Optimal. Leaf size=171 \[ -\frac{2 b \left (15 a^2-40 a b+24 b^2\right ) \tan (e+f x)}{15 a^4 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\left (15 a^2-40 a b+24 b^2\right ) \cot (e+f x)}{15 a^3 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{2 (5 a-3 b) \cot ^3(e+f x)}{15 a^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\cot ^5(e+f x)}{5 a f \sqrt{a+b \tan ^2(e+f x)}} \]
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Rubi [A] time = 0.178864, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3663, 462, 453, 271, 191} \[ -\frac{2 b \left (15 a^2-40 a b+24 b^2\right ) \tan (e+f x)}{15 a^4 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\left (15 a^2-40 a b+24 b^2\right ) \cot (e+f x)}{15 a^3 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{2 (5 a-3 b) \cot ^3(e+f x)}{15 a^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\cot ^5(e+f x)}{5 a f \sqrt{a+b \tan ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3663
Rule 462
Rule 453
Rule 271
Rule 191
Rubi steps
\begin{align*} \int \frac{\csc ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{x^6 \left (a+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{\cot ^5(e+f x)}{5 a f \sqrt{a+b \tan ^2(e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{2 (5 a-3 b)+5 a x^2}{x^4 \left (a+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{5 a f}\\ &=-\frac{2 (5 a-3 b) \cot ^3(e+f x)}{15 a^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\cot ^5(e+f x)}{5 a f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\left (-15 a^2+8 (5 a-3 b) b\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{15 a^2 f}\\ &=-\frac{\left (15 a^2-8 (5 a-3 b) b\right ) \cot (e+f x)}{15 a^3 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{2 (5 a-3 b) \cot ^3(e+f x)}{15 a^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\cot ^5(e+f x)}{5 a f \sqrt{a+b \tan ^2(e+f x)}}+\frac{\left (2 b \left (-15 a^2+8 (5 a-3 b) b\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{15 a^3 f}\\ &=-\frac{\left (15 a^2-8 (5 a-3 b) b\right ) \cot (e+f x)}{15 a^3 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{2 (5 a-3 b) \cot ^3(e+f x)}{15 a^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\cot ^5(e+f x)}{5 a f \sqrt{a+b \tan ^2(e+f x)}}-\frac{2 b \left (15 a^2-8 (5 a-3 b) b\right ) \tan (e+f x)}{15 a^4 f \sqrt{a+b \tan ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 1.28677, size = 135, normalized size = 0.79 \[ -\frac{\sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left (\cot (e+f x) \left (3 a^2 \csc ^4(e+f x)+8 a^2+a (4 a-9 b) \csc ^2(e+f x)-41 a b+33 b^2\right )+\frac{15 b (a-b)^2 \sin (2 (e+f x))}{(a-b) \cos (2 (e+f x))+a+b}\right )}{15 \sqrt{2} a^4 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.239, size = 264, normalized size = 1.5 \begin{align*} -{\frac{ \left ( 8\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}{a}^{3}-64\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}{a}^{2}b+104\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}a{b}^{2}-48\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}{b}^{3}-20\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}{a}^{3}+164\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}{a}^{2}b-288\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}a{b}^{2}+144\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}{b}^{3}+15\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}{a}^{3}-130\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}{a}^{2}b+264\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}a{b}^{2}-144\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}{b}^{3}+30\,{a}^{2}b-80\,a{b}^{2}+48\,{b}^{3} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{3}}{15\,f{a}^{4} \left ( a \left ( \cos \left ( fx+e \right ) \right ) ^{2}- \left ( \cos \left ( fx+e \right ) \right ) ^{2}b+b \right ) ^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{5}} \left ({\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}}} \right ) ^{{\frac{3}{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 [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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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}}{{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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