (\frac {\sqrt {b}-i \sqrt {a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a-b}}\right )-240 a^3 \arctan \left (\frac {\sqrt {b}+i \sqrt {a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a-b}}\right )-2 \sqrt {-a-b} \sqrt {b} \cos (c+d x) \left (120 a^2-100 a b+89 b^2+4 (5 a-7 b) b \cos (2 (c+d x))+3 b^2 \cos (4 (c+d x))\right )}{240 \sqrt {-a-b} b^{7/2} d} \]
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Time = 1.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(\frac {-\frac {\frac {\left (\cos ^{5}\left (d x +c \right )\right ) b^{2}}{5}+\frac {a b \left (\cos ^{3}\left (d x +c \right )\right )}{3}-\frac {2 b^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right ) a^{2}-a b \cos \left (d x +c \right )+b^{2} \cos \left (d x +c \right )}{b^{3}}+\frac {a^{3} \operatorname {arctanh}\left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (a +b \right ) b}}\right )}{b^{3} \sqrt {\left (a +b \right ) b}}}{d}\) | \(110\) |
default | \(\frac {-\frac {\frac {\left (\cos ^{5}\left (d x +c \right )\right ) b^{2}}{5}+\frac {a b \left (\cos ^{3}\left (d x +c \right )\right )}{3}-\frac {2 b^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right ) a^{2}-a b \cos \left (d x +c \right )+b^{2} \cos \left (d x +c \right )}{b^{3}}+\frac {a^{3} \operatorname {arctanh}\left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (a +b \right ) b}}\right )}{b^{3} \sqrt {\left (a +b \right ) b}}}{d}\) | \(110\) |
risch | \(-\frac {{\mathrm e}^{i \left (d x +c \right )} a^{2}}{2 b^{3} d}+\frac {3 a \,{\mathrm e}^{i \left (d x +c \right )}}{8 d \,b^{2}}-\frac {5 \,{\mathrm e}^{i \left (d x +c \right )}}{16 b d}-\frac {{\mathrm e}^{-i \left (d x +c \right )} a^{2}}{2 b^{3} d}+\frac {3 \,{\mathrm e}^{-i \left (d x +c \right )} a}{8 b^{2} d}-\frac {5 \,{\mathrm e}^{-i \left (d x +c \right )}}{16 b d}-\frac {i a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (d x +c \right )}}{\sqrt {-a b -b^{2}}}+1\right )}{2 \sqrt {-a b -b^{2}}\, d \,b^{3}}+\frac {i a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (d x +c \right )}}{\sqrt {-a b -b^{2}}}+1\right )}{2 \sqrt {-a b -b^{2}}\, d \,b^{3}}-\frac {\cos \left (5 d x +5 c \right )}{80 b d}-\frac {\cos \left (3 d x +3 c \right ) a}{12 d \,b^{2}}+\frac {5 \cos \left (3 d x +3 c \right )}{48 d b}\) | \(290\) |
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Time = 0.30 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.57 \[ \int \frac {\sin ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\left [-\frac {6 \, {\left (a b^{3} + b^{4}\right )} \cos \left (d x + c\right )^{5} - 15 \, \sqrt {a b + b^{2}} a^{3} \log \left (\frac {b \cos \left (d x + c\right )^{2} + 2 \, \sqrt {a b + b^{2}} \cos \left (d x + c\right ) + a + b}{b \cos \left (d x + c\right )^{2} - a - b}\right ) + 10 \, {\left (a^{2} b^{2} - a b^{3} - 2 \, b^{4}\right )} \cos \left (d x + c\right )^{3} + 30 \, {\left (a^{3} b + b^{4}\right )} \cos \left (d x + c\right )}{30 \, {\left (a b^{4} + b^{5}\right )} d}, -\frac {3 \, {\left (a b^{3} + b^{4}\right )} \cos \left (d x + c\right )^{5} + 15 \, \sqrt {-a b - b^{2}} a^{3} \arctan \left (\frac {\sqrt {-a b - b^{2}} \cos \left (d x + c\right )}{a + b}\right ) + 5 \, {\left (a^{2} b^{2} - a b^{3} - 2 \, b^{4}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (a^{3} b + b^{4}\right )} \cos \left (d x + c\right )}{15 \, {\left (a b^{4} + b^{5}\right )} d}\right ] \]
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Timed out. \[ \int \frac {\sin ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\text {Timed out} \]
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Time = 0.37 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.09 \[ \int \frac {\sin ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx=-\frac {\frac {15 \, a^{3} \log \left (\frac {b \cos \left (d x + c\right ) - \sqrt {{\left (a + b\right )} b}}{b \cos \left (d x + c\right ) + \sqrt {{\left (a + b\right )} b}}\right )}{\sqrt {{\left (a + b\right )} b} b^{3}} + \frac {2 \, {\left (3 \, b^{2} \cos \left (d x + c\right )^{5} + 5 \, {\left (a b - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (a^{2} - a b + b^{2}\right )} \cos \left (d x + c\right )\right )}}{b^{3}}}{30 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (94) = 188\).
Time = 0.43 (sec) , antiderivative size = 332, normalized size of antiderivative = 3.13 \[ \int \frac {\sin ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx=-\frac {\frac {15 \, a^{3} \arctan \left (\frac {b \cos \left (d x + c\right ) + a + b}{\sqrt {-a b - b^{2}} \cos \left (d x + c\right ) + \sqrt {-a b - b^{2}}}\right )}{\sqrt {-a b - b^{2}} b^{3}} - \frac {2 \, {\left (15 \, a^{2} - 10 \, a b + 8 \, b^{2} - \frac {60 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {50 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {40 \, b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {90 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {70 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {80 \, b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {60 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {30 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )}}{b^{3} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{5}}}{15 \, d} \]
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Time = 0.17 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.06 \[ \int \frac {\sin ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {a^3\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\cos \left (c+d\,x\right )}{\sqrt {a+b}}\right )}{b^{7/2}\,d\,\sqrt {a+b}}-\frac {{\cos \left (c+d\,x\right )}^5}{5\,b\,d}-\frac {{\cos \left (c+d\,x\right )}^3\,\left (\frac {a+b}{3\,b^2}-\frac {1}{b}\right )}{d}-\frac {\cos \left (c+d\,x\right )\,\left (\frac {3}{b}+\frac {\left (a+b\right )\,\left (\frac {a+b}{b^2}-\frac {3}{b}\right )}{b}\right )}{d} \]
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