Integrand size = 23, antiderivative size = 147 \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=-\frac {(3 a-4 b) \sqrt {b} \arctan \left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{2 a^3 \sqrt {a-b} f}-\frac {(a-4 b) \text {arctanh}(\cos (e+f x))}{2 a^3 f}-\frac {\cot (e+f x) \csc (e+f x)}{2 a f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {b \sec (e+f x)}{a^2 f \left (a-b+b \sec ^2(e+f x)\right )} \]
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Time = 0.23 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3745, 482, 541, 536, 213, 211} \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=-\frac {\sqrt {b} (3 a-4 b) \arctan \left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{2 a^3 f \sqrt {a-b}}-\frac {(a-4 b) \text {arctanh}(\cos (e+f x))}{2 a^3 f}-\frac {b \sec (e+f x)}{a^2 f \left (a+b \sec ^2(e+f x)-b\right )}-\frac {\cot (e+f x) \csc (e+f x)}{2 a f \left (a+b \sec ^2(e+f x)-b\right )} \]
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Rule 211
Rule 213
Rule 482
Rule 536
Rule 541
Rule 3745
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^2}{\left (-1+x^2\right )^2 \left (a-b+b x^2\right )^2} \, dx,x,\sec (e+f x)\right )}{f} \\ & = -\frac {\cot (e+f x) \csc (e+f x)}{2 a f \left (a-b+b \sec ^2(e+f x)\right )}+\frac {\text {Subst}\left (\int \frac {a-b-3 b x^2}{\left (-1+x^2\right ) \left (a-b+b x^2\right )^2} \, dx,x,\sec (e+f x)\right )}{2 a f} \\ & = -\frac {\cot (e+f x) \csc (e+f x)}{2 a f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {b \sec (e+f x)}{a^2 f \left (a-b+b \sec ^2(e+f x)\right )}+\frac {\text {Subst}\left (\int \frac {2 (a-2 b) (a-b)-4 (a-b) b x^2}{\left (-1+x^2\right ) \left (a-b+b x^2\right )} \, dx,x,\sec (e+f x)\right )}{4 a^2 (a-b) f} \\ & = -\frac {\cot (e+f x) \csc (e+f x)}{2 a f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {b \sec (e+f x)}{a^2 f \left (a-b+b \sec ^2(e+f x)\right )}+\frac {(a-4 b) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (e+f x)\right )}{2 a^3 f}-\frac {((3 a-4 b) b) \text {Subst}\left (\int \frac {1}{a-b+b x^2} \, dx,x,\sec (e+f x)\right )}{2 a^3 f} \\ & = -\frac {(3 a-4 b) \sqrt {b} \arctan \left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{2 a^3 \sqrt {a-b} f}-\frac {(a-4 b) \text {arctanh}(\cos (e+f x))}{2 a^3 f}-\frac {\cot (e+f x) \csc (e+f x)}{2 a f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {b \sec (e+f x)}{a^2 f \left (a-b+b \sec ^2(e+f x)\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(325\) vs. \(2(147)=294\).
Time = 6.94 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.21 \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=-\frac {(3 a-4 b) \sqrt {a-b} \sqrt {b} \arctan \left (\frac {\sec \left (\frac {1}{2} (e+f x)\right ) \left (\sqrt {a-b} \cos \left (\frac {1}{2} (e+f x)\right )-\sqrt {a} \sin \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {b}}\right )}{2 a^3 (-a+b) f}-\frac {(3 a-4 b) \sqrt {a-b} \sqrt {b} \arctan \left (\frac {\sec \left (\frac {1}{2} (e+f x)\right ) \left (\sqrt {a-b} \cos \left (\frac {1}{2} (e+f x)\right )+\sqrt {a} \sin \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {b}}\right )}{2 a^3 (-a+b) f}-\frac {b \cos (e+f x)}{a^2 f (a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x)))}-\frac {\csc ^2\left (\frac {1}{2} (e+f x)\right )}{8 a^2 f}+\frac {(-a+4 b) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{2 a^3 f}+\frac {(a-4 b) \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{2 a^3 f}+\frac {\sec ^2\left (\frac {1}{2} (e+f x)\right )}{8 a^2 f} \]
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Time = 0.61 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(\frac {\frac {b \left (-\frac {a \cos \left (f x +e \right )}{2 \left (a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b \right )}+\frac {\left (3 a -4 b \right ) \arctan \left (\frac {\left (a -b \right ) \cos \left (f x +e \right )}{\sqrt {b \left (a -b \right )}}\right )}{2 \sqrt {b \left (a -b \right )}}\right )}{a^{3}}+\frac {1}{4 a^{2} \left (\cos \left (f x +e \right )+1\right )}+\frac {\left (-a +4 b \right ) \ln \left (\cos \left (f x +e \right )+1\right )}{4 a^{3}}+\frac {1}{4 a^{2} \left (\cos \left (f x +e \right )-1\right )}+\frac {\left (a -4 b \right ) \ln \left (\cos \left (f x +e \right )-1\right )}{4 a^{3}}}{f}\) | \(156\) |
default | \(\frac {\frac {b \left (-\frac {a \cos \left (f x +e \right )}{2 \left (a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b \right )}+\frac {\left (3 a -4 b \right ) \arctan \left (\frac {\left (a -b \right ) \cos \left (f x +e \right )}{\sqrt {b \left (a -b \right )}}\right )}{2 \sqrt {b \left (a -b \right )}}\right )}{a^{3}}+\frac {1}{4 a^{2} \left (\cos \left (f x +e \right )+1\right )}+\frac {\left (-a +4 b \right ) \ln \left (\cos \left (f x +e \right )+1\right )}{4 a^{3}}+\frac {1}{4 a^{2} \left (\cos \left (f x +e \right )-1\right )}+\frac {\left (a -4 b \right ) \ln \left (\cos \left (f x +e \right )-1\right )}{4 a^{3}}}{f}\) | \(156\) |
risch | \(\frac {a \,{\mathrm e}^{7 i \left (f x +e \right )}-2 b \,{\mathrm e}^{7 i \left (f x +e \right )}+3 a \,{\mathrm e}^{5 i \left (f x +e \right )}+2 b \,{\mathrm e}^{5 i \left (f x +e \right )}+3 a \,{\mathrm e}^{3 i \left (f x +e \right )}+2 b \,{\mathrm e}^{3 i \left (f x +e \right )}+a \,{\mathrm e}^{i \left (f x +e \right )}-2 b \,{\mathrm e}^{i \left (f x +e \right )}}{f \,a^{2} \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{2} \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}-b \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+2 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a -b \right )}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{2 a^{2} f}+\frac {2 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) b}{a^{3} f}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{2 a^{2} f}-\frac {2 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) b}{a^{3} f}-\frac {3 i \sqrt {b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {b \left (a -b \right )}\, {\mathrm e}^{i \left (f x +e \right )}}{a -b}+1\right )}{4 \left (a -b \right ) f \,a^{2}}+\frac {i \sqrt {b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {b \left (a -b \right )}\, {\mathrm e}^{i \left (f x +e \right )}}{a -b}+1\right ) b}{\left (a -b \right ) f \,a^{3}}+\frac {3 i \sqrt {b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {b \left (a -b \right )}\, {\mathrm e}^{i \left (f x +e \right )}}{a -b}+1\right )}{4 \left (a -b \right ) f \,a^{2}}-\frac {i \sqrt {b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {b \left (a -b \right )}\, {\mathrm e}^{i \left (f x +e \right )}}{a -b}+1\right ) b}{\left (a -b \right ) f \,a^{3}}\) | \(515\) |
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Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (133) = 266\).
Time = 0.38 (sec) , antiderivative size = 672, normalized size of antiderivative = 4.57 \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\left [\frac {2 \, {\left (a^{2} - 2 \, a b\right )} \cos \left (f x + e\right )^{3} + 4 \, a b \cos \left (f x + e\right ) - {\left ({\left (3 \, a^{2} - 7 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - {\left (3 \, a^{2} - 10 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - 3 \, a b + 4 \, b^{2}\right )} \sqrt {-\frac {b}{a - b}} \log \left (-\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (a - b\right )} \sqrt {-\frac {b}{a - b}} \cos \left (f x + e\right ) - b}{{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}\right ) - {\left ({\left (a^{2} - 5 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - {\left (a^{2} - 6 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - a b + 4 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) + {\left ({\left (a^{2} - 5 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - {\left (a^{2} - 6 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - a b + 4 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right )}{4 \, {\left ({\left (a^{4} - a^{3} b\right )} f \cos \left (f x + e\right )^{4} - a^{3} b f - {\left (a^{4} - 2 \, a^{3} b\right )} f \cos \left (f x + e\right )^{2}\right )}}, \frac {2 \, {\left (a^{2} - 2 \, a b\right )} \cos \left (f x + e\right )^{3} + 4 \, a b \cos \left (f x + e\right ) - 2 \, {\left ({\left (3 \, a^{2} - 7 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - {\left (3 \, a^{2} - 10 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - 3 \, a b + 4 \, b^{2}\right )} \sqrt {\frac {b}{a - b}} \arctan \left (-\frac {{\left (a - b\right )} \sqrt {\frac {b}{a - b}} \cos \left (f x + e\right )}{b}\right ) - {\left ({\left (a^{2} - 5 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - {\left (a^{2} - 6 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - a b + 4 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) + {\left ({\left (a^{2} - 5 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - {\left (a^{2} - 6 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - a b + 4 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right )}{4 \, {\left ({\left (a^{4} - a^{3} b\right )} f \cos \left (f x + e\right )^{4} - a^{3} b f - {\left (a^{4} - 2 \, a^{3} b\right )} f \cos \left (f x + e\right )^{2}\right )}}\right ] \]
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\[ \int \frac {\csc ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\int \frac {\csc ^{3}{\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 390 vs. \(2 (133) = 266\).
Time = 0.57 (sec) , antiderivative size = 390, normalized size of antiderivative = 2.65 \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\frac {\frac {6 \, {\left (a - 4 \, b\right )} \log \left (\frac {{\left | -\cos \left (f x + e\right ) + 1 \right |}}{{\left | \cos \left (f x + e\right ) + 1 \right |}}\right )}{a^{3}} - \frac {12 \, {\left (3 \, a b - 4 \, b^{2}\right )} \arctan \left (-\frac {a \cos \left (f x + e\right ) - b \cos \left (f x + e\right ) - b}{\sqrt {a b - b^{2}} \cos \left (f x + e\right ) + \sqrt {a b - b^{2}}}\right )}{\sqrt {a b - b^{2}} a^{3}} - \frac {3 \, {\left (\cos \left (f x + e\right ) - 1\right )}}{a^{2} {\left (\cos \left (f x + e\right ) + 1\right )}} + \frac {3 \, a^{2} + \frac {4 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {28 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {16 \, b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {2 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {8 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{3} {\left (\frac {a {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {2 \, a {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {4 \, b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}}{24 \, f} \]
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Time = 11.39 (sec) , antiderivative size = 917, normalized size of antiderivative = 6.24 \[ \int \frac {\csc ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{8\,a^2\,f}-\frac {\frac {a}{2}-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (a-6\,b\right )+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (a^2-8\,a\,b+16\,b^2\right )}{2\,a}}{f\,\left (4\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+4\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (16\,a^2\,b-8\,a^3\right )\right )}+\frac {\ln \left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )\,\left (a-4\,b\right )}{2\,a^3\,f}+\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {2\,a^2\,\left (\frac {\sqrt {b}\,\left (3\,a-4\,b\right )\,\left (12\,a^6\,b-106\,a^5\,b^2+240\,a^4\,b^3-160\,a^3\,b^4\right )}{2\,a^9\,\sqrt {a-b}}+\frac {b^{3/2}\,{\left (3\,a-4\,b\right )}^3\,\left (8\,a^{11}-32\,a^{10}\,b+32\,a^9\,b^2\right )}{32\,a^{15}\,{\left (a-b\right )}^{3/2}}\right )\,\left (a-b\right )\,\left (15\,a^4-182\,a^3\,b+648\,a^2\,b^2-864\,a\,b^3+384\,b^4\right )}{\left (9\,a^2\,b-24\,a\,b^2+16\,b^3\right )\,\left (4\,a^3-27\,a^2\,b+72\,a\,b^2-48\,b^3\right )}-\frac {4\,a^7\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\left (a-b\right )}^{3/2}\,\left (\frac {\left (\frac {4\,\left (9\,a^2\,b^2-24\,a\,b^3+16\,b^4\right )}{a^5}-\frac {b\,{\left (3\,a-4\,b\right )}^2\,\left (2\,a^8-46\,a^7\,b+344\,a^6\,b^2-672\,a^5\,b^3+384\,a^4\,b^4\right )}{4\,a^{11}\,\left (a-b\right )}\right )\,\left (a^4-31\,a^3\,b+180\,a^2\,b^2-336\,a\,b^3+192\,b^4\right )}{\sqrt {b}\,\left (b\,\left (27\,a^7+b\,\left (48\,a^5\,b-72\,a^6\right )\right )-4\,a^8\right )}+\frac {\left (\frac {\sqrt {b}\,\left (3\,a-4\,b\right )\,\left (9\,a^5\,b-78\,a^4\,b^2+268\,a^3\,b^3-384\,a^2\,b^4+192\,a\,b^5\right )}{a^8\,\sqrt {a-b}}-\frac {b^{3/2}\,{\left (3\,a-4\,b\right )}^3\,\left (-4\,a^{10}+104\,a^9\,b-288\,a^8\,b^2+192\,a^7\,b^3\right )}{16\,a^{14}\,{\left (a-b\right )}^{3/2}}\right )\,\left (15\,a^4-182\,a^3\,b+648\,a^2\,b^2-864\,a\,b^3+384\,b^4\right )}{2\,a^5\,\sqrt {a-b}\,\left (4\,a^3-27\,a^2\,b+72\,a\,b^2-48\,b^3\right )}\right )}{9\,a^2\,b-24\,a\,b^2+16\,b^3}+\frac {4\,a^7\,{\left (a-b\right )}^{3/2}\,\left (\frac {2\,\left (9\,a^3\,b^2-60\,a^2\,b^3+112\,a\,b^4-64\,b^5\right )}{a^6}+\frac {b\,{\left (3\,a-4\,b\right )}^2\,\left (-4\,a^9+56\,a^8\,b-160\,a^7\,b^2+128\,a^6\,b^3\right )}{8\,a^{12}\,\left (a-b\right )}\right )\,\left (a^4-31\,a^3\,b+180\,a^2\,b^2-336\,a\,b^3+192\,b^4\right )}{\sqrt {b}\,\left (b\,\left (27\,a^7+b\,\left (48\,a^5\,b-72\,a^6\right )\right )-4\,a^8\right )\,\left (9\,a^2\,b-24\,a\,b^2+16\,b^3\right )}\right )\,\left (3\,a-4\,b\right )}{2\,a^3\,f\,\sqrt {a-b}} \]
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