Integrand size = 23, antiderivative size = 102 \[ \int \frac {\sin ^5(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=\frac {a (3 a+4 b) \text {arctanh}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{2 b^{5/2} (a+b)^{3/2} d}-\frac {\cos (c+d x)}{b^2 d}-\frac {a^2 \cos (c+d x)}{2 b^2 (a+b) d \left (a+b-b \cos ^2(c+d x)\right )} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3265, 398, 393, 214} \[ \int \frac {\sin ^5(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=-\frac {a^2 \cos (c+d x)}{2 b^2 d (a+b) \left (a-b \cos ^2(c+d x)+b\right )}+\frac {a (3 a+4 b) \text {arctanh}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{2 b^{5/2} d (a+b)^{3/2}}-\frac {\cos (c+d x)}{b^2 d} \]
[In]
[Out]
Rule 214
Rule 393
Rule 398
Rule 3265
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{\left (a+b-b x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {1}{b^2}-\frac {a (a+2 b)-2 a b x^2}{b^2 \left (a+b-b x^2\right )^2}\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {\cos (c+d x)}{b^2 d}+\frac {\text {Subst}\left (\int \frac {a (a+2 b)-2 a b x^2}{\left (a+b-b x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{b^2 d} \\ & = -\frac {\cos (c+d x)}{b^2 d}-\frac {a^2 \cos (c+d x)}{2 b^2 (a+b) d \left (a+b-b \cos ^2(c+d x)\right )}+\frac {(a (3 a+4 b)) \text {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\cos (c+d x)\right )}{2 b^2 (a+b) d} \\ & = \frac {a (3 a+4 b) \text {arctanh}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{2 b^{5/2} (a+b)^{3/2} d}-\frac {\cos (c+d x)}{b^2 d}-\frac {a^2 \cos (c+d x)}{2 b^2 (a+b) d \left (a+b-b \cos ^2(c+d x)\right )} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.90 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.69 \[ \int \frac {\sin ^5(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=\frac {\frac {a (3 a+4 b) \arctan \left (\frac {\sqrt {b}-i \sqrt {a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a-b}}\right )}{(-a-b)^{3/2}}+\frac {a (3 a+4 b) \arctan \left (\frac {\sqrt {b}+i \sqrt {a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a-b}}\right )}{(-a-b)^{3/2}}+2 \sqrt {b} \cos (c+d x) \left (-1-\frac {a^2}{(a+b) (2 a+b-b \cos (2 (c+d x)))}\right )}{2 b^{5/2} d} \]
[In]
[Out]
Time = 1.10 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {-\frac {\cos \left (d x +c \right )}{b^{2}}+\frac {a \left (-\frac {a \cos \left (d x +c \right )}{2 \left (a +b \right ) \left (a +b -b \left (\cos ^{2}\left (d x +c \right )\right )\right )}+\frac {\left (3 a +4 b \right ) \operatorname {arctanh}\left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (a +b \right ) b}}\right )}{2 \left (a +b \right ) \sqrt {\left (a +b \right ) b}}\right )}{b^{2}}}{d}\) | \(90\) |
default | \(\frac {-\frac {\cos \left (d x +c \right )}{b^{2}}+\frac {a \left (-\frac {a \cos \left (d x +c \right )}{2 \left (a +b \right ) \left (a +b -b \left (\cos ^{2}\left (d x +c \right )\right )\right )}+\frac {\left (3 a +4 b \right ) \operatorname {arctanh}\left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (a +b \right ) b}}\right )}{2 \left (a +b \right ) \sqrt {\left (a +b \right ) b}}\right )}{b^{2}}}{d}\) | \(90\) |
risch | \(-\frac {{\mathrm e}^{i \left (d x +c \right )}}{2 b^{2} d}-\frac {{\mathrm e}^{-i \left (d x +c \right )}}{2 b^{2} d}+\frac {a^{2} \left ({\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}\right )}{b^{2} \left (a +b \right ) d \left (b \,{\mathrm e}^{4 i \left (d x +c \right )}-4 a \,{\mathrm e}^{2 i \left (d x +c \right )}-2 b \,{\mathrm e}^{2 i \left (d x +c \right )}+b \right )}+\frac {3 i a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (d x +c \right )}}{\sqrt {-a b -b^{2}}}+1\right )}{4 \sqrt {-a b -b^{2}}\, \left (a +b \right ) d \,b^{2}}+\frac {i a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (d x +c \right )}}{\sqrt {-a b -b^{2}}}+1\right )}{\sqrt {-a b -b^{2}}\, \left (a +b \right ) d b}-\frac {3 i a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (d x +c \right )}}{\sqrt {-a b -b^{2}}}+1\right )}{4 \sqrt {-a b -b^{2}}\, \left (a +b \right ) d \,b^{2}}-\frac {i a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (d x +c \right )}}{\sqrt {-a b -b^{2}}}+1\right )}{\sqrt {-a b -b^{2}}\, \left (a +b \right ) d b}\) | \(377\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (93) = 186\).
Time = 0.28 (sec) , antiderivative size = 427, normalized size of antiderivative = 4.19 \[ \int \frac {\sin ^5(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=\left [-\frac {4 \, {\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} \cos \left (d x + c\right )^{3} + {\left (3 \, a^{3} + 7 \, a^{2} b + 4 \, a b^{2} - {\left (3 \, a^{2} b + 4 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {a b + b^{2}} \log \left (\frac {b \cos \left (d x + c\right )^{2} + 2 \, \sqrt {a b + b^{2}} \cos \left (d x + c\right ) + a + b}{b \cos \left (d x + c\right )^{2} - a - b}\right ) - 2 \, {\left (3 \, a^{3} b + 7 \, a^{2} b^{2} + 6 \, a b^{3} + 2 \, b^{4}\right )} \cos \left (d x + c\right )}{4 \, {\left ({\left (a^{2} b^{4} + 2 \, a b^{5} + b^{6}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{3} b^{3} + 3 \, a^{2} b^{4} + 3 \, a b^{5} + b^{6}\right )} d\right )}}, -\frac {2 \, {\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} \cos \left (d x + c\right )^{3} - {\left (3 \, a^{3} + 7 \, a^{2} b + 4 \, a b^{2} - {\left (3 \, a^{2} b + 4 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {-a b - b^{2}} \arctan \left (\frac {\sqrt {-a b - b^{2}} \cos \left (d x + c\right )}{a + b}\right ) - {\left (3 \, a^{3} b + 7 \, a^{2} b^{2} + 6 \, a b^{3} + 2 \, b^{4}\right )} \cos \left (d x + c\right )}{2 \, {\left ({\left (a^{2} b^{4} + 2 \, a b^{5} + b^{6}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{3} b^{3} + 3 \, a^{2} b^{4} + 3 \, a b^{5} + b^{6}\right )} d\right )}}\right ] \]
[In]
[Out]
Timed out. \[ \int \frac {\sin ^5(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.28 \[ \int \frac {\sin ^5(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=-\frac {\frac {2 \, a^{2} \cos \left (d x + c\right )}{a^{2} b^{2} + 2 \, a b^{3} + b^{4} - {\left (a b^{3} + b^{4}\right )} \cos \left (d x + c\right )^{2}} + \frac {{\left (3 \, a + 4 \, b\right )} a \log \left (\frac {b \cos \left (d x + c\right ) - \sqrt {{\left (a + b\right )} b}}{b \cos \left (d x + c\right ) + \sqrt {{\left (a + b\right )} b}}\right )}{{\left (a b^{2} + b^{3}\right )} \sqrt {{\left (a + b\right )} b}} + \frac {4 \, \cos \left (d x + c\right )}{b^{2}}}{4 \, d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 342 vs. \(2 (93) = 186\).
Time = 0.36 (sec) , antiderivative size = 342, normalized size of antiderivative = 3.35 \[ \int \frac {\sin ^5(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=-\frac {\frac {{\left (3 \, a^{2} + 4 \, a b\right )} \arctan \left (\frac {b \cos \left (d x + c\right ) + a + b}{\sqrt {-a b - b^{2}} \cos \left (d x + c\right ) + \sqrt {-a b - b^{2}}}\right )}{{\left (a b^{2} + b^{3}\right )} \sqrt {-a b - b^{2}}} + \frac {2 \, {\left (3 \, a^{2} + 2 \, a b - \frac {6 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {14 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {8 \, b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {4 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{{\left (a b^{2} + b^{3}\right )} {\left (a - \frac {3 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {4 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {4 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}}{2 \, d} \]
[In]
[Out]
Time = 0.19 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.93 \[ \int \frac {\sin ^5(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=\frac {a\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\cos \left (c+d\,x\right )}{\sqrt {a+b}}\right )\,\left (3\,a+4\,b\right )}{2\,b^{5/2}\,d\,{\left (a+b\right )}^{3/2}}-\frac {a^2\,\cos \left (c+d\,x\right )}{2\,d\,\left (a+b\right )\,\left (-b^3\,{\cos \left (c+d\,x\right )}^2+b^3+a\,b^2\right )}-\frac {\cos \left (c+d\,x\right )}{b^2\,d} \]
[In]
[Out]