Optimal. Leaf size=123 \[ -\frac{\left (15 a^2-20 a b+8 b^2\right ) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^3 f}-\frac{2 (5 a-2 b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^2 f}-\frac{\cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a f} \]
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Rubi [A] time = 0.136504, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3663, 462, 453, 264} \[ -\frac{\left (15 a^2-20 a b+8 b^2\right ) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^3 f}-\frac{2 (5 a-2 b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^2 f}-\frac{\cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a f} \]
Antiderivative was successfully verified.
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Rule 3663
Rule 462
Rule 453
Rule 264
Rubi steps
\begin{align*} \int \frac{\csc ^6(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{x^6 \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{\cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a f}+\frac{\operatorname{Subst}\left (\int \frac{2 (5 a-2 b)+5 a x^2}{x^4 \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{5 a f}\\ &=-\frac{2 (5 a-2 b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^2 f}-\frac{\cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a f}+\frac{\left (15 a^2-20 a b+8 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{15 a^2 f}\\ &=-\frac{\left (15 a^2-20 a b+8 b^2\right ) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^3 f}-\frac{2 (5 a-2 b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^2 f}-\frac{\cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a f}\\ \end{align*}
Mathematica [A] time = 1.81809, size = 90, normalized size = 0.73 \[ -\frac{\cot (e+f x) \left (3 a^2 \csc ^4(e+f x)+4 a (a-b) \csc ^2(e+f x)+8 (a-b)^2\right ) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}{15 \sqrt{2} a^3 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.214, size = 148, normalized size = 1.2 \begin{align*} -{\frac{ \left ( 8\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}{a}^{2}-16\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}ab+8\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}{b}^{2}-20\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}{a}^{2}+36\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}ab-16\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}{b}^{2}+15\,{a}^{2}-20\,ab+8\,{b}^{2} \right ) \cos \left ( fx+e \right ) }{15\,f{a}^{3} \left ( \sin \left ( fx+e \right ) \right ) ^{5}}\sqrt{{\frac{a \left ( \cos \left ( fx+e \right ) \right ) ^{2}- \left ( \cos \left ( fx+e \right ) \right ) ^{2}b+b}{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 6.82206, size = 339, normalized size = 2.76 \begin{align*} -\frac{{\left (8 \,{\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{5} - 4 \,{\left (5 \, a^{2} - 9 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{3} +{\left (15 \, a^{2} - 20 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt{\frac{{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{15 \,{\left (a^{3} f \cos \left (f x + e\right )^{4} - 2 \, a^{3} f \cos \left (f x + e\right )^{2} + a^{3} f\right )} \sin \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (f x + e\right )^{6}}{\sqrt{b \tan \left (f x + e\right )^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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