Integrand size = 23, antiderivative size = 210 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=-\frac {3 (a-2 b) \sqrt {a-b} \sqrt {b} \arctan \left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{2 a^4 f}-\frac {3 \left (a^2-8 a b+8 b^2\right ) \text {arctanh}(\cos (e+f x))}{8 a^4 f}-\frac {(5 a-6 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {3 (3 a-4 b) b \sec (e+f x)}{8 a^3 f \left (a-b+b \sec ^2(e+f x)\right )} \]
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Time = 0.35 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3745, 481, 541, 536, 213, 211} \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=-\frac {3 \sqrt {b} (a-2 b) \sqrt {a-b} \arctan \left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{2 a^4 f}-\frac {3 b (3 a-4 b) \sec (e+f x)}{8 a^3 f \left (a+b \sec ^2(e+f x)-b\right )}-\frac {(5 a-6 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a+b \sec ^2(e+f x)-b\right )}-\frac {3 \left (a^2-8 a b+8 b^2\right ) \text {arctanh}(\cos (e+f x))}{8 a^4 f}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a+b \sec ^2(e+f x)-b\right )} \]
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Rule 211
Rule 213
Rule 481
Rule 536
Rule 541
Rule 3745
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^3 \left (a-b+b x^2\right )^2} \, dx,x,\sec (e+f x)\right )}{f} \\ & = -\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {\text {Subst}\left (\int \frac {-a+b+(-4 a+5 b) x^2}{\left (-1+x^2\right )^2 \left (a-b+b x^2\right )^2} \, dx,x,\sec (e+f x)\right )}{4 a f} \\ & = -\frac {(5 a-6 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {\text {Subst}\left (\int \frac {-3 (a-2 b) (a-b)+3 (5 a-6 b) b x^2}{\left (-1+x^2\right ) \left (a-b+b x^2\right )^2} \, dx,x,\sec (e+f x)\right )}{8 a^2 f} \\ & = -\frac {(5 a-6 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {3 (3 a-4 b) b \sec (e+f x)}{8 a^3 f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {\text {Subst}\left (\int \frac {-6 (a-4 b) (a-b)^2+6 (3 a-4 b) (a-b) b x^2}{\left (-1+x^2\right ) \left (a-b+b x^2\right )} \, dx,x,\sec (e+f x)\right )}{16 a^3 (a-b) f} \\ & = -\frac {(5 a-6 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {3 (3 a-4 b) b \sec (e+f x)}{8 a^3 f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {(3 (a-2 b) (a-b) b) \text {Subst}\left (\int \frac {1}{a-b+b x^2} \, dx,x,\sec (e+f x)\right )}{2 a^4 f}+\frac {\left (3 \left (a^2-8 a b+8 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (e+f x)\right )}{8 a^4 f} \\ & = -\frac {3 (a-2 b) \sqrt {a-b} \sqrt {b} \arctan \left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{2 a^4 f}-\frac {3 \left (a^2-8 a b+8 b^2\right ) \text {arctanh}(\cos (e+f x))}{8 a^4 f}-\frac {(5 a-6 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {3 (3 a-4 b) b \sec (e+f x)}{8 a^3 f \left (a-b+b \sec ^2(e+f x)\right )} \\ \end{align*}
Time = 7.03 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.87 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\frac {3 (a-2 b) \sqrt {a-b} \sqrt {b} \arctan \left (\frac {\sec \left (\frac {1}{2} (e+f x)\right ) \left (\sqrt {a-b} \cos \left (\frac {1}{2} (e+f x)\right )-\sqrt {a} \sin \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {b}}\right )}{2 a^4 f}+\frac {3 (a-2 b) \sqrt {a-b} \sqrt {b} \arctan \left (\frac {\sec \left (\frac {1}{2} (e+f x)\right ) \left (\sqrt {a-b} \cos \left (\frac {1}{2} (e+f x)\right )+\sqrt {a} \sin \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {b}}\right )}{2 a^4 f}+\frac {-a b \cos (e+f x)+b^2 \cos (e+f x)}{a^3 f (a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x)))}+\frac {(-3 a+8 b) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{32 a^3 f}-\frac {\csc ^4\left (\frac {1}{2} (e+f x)\right )}{64 a^2 f}-\frac {3 \left (a^2-8 a b+8 b^2\right ) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{8 a^4 f}+\frac {3 \left (a^2-8 a b+8 b^2\right ) \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{8 a^4 f}+\frac {(3 a-8 b) \sec ^2\left (\frac {1}{2} (e+f x)\right )}{32 a^3 f}+\frac {\sec ^4\left (\frac {1}{2} (e+f x)\right )}{64 a^2 f} \]
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Time = 1.23 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.10
method | result | size |
derivativedivides | \(\frac {\frac {b \left (\frac {\left (-\frac {1}{2} a^{2}+\frac {1}{2} a b \right ) \cos \left (f x +e \right )}{a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}+\frac {3 \left (a^{2}-3 a b +2 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \cos \left (f x +e \right )}{\sqrt {b \left (a -b \right )}}\right )}{2 \sqrt {b \left (a -b \right )}}\right )}{a^{4}}+\frac {1}{16 a^{2} \left (\cos \left (f x +e \right )+1\right )^{2}}-\frac {-3 a +8 b}{16 a^{3} \left (\cos \left (f x +e \right )+1\right )}+\frac {\left (-3 a^{2}+24 a b -24 b^{2}\right ) \ln \left (\cos \left (f x +e \right )+1\right )}{16 a^{4}}-\frac {1}{16 a^{2} \left (\cos \left (f x +e \right )-1\right )^{2}}-\frac {-3 a +8 b}{16 a^{3} \left (\cos \left (f x +e \right )-1\right )}+\frac {\left (3 a^{2}-24 a b +24 b^{2}\right ) \ln \left (\cos \left (f x +e \right )-1\right )}{16 a^{4}}}{f}\) | \(232\) |
default | \(\frac {\frac {b \left (\frac {\left (-\frac {1}{2} a^{2}+\frac {1}{2} a b \right ) \cos \left (f x +e \right )}{a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}+\frac {3 \left (a^{2}-3 a b +2 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \cos \left (f x +e \right )}{\sqrt {b \left (a -b \right )}}\right )}{2 \sqrt {b \left (a -b \right )}}\right )}{a^{4}}+\frac {1}{16 a^{2} \left (\cos \left (f x +e \right )+1\right )^{2}}-\frac {-3 a +8 b}{16 a^{3} \left (\cos \left (f x +e \right )+1\right )}+\frac {\left (-3 a^{2}+24 a b -24 b^{2}\right ) \ln \left (\cos \left (f x +e \right )+1\right )}{16 a^{4}}-\frac {1}{16 a^{2} \left (\cos \left (f x +e \right )-1\right )^{2}}-\frac {-3 a +8 b}{16 a^{3} \left (\cos \left (f x +e \right )-1\right )}+\frac {\left (3 a^{2}-24 a b +24 b^{2}\right ) \ln \left (\cos \left (f x +e \right )-1\right )}{16 a^{4}}}{f}\) | \(232\) |
risch | \(\frac {3 a^{2} {\mathrm e}^{11 i \left (f x +e \right )}-15 a b \,{\mathrm e}^{11 i \left (f x +e \right )}+12 b^{2} {\mathrm e}^{11 i \left (f x +e \right )}-5 a^{2} {\mathrm e}^{9 i \left (f x +e \right )}+21 a b \,{\mathrm e}^{9 i \left (f x +e \right )}-36 b^{2} {\mathrm e}^{9 i \left (f x +e \right )}-30 a^{2} {\mathrm e}^{7 i \left (f x +e \right )}-6 a b \,{\mathrm e}^{7 i \left (f x +e \right )}+24 b^{2} {\mathrm e}^{7 i \left (f x +e \right )}-30 a^{2} {\mathrm e}^{5 i \left (f x +e \right )}-6 a b \,{\mathrm e}^{5 i \left (f x +e \right )}+24 b^{2} {\mathrm e}^{5 i \left (f x +e \right )}-5 a^{2} {\mathrm e}^{3 i \left (f x +e \right )}+21 a b \,{\mathrm e}^{3 i \left (f x +e \right )}-36 b^{2} {\mathrm e}^{3 i \left (f x +e \right )}+3 a^{2} {\mathrm e}^{i \left (f x +e \right )}-15 a b \,{\mathrm e}^{i \left (f x +e \right )}+12 b^{2} {\mathrm e}^{i \left (f x +e \right )}}{4 f \,a^{3} \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{4} \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}-b \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+2 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a -b \right )}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{8 a^{2} f}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) b}{a^{3} f}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) b^{2}}{a^{4} f}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{8 a^{2} f}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) b}{a^{3} f}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) b^{2}}{a^{4} f}+\frac {3 i \sqrt {a b -b^{2}}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {a b -b^{2}}\, {\mathrm e}^{i \left (f x +e \right )}}{a -b}+1\right )}{4 f \,a^{3}}-\frac {3 i \sqrt {a b -b^{2}}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {a b -b^{2}}\, {\mathrm e}^{i \left (f x +e \right )}}{a -b}+1\right ) b}{2 f \,a^{4}}-\frac {3 i \sqrt {a b -b^{2}}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {a b -b^{2}}\, {\mathrm e}^{i \left (f x +e \right )}}{a -b}+1\right )}{4 f \,a^{3}}+\frac {3 i \sqrt {a b -b^{2}}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {a b -b^{2}}\, {\mathrm e}^{i \left (f x +e \right )}}{a -b}+1\right ) b}{2 f \,a^{4}}\) | \(702\) |
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Leaf count of result is larger than twice the leaf count of optimal. 509 vs. \(2 (192) = 384\).
Time = 0.38 (sec) , antiderivative size = 1052, normalized size of antiderivative = 5.01 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 516 vs. \(2 (192) = 384\).
Time = 0.59 (sec) , antiderivative size = 516, normalized size of antiderivative = 2.46 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\frac {\frac {12 \, {\left (a^{2} - 8 \, a b + 8 \, b^{2}\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}} - \frac {96 \, {\left (a^{2} b - 3 \, a b^{2} + 2 \, b^{3}\right )} \arctan \left (-\frac {a \cos \left (f x + e\right ) - b \cos \left (f x + e\right ) - b}{\sqrt {a b - b^{2}} \cos \left (f x + e\right ) + \sqrt {a b - b^{2}}}\right )}{\sqrt {a b - b^{2}} a^{4}} - \frac {\frac {8 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {16 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}}{a^{4}} - \frac {{\left (a^{2} - \frac {8 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {16 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {18 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {144 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {144 \, b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{2}}{a^{4} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}} - \frac {64 \, {\left (a^{2} b - a b^{2} + \frac {a^{2} b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {3 \, a b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {2 \, b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1}\right )}}{{\left (a + \frac {2 \, a {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {4 \, 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}}\right )} a^{4}}}{64 \, f} \]
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Time = 11.14 (sec) , antiderivative size = 1113, normalized size of antiderivative = 5.30 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\text {Too large to display} \]
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