Optimal. Leaf size=114 \[ -\frac{2 b (3 a-4 b) \tan (e+f x)}{3 a^3 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{(3 a-4 b) \cot (e+f x)}{3 a^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\cot ^3(e+f x)}{3 a f \sqrt{a+b \tan ^2(e+f x)}} \]
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Rubi [A] time = 0.121558, antiderivative size = 114, 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, 453, 271, 191} \[ -\frac{2 b (3 a-4 b) \tan (e+f x)}{3 a^3 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{(3 a-4 b) \cot (e+f x)}{3 a^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\cot ^3(e+f x)}{3 a f \sqrt{a+b \tan ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3663
Rule 453
Rule 271
Rule 191
Rubi steps
\begin{align*} \int \frac{\csc ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1+x^2}{x^4 \left (a+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{\cot ^3(e+f x)}{3 a f \sqrt{a+b \tan ^2(e+f x)}}+\frac{(3 a-4 b) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 a f}\\ &=-\frac{(3 a-4 b) \cot (e+f x)}{3 a^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\cot ^3(e+f x)}{3 a f \sqrt{a+b \tan ^2(e+f x)}}-\frac{(2 (3 a-4 b) b) \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 a^2 f}\\ &=-\frac{(3 a-4 b) \cot (e+f x)}{3 a^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\cot ^3(e+f x)}{3 a f \sqrt{a+b \tan ^2(e+f x)}}-\frac{2 (3 a-4 b) b \tan (e+f x)}{3 a^3 f \sqrt{a+b \tan ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.868828, size = 119, normalized size = 1.04 \[ \frac{\csc ^3(e+f x) \sec (e+f x) \left (-2 \left (a^2-6 a b+8 b^2\right ) \cos (2 (e+f x))+\left (a^2-5 a b+4 b^2\right ) \cos (4 (e+f x))-3 a^2-7 a b+12 b^2\right )}{6 \sqrt{2} a^3 f \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.194, size = 170, normalized size = 1.5 \begin{align*}{\frac{ \left ( 2\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}{a}^{2}-10\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}ab+8\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}{b}^{2}-3\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}{a}^{2}+16\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}ab-16\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}{b}^{2}-6\,ab+8\,{b}^{2} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{3}}{3\,f{a}^{3} \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 ) ^{3}} \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 [A] time = 27.5993, size = 359, normalized size = 3.15 \begin{align*} -\frac{{\left (2 \,{\left (a^{2} - 5 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{5} -{\left (3 \, a^{2} - 16 \, a b + 16 \, b^{2}\right )} \cos \left (f x + e\right )^{3} - 2 \,{\left (3 \, a b - 4 \, 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}}}}{3 \,{\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 )} \sin \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc ^{4}{\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx \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 )^{4}}{{\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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