Integrand size = 23, antiderivative size = 162 \[ \int \frac {\csc ^4(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=\frac {b^2 (6 a+5 b) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{2 a^{7/2} (a+b)^{3/2} d}-\frac {\left (2 a^2-a b-5 b^2\right ) \cot (c+d x)}{2 a^3 (a+b) d}-\frac {(2 a+5 b) \cot ^3(c+d x)}{6 a^2 (a+b) d}+\frac {b \csc ^3(c+d x) \sec (c+d x)}{2 a (a+b) d \left (a+(a+b) \tan ^2(c+d x)\right )} \]
[Out]
Time = 0.14 (sec) , antiderivative size = 162, 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 = {3266, 479, 584, 211} \[ \int \frac {\csc ^4(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=\frac {b^2 (6 a+5 b) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{2 a^{7/2} d (a+b)^{3/2}}-\frac {(2 a+5 b) \cot ^3(c+d x)}{6 a^2 d (a+b)}-\frac {\left (2 a^2-a b-5 b^2\right ) \cot (c+d x)}{2 a^3 d (a+b)}+\frac {b \csc ^3(c+d x) \sec (c+d x)}{2 a d (a+b) \left ((a+b) \tan ^2(c+d x)+a\right )} \]
[In]
[Out]
Rule 211
Rule 479
Rule 584
Rule 3266
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^4 \left (a+(a+b) x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {b \csc ^3(c+d x) \sec (c+d x)}{2 a (a+b) d \left (a+(a+b) \tan ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right ) \left (-2 a-5 b+(-2 a-b) x^2\right )}{x^4 \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{2 a (a+b) d} \\ & = \frac {b \csc ^3(c+d x) \sec (c+d x)}{2 a (a+b) d \left (a+(a+b) \tan ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \left (\frac {-2 a-5 b}{a x^4}+\frac {-2 a^2+a b+5 b^2}{a^2 x^2}+\frac {(-6 a-5 b) b^2}{a^2 \left (a+(a+b) x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{2 a (a+b) d} \\ & = -\frac {\left (2 a^2-a b-5 b^2\right ) \cot (c+d x)}{2 a^3 (a+b) d}-\frac {(2 a+5 b) \cot ^3(c+d x)}{6 a^2 (a+b) d}+\frac {b \csc ^3(c+d x) \sec (c+d x)}{2 a (a+b) d \left (a+(a+b) \tan ^2(c+d x)\right )}+\frac {\left (b^2 (6 a+5 b)\right ) \text {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\tan (c+d x)\right )}{2 a^3 (a+b) d} \\ & = \frac {b^2 (6 a+5 b) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{2 a^{7/2} (a+b)^{3/2} d}-\frac {\left (2 a^2-a b-5 b^2\right ) \cot (c+d x)}{2 a^3 (a+b) d}-\frac {(2 a+5 b) \cot ^3(c+d x)}{6 a^2 (a+b) d}+\frac {b \csc ^3(c+d x) \sec (c+d x)}{2 a (a+b) d \left (a+(a+b) \tan ^2(c+d x)\right )} \\ \end{align*}
Time = 2.33 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.25 \[ \int \frac {\csc ^4(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=\frac {(-2 a-b+b \cos (2 (c+d x))) \csc ^4(c+d x) \left (\frac {3 b^2 (6 a+5 b) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right ) (-2 a-b+b \cos (2 (c+d x)))}{(a+b)^{3/2}}+4 \sqrt {a} (a-3 b) (2 a+b-b \cos (2 (c+d x))) \cot (c+d x)+2 a^{3/2} (2 a+b-b \cos (2 (c+d x))) \cot (c+d x) \csc ^2(c+d x)-\frac {3 \sqrt {a} b^3 \sin (2 (c+d x))}{a+b}\right )}{24 a^{7/2} d \left (b+a \csc ^2(c+d x)\right )^2} \]
[In]
[Out]
Time = 1.25 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {\frac {b^{2} \left (\frac {b \tan \left (d x +c \right )}{2 \left (a +b \right ) \left (a \left (\tan ^{2}\left (d x +c \right )\right )+\left (\tan ^{2}\left (d x +c \right )\right ) b +a \right )}+\frac {\left (6 a +5 b \right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{2 \left (a +b \right ) \sqrt {a \left (a +b \right )}}\right )}{a^{3}}-\frac {1}{3 a^{2} \tan \left (d x +c \right )^{3}}-\frac {a -2 b}{a^{3} \tan \left (d x +c \right )}}{d}\) | \(122\) |
default | \(\frac {\frac {b^{2} \left (\frac {b \tan \left (d x +c \right )}{2 \left (a +b \right ) \left (a \left (\tan ^{2}\left (d x +c \right )\right )+\left (\tan ^{2}\left (d x +c \right )\right ) b +a \right )}+\frac {\left (6 a +5 b \right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{2 \left (a +b \right ) \sqrt {a \left (a +b \right )}}\right )}{a^{3}}-\frac {1}{3 a^{2} \tan \left (d x +c \right )^{3}}-\frac {a -2 b}{a^{3} \tan \left (d x +c \right )}}{d}\) | \(122\) |
risch | \(\frac {i \left (-18 a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-15 b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+36 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+102 a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+60 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+48 a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-20 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-158 a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-90 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-16 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+12 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+82 a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+60 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+4 a^{2} b -8 a \,b^{2}-15 b^{3}\right )}{3 d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left (a +b \right ) \left (-b \,{\mathrm e}^{4 i \left (d x +c \right )}+4 a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{2 i \left (d x +c \right )}-b \right )}-\frac {3 b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i a^{2}+2 i a b +2 a \sqrt {-a^{2}-a b}+b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right ) d \,a^{2}}-\frac {5 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i a^{2}+2 i a b +2 a \sqrt {-a^{2}-a b}+b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right ) d \,a^{3}}+\frac {3 b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i a^{2}+2 i a b -2 a \sqrt {-a^{2}-a b}-b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right ) d \,a^{2}}+\frac {5 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i a^{2}+2 i a b -2 a \sqrt {-a^{2}-a b}-b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right ) d \,a^{3}}\) | \(666\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 371 vs. \(2 (146) = 292\).
Time = 0.30 (sec) , antiderivative size = 843, normalized size of antiderivative = 5.20 \[ \int \frac {\csc ^4(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=\left [-\frac {4 \, {\left (4 \, a^{4} b - 4 \, a^{3} b^{2} - 23 \, a^{2} b^{3} - 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} - 8 \, {\left (2 \, a^{5} + 3 \, a^{4} b - 12 \, a^{3} b^{2} - 28 \, a^{2} b^{3} - 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left ({\left (6 \, a b^{3} + 5 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 6 \, a^{2} b^{2} + 11 \, a b^{3} + 5 \, b^{4} - {\left (6 \, a^{2} b^{2} + 17 \, a b^{3} + 10 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {-a^{2} - a b} \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (4 \, a^{2} + 5 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left ({\left (2 \, a + b\right )} \cos \left (d x + c\right )^{3} - {\left (a + b\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} - a b} \sin \left (d x + c\right ) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos \left (d x + c\right )^{4} - 2 \, {\left (a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}}\right ) \sin \left (d x + c\right ) + 12 \, {\left (2 \, a^{5} + 2 \, a^{4} b - 6 \, a^{3} b^{2} - 11 \, a^{2} b^{3} - 5 \, a b^{4}\right )} \cos \left (d x + c\right )}{24 \, {\left ({\left (a^{6} b + 2 \, a^{5} b^{2} + a^{4} b^{3}\right )} d \cos \left (d x + c\right )^{4} - {\left (a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} + 2 \, a^{4} b^{3}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} d\right )} \sin \left (d x + c\right )}, -\frac {2 \, {\left (4 \, a^{4} b - 4 \, a^{3} b^{2} - 23 \, a^{2} b^{3} - 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} - 4 \, {\left (2 \, a^{5} + 3 \, a^{4} b - 12 \, a^{3} b^{2} - 28 \, a^{2} b^{3} - 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left ({\left (6 \, a b^{3} + 5 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 6 \, a^{2} b^{2} + 11 \, a b^{3} + 5 \, b^{4} - {\left (6 \, a^{2} b^{2} + 17 \, a b^{3} + 10 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {a^{2} + a b} \arctan \left (\frac {{\left (2 \, a + b\right )} \cos \left (d x + c\right )^{2} - a - b}{2 \, \sqrt {a^{2} + a b} \cos \left (d x + c\right ) \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) + 6 \, {\left (2 \, a^{5} + 2 \, a^{4} b - 6 \, a^{3} b^{2} - 11 \, a^{2} b^{3} - 5 \, a b^{4}\right )} \cos \left (d x + c\right )}{12 \, {\left ({\left (a^{6} b + 2 \, a^{5} b^{2} + a^{4} b^{3}\right )} d \cos \left (d x + c\right )^{4} - {\left (a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} + 2 \, a^{4} b^{3}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} d\right )} \sin \left (d x + c\right )}\right ] \]
[In]
[Out]
\[ \int \frac {\csc ^4(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=\int \frac {\csc ^{4}{\left (c + d x \right )}}{\left (a + b \sin ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \]
[In]
[Out]
none
Time = 0.38 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.06 \[ \int \frac {\csc ^4(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=\frac {\frac {3 \, {\left (6 \, a b^{2} + 5 \, b^{3}\right )} \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{{\left (a^{4} + a^{3} b\right )} \sqrt {{\left (a + b\right )} a}} - \frac {3 \, {\left (2 \, a^{3} - 6 \, a b^{2} - 5 \, b^{3}\right )} \tan \left (d x + c\right )^{4} + 2 \, a^{3} + 2 \, a^{2} b + 2 \, {\left (4 \, a^{3} - a^{2} b - 5 \, a b^{2}\right )} \tan \left (d x + c\right )^{2}}{{\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} \tan \left (d x + c\right )^{5} + {\left (a^{5} + a^{4} b\right )} \tan \left (d x + c\right )^{3}}}{6 \, d} \]
[In]
[Out]
none
Time = 0.36 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.07 \[ \int \frac {\csc ^4(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=\frac {\frac {3 \, b^{3} \tan \left (d x + c\right )}{{\left (a^{4} + a^{3} b\right )} {\left (a \tan \left (d x + c\right )^{2} + b \tan \left (d x + c\right )^{2} + a\right )}} + \frac {3 \, {\left (6 \, a b^{2} + 5 \, b^{3}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a^{2} + a b}}\right )\right )}}{{\left (a^{4} + a^{3} b\right )} \sqrt {a^{2} + a b}} - \frac {2 \, {\left (3 \, a \tan \left (d x + c\right )^{2} - 6 \, b \tan \left (d x + c\right )^{2} + a\right )}}{a^{3} \tan \left (d x + c\right )^{3}}}{6 \, d} \]
[In]
[Out]
Time = 14.87 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.01 \[ \int \frac {\csc ^4(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx=\frac {b^2\,\mathrm {atan}\left (\frac {b^2\,\mathrm {tan}\left (c+d\,x\right )\,\left (a^4+b\,a^3\right )\,\left (6\,a+5\,b\right )}{a^{7/2}\,\left (5\,b^3+6\,a\,b^2\right )\,\sqrt {a+b}}\right )\,\left (6\,a+5\,b\right )}{2\,a^{7/2}\,d\,{\left (a+b\right )}^{3/2}}-\frac {\frac {1}{3\,a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (4\,a-5\,b\right )}{3\,a^2}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (-2\,a^3+6\,a\,b^2+5\,b^3\right )}{2\,a^3\,\left (a+b\right )}}{d\,\left (\left (a+b\right )\,{\mathrm {tan}\left (c+d\,x\right )}^5+a\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )} \]
[In]
[Out]