Optimal. Leaf size=132 \[ -\frac{\left (15 a^2+10 a b+3 b^2\right ) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 f (a+b)^3}-\frac{\cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{5 f (a+b)}-\frac{2 (5 a+3 b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 f (a+b)^2} \]
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Rubi [A] time = 0.139957, antiderivative size = 132, 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 = {4132, 462, 453, 264} \[ -\frac{\left (15 a^2+10 a b+3 b^2\right ) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 f (a+b)^3}-\frac{\cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{5 f (a+b)}-\frac{2 (5 a+3 b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 f (a+b)^2} \]
Antiderivative was successfully verified.
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Rule 4132
Rule 462
Rule 453
Rule 264
Rubi steps
\begin{align*} \int \frac{\csc ^6(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{x^6 \sqrt{a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{\cot ^5(e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{5 (a+b) f}+\frac{\operatorname{Subst}\left (\int \frac{2 (5 a+3 b)+5 (a+b) x^2}{x^4 \sqrt{a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{5 (a+b) f}\\ &=-\frac{2 (5 a+3 b) \cot ^3(e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{15 (a+b)^2 f}-\frac{\cot ^5(e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{5 (a+b) f}+\frac{\left (15 a^2+10 a b+3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{15 (a+b)^2 f}\\ &=-\frac{\left (15 a^2+10 a b+3 b^2\right ) \cot (e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{15 (a+b)^3 f}-\frac{2 (5 a+3 b) \cot ^3(e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{15 (a+b)^2 f}-\frac{\cot ^5(e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{5 (a+b) f}\\ \end{align*}
Mathematica [A] time = 0.350556, size = 100, normalized size = 0.76 \[ -\frac{\csc ^5(e+f x) \sec (e+f x) (a \cos (2 (e+f x))+a+2 b) \left (a^2 \cos (4 (e+f x))+8 a^2-2 a (3 a+b) \cos (2 (e+f x))+8 a b+3 b^2\right )}{30 f (a+b)^3 \sqrt{a+b \sec ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.383, size = 101, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 8\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}{a}^{2}-20\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}{a}^{2}-4\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}ab+15\,{a}^{2}+10\,ab+3\,{b}^{2} \right ) \cos \left ( fx+e \right ) }{15\,f \left ( a+b \right ) ^{3} \left ( \sin \left ( fx+e \right ) \right ) ^{5}}\sqrt{{\frac{b+a \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{ \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 = 1.81973, size = 409, normalized size = 3.1 \begin{align*} -\frac{{\left (8 \, a^{2} \cos \left (f x + e\right )^{5} - 4 \,{\left (5 \, a^{2} + a b\right )} \cos \left (f x + e\right )^{3} +{\left (15 \, a^{2} + 10 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt{\frac{a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{15 \,{\left ({\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} f \cos \left (f x + e\right )^{4} - 2 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} f \cos \left (f x + e\right )^{2} +{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} 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 \sec \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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