Integrand size = 23, antiderivative size = 103 \[ \int \csc ^6(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=-\frac {\left (a^2+6 a b+6 b^2\right ) \cot (e+f x)}{f}-\frac {2 (a+b) (a+2 b) \cot ^3(e+f x)}{3 f}-\frac {(a+b)^2 \cot ^5(e+f x)}{5 f}+\frac {2 b (a+2 b) \tan (e+f x)}{f}+\frac {b^2 \tan ^3(e+f x)}{3 f} \]
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Time = 0.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {4217, 459} \[ \int \csc ^6(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=-\frac {\left (a^2+6 a b+6 b^2\right ) \cot (e+f x)}{f}+\frac {2 b (a+2 b) \tan (e+f x)}{f}-\frac {(a+b)^2 \cot ^5(e+f x)}{5 f}-\frac {2 (a+b) (a+2 b) \cot ^3(e+f x)}{3 f}+\frac {b^2 \tan ^3(e+f x)}{3 f} \]
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Rule 459
Rule 4217
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}{x^6} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \left (2 b (a+2 b)+\frac {(a+b)^2}{x^6}+\frac {2 (a+b) (a+2 b)}{x^4}+\frac {a^2+6 a b+6 b^2}{x^2}+b^2 x^2\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {\left (a^2+6 a b+6 b^2\right ) \cot (e+f x)}{f}-\frac {2 (a+b) (a+2 b) \cot ^3(e+f x)}{3 f}-\frac {(a+b)^2 \cot ^5(e+f x)}{5 f}+\frac {2 b (a+2 b) \tan (e+f x)}{f}+\frac {b^2 \tan ^3(e+f x)}{3 f} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(353\) vs. \(2(103)=206\).
Time = 2.67 (sec) , antiderivative size = 353, normalized size of antiderivative = 3.43 \[ \int \csc ^6(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=-\frac {\csc (e) \csc ^5(e+f x) \sec (e) \sec ^3(e+f x) \left (20 a (5 a+12 b) \sin (2 e)-32 \left (2 a^2+9 a b+12 b^2\right ) \sin (2 f x)-24 a^2 \sin (2 (e+f x))-108 a b \sin (2 (e+f x))-54 b^2 \sin (2 (e+f x))+8 a^2 \sin (4 (e+f x))+36 a b \sin (4 (e+f x))+18 b^2 \sin (4 (e+f x))+8 a^2 \sin (6 (e+f x))+36 a b \sin (6 (e+f x))+18 b^2 \sin (6 (e+f x))-4 a^2 \sin (8 (e+f x))-18 a b \sin (8 (e+f x))-9 b^2 \sin (8 (e+f x))+8 a^2 \sin (2 (e+2 f x))+96 a b \sin (2 (e+2 f x))+128 b^2 \sin (2 (e+2 f x))+40 a^2 \sin (4 e+2 f x)+8 a^2 \sin (4 e+6 f x)+96 a b \sin (4 e+6 f x)+128 b^2 \sin (4 e+6 f x)-4 a^2 \sin (6 e+8 f x)-48 a b \sin (6 e+8 f x)-64 b^2 \sin (6 e+8 f x)\right )}{1920 f} \]
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Time = 0.45 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.50
method | result | size |
parallelrisch | \(\frac {\left (\left (-13 a^{2}-36 a b -48 b^{2}\right ) \cos \left (2 f x +2 e \right )+\left (a^{2}+12 a b +16 b^{2}\right ) \cos \left (4 f x +4 e \right )+\left (a^{2}+12 a b +16 b^{2}\right ) \cos \left (6 f x +6 e \right )+\left (-\frac {1}{2} a^{2}-6 a b -8 b^{2}\right ) \cos \left (8 f x +8 e \right )-\frac {25 a^{2}}{2}-30 a b \right ) \csc \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} \sec \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{960 f \left (\cos \left (3 f x +3 e \right )+3 \cos \left (f x +e \right )\right )}\) | \(154\) |
derivativedivides | \(\frac {a^{2} \left (-\frac {8}{15}-\frac {\csc \left (f x +e \right )^{4}}{5}-\frac {4 \csc \left (f x +e \right )^{2}}{15}\right ) \cot \left (f x +e \right )+2 a b \left (-\frac {1}{5 \sin \left (f x +e \right )^{5} \cos \left (f x +e \right )}-\frac {2}{5 \sin \left (f x +e \right )^{3} \cos \left (f x +e \right )}+\frac {8}{5 \sin \left (f x +e \right ) \cos \left (f x +e \right )}-\frac {16 \cot \left (f x +e \right )}{5}\right )+b^{2} \left (-\frac {1}{5 \sin \left (f x +e \right )^{5} \cos \left (f x +e \right )^{3}}+\frac {8}{15 \sin \left (f x +e \right )^{3} \cos \left (f x +e \right )^{3}}-\frac {16}{15 \sin \left (f x +e \right )^{3} \cos \left (f x +e \right )}+\frac {64}{15 \sin \left (f x +e \right ) \cos \left (f x +e \right )}-\frac {128 \cot \left (f x +e \right )}{15}\right )}{f}\) | \(190\) |
default | \(\frac {a^{2} \left (-\frac {8}{15}-\frac {\csc \left (f x +e \right )^{4}}{5}-\frac {4 \csc \left (f x +e \right )^{2}}{15}\right ) \cot \left (f x +e \right )+2 a b \left (-\frac {1}{5 \sin \left (f x +e \right )^{5} \cos \left (f x +e \right )}-\frac {2}{5 \sin \left (f x +e \right )^{3} \cos \left (f x +e \right )}+\frac {8}{5 \sin \left (f x +e \right ) \cos \left (f x +e \right )}-\frac {16 \cot \left (f x +e \right )}{5}\right )+b^{2} \left (-\frac {1}{5 \sin \left (f x +e \right )^{5} \cos \left (f x +e \right )^{3}}+\frac {8}{15 \sin \left (f x +e \right )^{3} \cos \left (f x +e \right )^{3}}-\frac {16}{15 \sin \left (f x +e \right )^{3} \cos \left (f x +e \right )}+\frac {64}{15 \sin \left (f x +e \right ) \cos \left (f x +e \right )}-\frac {128 \cot \left (f x +e \right )}{15}\right )}{f}\) | \(190\) |
risch | \(-\frac {16 i \left (10 a^{2} {\mathrm e}^{10 i \left (f x +e \right )}+25 a^{2} {\mathrm e}^{8 i \left (f x +e \right )}+60 a b \,{\mathrm e}^{8 i \left (f x +e \right )}+16 a^{2} {\mathrm e}^{6 i \left (f x +e \right )}+72 a b \,{\mathrm e}^{6 i \left (f x +e \right )}+96 b^{2} {\mathrm e}^{6 i \left (f x +e \right )}-2 a^{2} {\mathrm e}^{4 i \left (f x +e \right )}-24 a b \,{\mathrm e}^{4 i \left (f x +e \right )}-32 b^{2} {\mathrm e}^{4 i \left (f x +e \right )}-2 a^{2} {\mathrm e}^{2 i \left (f x +e \right )}-24 a b \,{\mathrm e}^{2 i \left (f x +e \right )}-32 b^{2} {\mathrm e}^{2 i \left (f x +e \right )}+a^{2}+12 a b +16 b^{2}\right )}{15 f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{5} \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{3}}\) | \(210\) |
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Time = 0.25 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.35 \[ \int \csc ^6(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=-\frac {8 \, {\left (a^{2} + 12 \, a b + 16 \, b^{2}\right )} \cos \left (f x + e\right )^{8} - 20 \, {\left (a^{2} + 12 \, a b + 16 \, b^{2}\right )} \cos \left (f x + e\right )^{6} + 15 \, {\left (a^{2} + 12 \, a b + 16 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 10 \, {\left (3 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - 5 \, b^{2}}{15 \, {\left (f \cos \left (f x + e\right )^{7} - 2 \, f \cos \left (f x + e\right )^{5} + f \cos \left (f x + e\right )^{3}\right )} \sin \left (f x + e\right )} \]
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Timed out. \[ \int \csc ^6(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.04 \[ \int \csc ^6(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {5 \, b^{2} \tan \left (f x + e\right )^{3} + 30 \, {\left (a b + 2 \, b^{2}\right )} \tan \left (f x + e\right ) - \frac {15 \, {\left (a^{2} + 6 \, a b + 6 \, b^{2}\right )} \tan \left (f x + e\right )^{4} + 10 \, {\left (a^{2} + 3 \, a b + 2 \, b^{2}\right )} \tan \left (f x + e\right )^{2} + 3 \, a^{2} + 6 \, a b + 3 \, b^{2}}{\tan \left (f x + e\right )^{5}}}{15 \, f} \]
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Time = 0.39 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.37 \[ \int \csc ^6(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {5 \, b^{2} \tan \left (f x + e\right )^{3} + 30 \, a b \tan \left (f x + e\right ) + 60 \, b^{2} \tan \left (f x + e\right ) - \frac {15 \, a^{2} \tan \left (f x + e\right )^{4} + 90 \, a b \tan \left (f x + e\right )^{4} + 90 \, b^{2} \tan \left (f x + e\right )^{4} + 10 \, a^{2} \tan \left (f x + e\right )^{2} + 30 \, a b \tan \left (f x + e\right )^{2} + 20 \, b^{2} \tan \left (f x + e\right )^{2} + 3 \, a^{2} + 6 \, a b + 3 \, b^{2}}{\tan \left (f x + e\right )^{5}}}{15 \, f} \]
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Time = 18.75 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.05 \[ \int \csc ^6(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {b^2\,{\mathrm {tan}\left (e+f\,x\right )}^3}{3\,f}-\frac {\frac {2\,a\,b}{5}+{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (a^2+6\,a\,b+6\,b^2\right )+\frac {a^2}{5}+\frac {b^2}{5}+{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {2\,a^2}{3}+2\,a\,b+\frac {4\,b^2}{3}\right )}{f\,{\mathrm {tan}\left (e+f\,x\right )}^5}+\frac {2\,b\,\mathrm {tan}\left (e+f\,x\right )\,\left (a+2\,b\right )}{f} \]
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