Integrand size = 23, antiderivative size = 213 \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {\sqrt {b} \left (15 a^2-10 a b-b^2\right ) \arctan \left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{8 a^{3/2} (a+b)^4 f}-\frac {(a-5 b) \text {arctanh}(\cos (e+f x))}{2 (a+b)^4 f}-\frac {(2 a-b) b \cos (e+f x)}{4 a (a+b)^2 f \left (b+a \cos ^2(e+f x)\right )^2}+\frac {\left (4 a^2-9 a b-b^2\right ) \cos (e+f x)}{8 a (a+b)^3 f \left (b+a \cos ^2(e+f x)\right )}-\frac {\cos (e+f x) \cot ^2(e+f x)}{2 (a+b) f \left (b+a \cos ^2(e+f x)\right )^2} \]
[Out]
Time = 0.35 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {4218, 481, 592, 541, 536, 212, 211} \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {\left (4 a^2-9 a b-b^2\right ) \cos (e+f x)}{8 a f (a+b)^3 \left (a \cos ^2(e+f x)+b\right )}+\frac {\sqrt {b} \left (15 a^2-10 a b-b^2\right ) \arctan \left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{8 a^{3/2} f (a+b)^4}-\frac {(a-5 b) \text {arctanh}(\cos (e+f x))}{2 f (a+b)^4}-\frac {b (2 a-b) \cos (e+f x)}{4 a f (a+b)^2 \left (a \cos ^2(e+f x)+b\right )^2}-\frac {\cos (e+f x) \cot ^2(e+f x)}{2 f (a+b) \left (a \cos ^2(e+f x)+b\right )^2} \]
[In]
[Out]
Rule 211
Rule 212
Rule 481
Rule 536
Rule 541
Rule 592
Rule 4218
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^2 \left (b+a x^2\right )^3} \, dx,x,\cos (e+f x)\right )}{f} \\ & = -\frac {\cos (e+f x) \cot ^2(e+f x)}{2 (a+b) f \left (b+a \cos ^2(e+f x)\right )^2}+\frac {\text {Subst}\left (\int \frac {x^2 \left (3 b+(-a+2 b) x^2\right )}{\left (1-x^2\right ) \left (b+a x^2\right )^3} \, dx,x,\cos (e+f x)\right )}{2 (a+b) f} \\ & = -\frac {(2 a-b) b \cos (e+f x)}{4 a (a+b)^2 f \left (b+a \cos ^2(e+f x)\right )^2}-\frac {\cos (e+f x) \cot ^2(e+f x)}{2 (a+b) f \left (b+a \cos ^2(e+f x)\right )^2}+\frac {\text {Subst}\left (\int \frac {2 (2 a-b) b-2 \left (2 a^2-8 a b-b^2\right ) x^2}{\left (1-x^2\right ) \left (b+a x^2\right )^2} \, dx,x,\cos (e+f x)\right )}{8 a (a+b)^2 f} \\ & = -\frac {(2 a-b) b \cos (e+f x)}{4 a (a+b)^2 f \left (b+a \cos ^2(e+f x)\right )^2}+\frac {\left (4 a^2-9 a b-b^2\right ) \cos (e+f x)}{8 a (a+b)^3 f \left (b+a \cos ^2(e+f x)\right )}-\frac {\cos (e+f x) \cot ^2(e+f x)}{2 (a+b) f \left (b+a \cos ^2(e+f x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-2 (11 a-b) b^2+2 b \left (4 a^2-9 a b-b^2\right ) x^2}{\left (1-x^2\right ) \left (b+a x^2\right )} \, dx,x,\cos (e+f x)\right )}{16 a b (a+b)^3 f} \\ & = -\frac {(2 a-b) b \cos (e+f x)}{4 a (a+b)^2 f \left (b+a \cos ^2(e+f x)\right )^2}+\frac {\left (4 a^2-9 a b-b^2\right ) \cos (e+f x)}{8 a (a+b)^3 f \left (b+a \cos ^2(e+f x)\right )}-\frac {\cos (e+f x) \cot ^2(e+f x)}{2 (a+b) f \left (b+a \cos ^2(e+f x)\right )^2}-\frac {(a-5 b) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (e+f x)\right )}{2 (a+b)^4 f}+\frac {\left (b \left (15 a^2-10 a b-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\cos (e+f x)\right )}{8 a (a+b)^4 f} \\ & = \frac {\sqrt {b} \left (15 a^2-10 a b-b^2\right ) \arctan \left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{8 a^{3/2} (a+b)^4 f}-\frac {(a-5 b) \text {arctanh}(\cos (e+f x))}{2 (a+b)^4 f}-\frac {(2 a-b) b \cos (e+f x)}{4 a (a+b)^2 f \left (b+a \cos ^2(e+f x)\right )^2}+\frac {\left (4 a^2-9 a b-b^2\right ) \cos (e+f x)}{8 a (a+b)^3 f \left (b+a \cos ^2(e+f x)\right )}-\frac {\cos (e+f x) \cot ^2(e+f x)}{2 (a+b) f \left (b+a \cos ^2(e+f x)\right )^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 5.36 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.50 \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {(a+2 b+a \cos (2 (e+f x))) \sec ^5(e+f x) \left (\frac {8 b^2 (a+b)^2}{a}-\frac {2 b (a+b) (9 a+b) (a+2 b+a \cos (2 (e+f x)))}{a}-\frac {\sqrt {b} \left (-15 a^2+10 a b+b^2\right ) \arctan \left (\frac {\left (-\sqrt {a}-i \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2}\right ) \sin (e) \tan \left (\frac {f x}{2}\right )+\cos (e) \left (\sqrt {a}-\sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {b}}\right ) (a+2 b+a \cos (2 (e+f x)))^2 \sec (e+f x)}{a^{3/2}}-\frac {\sqrt {b} \left (-15 a^2+10 a b+b^2\right ) \arctan \left (\frac {\left (-\sqrt {a}+i \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2}\right ) \sin (e) \tan \left (\frac {f x}{2}\right )+\cos (e) \left (\sqrt {a}+\sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {b}}\right ) (a+2 b+a \cos (2 (e+f x)))^2 \sec (e+f x)}{a^{3/2}}-(a+b) (a+2 b+a \cos (2 (e+f x)))^2 \csc ^2\left (\frac {1}{2} (e+f x)\right ) \sec (e+f x)-4 (a-5 b) (a+2 b+a \cos (2 (e+f x)))^2 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right ) \sec (e+f x)+4 (a-5 b) (a+2 b+a \cos (2 (e+f x)))^2 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sec (e+f x)+(a+b) (a+2 b+a \cos (2 (e+f x)))^2 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sec (e+f x)\right )}{64 (a+b)^4 f \left (a+b \sec ^2(e+f x)\right )^3} \]
[In]
[Out]
Time = 1.76 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {\frac {1}{4 \left (a +b \right )^{3} \left (-1+\cos \left (f x +e \right )\right )}+\frac {\left (a -5 b \right ) \ln \left (-1+\cos \left (f x +e \right )\right )}{4 \left (a +b \right )^{4}}+\frac {b \left (\frac {\left (-\frac {9}{8} a^{2}-\frac {5}{4} a b -\frac {1}{8} b^{2}\right ) \cos \left (f x +e \right )^{3}-\frac {b \left (7 a^{2}+6 a b -b^{2}\right ) \cos \left (f x +e \right )}{8 a}}{\left (b +a \cos \left (f x +e \right )^{2}\right )^{2}}+\frac {\left (15 a^{2}-10 a b -b^{2}\right ) \arctan \left (\frac {a \cos \left (f x +e \right )}{\sqrt {a b}}\right )}{8 a \sqrt {a b}}\right )}{\left (a +b \right )^{4}}+\frac {1}{4 \left (a +b \right )^{3} \left (1+\cos \left (f x +e \right )\right )}+\frac {\left (-a +5 b \right ) \ln \left (1+\cos \left (f x +e \right )\right )}{4 \left (a +b \right )^{4}}}{f}\) | \(198\) |
default | \(\frac {\frac {1}{4 \left (a +b \right )^{3} \left (-1+\cos \left (f x +e \right )\right )}+\frac {\left (a -5 b \right ) \ln \left (-1+\cos \left (f x +e \right )\right )}{4 \left (a +b \right )^{4}}+\frac {b \left (\frac {\left (-\frac {9}{8} a^{2}-\frac {5}{4} a b -\frac {1}{8} b^{2}\right ) \cos \left (f x +e \right )^{3}-\frac {b \left (7 a^{2}+6 a b -b^{2}\right ) \cos \left (f x +e \right )}{8 a}}{\left (b +a \cos \left (f x +e \right )^{2}\right )^{2}}+\frac {\left (15 a^{2}-10 a b -b^{2}\right ) \arctan \left (\frac {a \cos \left (f x +e \right )}{\sqrt {a b}}\right )}{8 a \sqrt {a b}}\right )}{\left (a +b \right )^{4}}+\frac {1}{4 \left (a +b \right )^{3} \left (1+\cos \left (f x +e \right )\right )}+\frac {\left (-a +5 b \right ) \ln \left (1+\cos \left (f x +e \right )\right )}{4 \left (a +b \right )^{4}}}{f}\) | \(198\) |
risch | \(-\frac {-4 a^{3} {\mathrm e}^{11 i \left (f x +e \right )}+9 a^{2} b \,{\mathrm e}^{11 i \left (f x +e \right )}+a \,b^{2} {\mathrm e}^{11 i \left (f x +e \right )}-20 a^{3} {\mathrm e}^{9 i \left (f x +e \right )}-23 a^{2} b \,{\mathrm e}^{9 i \left (f x +e \right )}+29 a \,b^{2} {\mathrm e}^{9 i \left (f x +e \right )}-4 b^{3} {\mathrm e}^{9 i \left (f x +e \right )}-40 a^{3} {\mathrm e}^{7 i \left (f x +e \right )}-114 a^{2} b \,{\mathrm e}^{7 i \left (f x +e \right )}-94 a \,b^{2} {\mathrm e}^{7 i \left (f x +e \right )}+4 b^{3} {\mathrm e}^{7 i \left (f x +e \right )}-40 a^{3} {\mathrm e}^{5 i \left (f x +e \right )}-114 a^{2} b \,{\mathrm e}^{5 i \left (f x +e \right )}-94 a \,b^{2} {\mathrm e}^{5 i \left (f x +e \right )}+4 b^{3} {\mathrm e}^{5 i \left (f x +e \right )}-20 a^{3} {\mathrm e}^{3 i \left (f x +e \right )}-23 a^{2} b \,{\mathrm e}^{3 i \left (f x +e \right )}+29 a \,b^{2} {\mathrm e}^{3 i \left (f x +e \right )}-4 b^{3} {\mathrm e}^{3 i \left (f x +e \right )}-4 a^{3} {\mathrm e}^{i \left (f x +e \right )}+9 a^{2} b \,{\mathrm e}^{i \left (f x +e \right )}+a \,b^{2} {\mathrm e}^{i \left (f x +e \right )}}{4 a f \left (a +b \right )^{3} \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{2} \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+4 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a \right )^{2}}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) a}{2 f \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}-\frac {5 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) b}{2 f \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) a}{2 f \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}+\frac {5 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) b}{2 f \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}-\frac {15 i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right )}{16 \left (a +b \right )^{4} f}+\frac {5 i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right ) b}{8 a \left (a +b \right )^{4} f}+\frac {i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right ) b^{2}}{16 a^{2} \left (a +b \right )^{4} f}+\frac {15 i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right )}{16 \left (a +b \right )^{4} f}-\frac {5 i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right ) b}{8 a \left (a +b \right )^{4} f}-\frac {i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right ) b^{2}}{16 a^{2} \left (a +b \right )^{4} f}\) | \(882\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (195) = 390\).
Time = 0.43 (sec) , antiderivative size = 1332, normalized size of antiderivative = 6.25 \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 399 vs. \(2 (195) = 390\).
Time = 0.27 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.87 \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=-\frac {\frac {2 \, {\left (a - 5 \, b\right )} \log \left (\cos \left (f x + e\right ) + 1\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} - \frac {2 \, {\left (a - 5 \, b\right )} \log \left (\cos \left (f x + e\right ) - 1\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} - \frac {{\left (15 \, a^{2} b - 10 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {a \cos \left (f x + e\right )}{\sqrt {a b}}\right )}{{\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} \sqrt {a b}} - \frac {{\left (4 \, a^{3} - 9 \, a^{2} b - a b^{2}\right )} \cos \left (f x + e\right )^{5} + {\left (17 \, a^{2} b - 6 \, a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{3} + {\left (11 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )}{{\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )} \cos \left (f x + e\right )^{6} - a^{4} b^{2} - 3 \, a^{3} b^{3} - 3 \, a^{2} b^{4} - a b^{5} - {\left (a^{6} + a^{5} b - 3 \, a^{4} b^{2} - 5 \, a^{3} b^{3} - 2 \, a^{2} b^{4}\right )} \cos \left (f x + e\right )^{4} - {\left (2 \, a^{5} b + 5 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4} - a b^{5}\right )} \cos \left (f x + e\right )^{2}}}{8 \, f} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 750 vs. \(2 (195) = 390\).
Time = 0.52 (sec) , antiderivative size = 750, normalized size of antiderivative = 3.52 \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {\frac {2 \, {\left (a - 5 \, b\right )} \log \left (\frac {{\left | -\cos \left (f x + e\right ) + 1 \right |}}{{\left | \cos \left (f x + e\right ) + 1 \right |}}\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} - \frac {{\left (15 \, a^{2} b - 10 \, a b^{2} - b^{3}\right )} \arctan \left (-\frac {a \cos \left (f x + e\right ) - b}{\sqrt {a b} \cos \left (f x + e\right ) + \sqrt {a b}}\right )}{{\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} \sqrt {a b}} + \frac {{\left (a + b - \frac {2 \, a {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {10 \, b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} {\left (\cos \left (f x + e\right ) - 1\right )}} - \frac {\cos \left (f x + e\right ) - 1}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}} - \frac {2 \, {\left (9 \, a^{3} b + 17 \, a^{2} b^{2} + 7 \, a b^{3} - b^{4} + \frac {27 \, a^{3} b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {a^{2} b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {23 \, a b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {3 \, b^{4} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {27 \, a^{3} b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {21 \, a^{2} b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {29 \, a b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {3 \, b^{4} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {9 \, a^{3} b {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {5 \, a^{2} b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {13 \, a b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {b^{4} {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{{\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} {\left (a + b + \frac {2 \, a {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {2 \, b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {a {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}^{2}}}{8 \, f} \]
[In]
[Out]
Time = 19.87 (sec) , antiderivative size = 2728, normalized size of antiderivative = 12.81 \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\text {Too large to display} \]
[In]
[Out]