Integrand size = 23, antiderivative size = 83 \[ \int \frac {\sin ^3(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=-\frac {(a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{2 b^{3/2} (a+b)^{3/2} d}+\frac {a \cos (c+d x)}{2 b (a+b) d \left (a+b-b \cos ^2(c+d x)\right )} \]
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Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3265, 393, 214} \[ \int \frac {\sin ^3(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=\frac {a \cos (c+d x)}{2 b d (a+b) \left (a-b \cos ^2(c+d x)+b\right )}-\frac {(a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{2 b^{3/2} d (a+b)^{3/2}} \]
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Rule 214
Rule 393
Rule 3265
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1-x^2}{\left (a+b-b x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{d} \\ & = \frac {a \cos (c+d x)}{2 b (a+b) d \left (a+b-b \cos ^2(c+d x)\right )}-\frac {(a+2 b) \text {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\cos (c+d x)\right )}{2 b (a+b) d} \\ & = -\frac {(a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{2 b^{3/2} (a+b)^{3/2} d}+\frac {a \cos (c+d x)}{2 b (a+b) d \left (a+b-b \cos ^2(c+d x)\right )} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.53 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.93 \[ \int \frac {\sin ^3(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=\frac {\frac {(a+2 b) \arctan \left (\frac {\sqrt {b}-i \sqrt {a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a-b}}\right )}{\sqrt {-a-b}}+\frac {(a+2 b) \arctan \left (\frac {\sqrt {b}+i \sqrt {a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a-b}}\right )}{\sqrt {-a-b}}+\frac {2 a \sqrt {b} \cos (c+d x)}{2 a+b-b \cos (2 (c+d x))}}{2 b^{3/2} (a+b) d} \]
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Time = 0.72 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {\frac {a \cos \left (d x +c \right )}{2 \left (a +b \right ) b \left (a +b -b \left (\cos ^{2}\left (d x +c \right )\right )\right )}-\frac {\left (a +2 b \right ) \operatorname {arctanh}\left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (a +b \right ) b}}\right )}{2 \left (a +b \right ) b \sqrt {\left (a +b \right ) b}}}{d}\) | \(77\) |
default | \(\frac {\frac {a \cos \left (d x +c \right )}{2 \left (a +b \right ) b \left (a +b -b \left (\cos ^{2}\left (d x +c \right )\right )\right )}-\frac {\left (a +2 b \right ) \operatorname {arctanh}\left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (a +b \right ) b}}\right )}{2 \left (a +b \right ) b \sqrt {\left (a +b \right ) b}}}{d}\) | \(77\) |
risch | \(-\frac {a \left ({\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}\right )}{b \left (a +b \right ) d \left (b \,{\mathrm e}^{4 i \left (d x +c \right )}-4 a \,{\mathrm e}^{2 i \left (d x +c \right )}-2 b \,{\mathrm e}^{2 i \left (d x +c \right )}+b \right )}-\frac {i a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (d x +c \right )}}{\sqrt {-a b -b^{2}}}+1\right )}{4 \sqrt {-a b -b^{2}}\, \left (a +b \right ) d b}-\frac {i \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (d x +c \right )}}{\sqrt {-a b -b^{2}}}+1\right )}{2 \sqrt {-a b -b^{2}}\, \left (a +b \right ) d}+\frac {i a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (d x +c \right )}}{\sqrt {-a b -b^{2}}}+1\right )}{4 \sqrt {-a b -b^{2}}\, \left (a +b \right ) d b}+\frac {i \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (d x +c \right )}}{\sqrt {-a b -b^{2}}}+1\right )}{2 \sqrt {-a b -b^{2}}\, \left (a +b \right ) d}\) | \(330\) |
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Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (74) = 148\).
Time = 0.28 (sec) , antiderivative size = 327, normalized size of antiderivative = 3.94 \[ \int \frac {\sin ^3(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=\left [\frac {{\left ({\left (a b + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} - 3 \, a b - 2 \, b^{2}\right )} \sqrt {a b + b^{2}} \log \left (-\frac {b \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a b + b^{2}} \cos \left (d x + c\right ) + a + b}{b \cos \left (d x + c\right )^{2} - a - b}\right ) - 2 \, {\left (a^{2} b + a b^{2}\right )} \cos \left (d x + c\right )}{4 \, {\left ({\left (a^{2} b^{3} + 2 \, a b^{4} + b^{5}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5}\right )} d\right )}}, \frac {{\left ({\left (a b + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} - 3 \, a b - 2 \, b^{2}\right )} \sqrt {-a b - b^{2}} \arctan \left (\frac {\sqrt {-a b - b^{2}} \cos \left (d x + c\right )}{a + b}\right ) - {\left (a^{2} b + a b^{2}\right )} \cos \left (d x + c\right )}{2 \, {\left ({\left (a^{2} b^{3} + 2 \, a b^{4} + b^{5}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5}\right )} d\right )}}\right ] \]
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Timed out. \[ \int \frac {\sin ^3(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=\text {Timed out} \]
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Time = 0.33 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.34 \[ \int \frac {\sin ^3(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=\frac {\frac {2 \, a \cos \left (d x + c\right )}{a^{2} b + 2 \, a b^{2} + b^{3} - {\left (a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{2}} + \frac {{\left (a + 2 \, b\right )} \log \left (\frac {b \cos \left (d x + c\right ) - \sqrt {{\left (a + b\right )} b}}{b \cos \left (d x + c\right ) + \sqrt {{\left (a + b\right )} b}}\right )}{\sqrt {{\left (a + b\right )} b} {\left (a b + b^{2}\right )}}}{4 \, d} \]
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Time = 0.43 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.12 \[ \int \frac {\sin ^3(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=\frac {{\left (a + 2 \, b\right )} \arctan \left (\frac {b \cos \left (d x + c\right )}{\sqrt {-a b - b^{2}}}\right )}{2 \, {\left (a b + b^{2}\right )} \sqrt {-a b - b^{2}} d} - \frac {a \cos \left (d x + c\right )}{2 \, {\left (b \cos \left (d x + c\right )^{2} - a - b\right )} {\left (a b + b^{2}\right )} d} \]
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Time = 13.53 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.86 \[ \int \frac {\sin ^3(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=\frac {a\,\cos \left (c+d\,x\right )}{2\,b\,d\,\left (a+b\right )\,\left (-b\,{\cos \left (c+d\,x\right )}^2+a+b\right )}-\frac {\mathrm {atanh}\left (\frac {\sqrt {b}\,\cos \left (c+d\,x\right )}{\sqrt {a+b}}\right )\,\left (a+2\,b\right )}{2\,b^{3/2}\,d\,{\left (a+b\right )}^{3/2}} \]
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