Optimal. Leaf size=123 \[ -\frac{2 b (3 a-b) \tan (e+f x)}{3 f (a+b)^3 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{\cot ^3(e+f x)}{3 f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{(3 a-b) \cot (e+f x)}{3 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}} \]
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Rubi [A] time = 0.127824, 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 = {4132, 453, 271, 191} \[ -\frac{2 b (3 a-b) \tan (e+f x)}{3 f (a+b)^3 \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{\cot ^3(e+f x)}{3 f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}}-\frac{(3 a-b) \cot (e+f x)}{3 f (a+b)^2 \sqrt{a+b \tan ^2(e+f x)+b}} \]
Antiderivative was successfully verified.
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Rule 4132
Rule 453
Rule 271
Rule 191
Rubi steps
\begin{align*} \int \frac{\csc ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1+x^2}{x^4 \left (a+b+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{\cot ^3(e+f x)}{3 (a+b) f \sqrt{a+b+b \tan ^2(e+f x)}}+\frac{(3 a-b) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 (a+b) f}\\ &=-\frac{(3 a-b) \cot (e+f x)}{3 (a+b)^2 f \sqrt{a+b+b \tan ^2(e+f x)}}-\frac{\cot ^3(e+f x)}{3 (a+b) f \sqrt{a+b+b \tan ^2(e+f x)}}-\frac{(2 (3 a-b) b) \operatorname{Subst}\left (\int \frac{1}{\left (a+b+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 (a+b)^2 f}\\ &=-\frac{(3 a-b) \cot (e+f x)}{3 (a+b)^2 f \sqrt{a+b+b \tan ^2(e+f x)}}-\frac{\cot ^3(e+f x)}{3 (a+b) f \sqrt{a+b+b \tan ^2(e+f x)}}-\frac{2 (3 a-b) b \tan (e+f x)}{3 (a+b)^3 f \sqrt{a+b+b \tan ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.668803, size = 102, normalized size = 0.83 \[ -\frac{\tan (e+f x) \sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left (\left (a^2-2 a b-3 b^2\right ) \csc ^2(e+f x)+(a+b)^2 \csc ^4(e+f x)-2 a (a-3 b)\right )}{6 f (a+b)^3 \left (a+b \sec ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.316, size = 137, normalized size = 1.1 \begin{align*}{\frac{ \left ( 2\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}{a}^{2}-6\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}ab-3\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}{a}^{2}+10\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}ab-3\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}{b}^{2}-6\,ab+2\,{b}^{2} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{3}}{3\,f \left ( a+b \right ) ^{3} \left ( b+a \left ( \cos \left ( fx+e \right ) \right ) ^{2} \right ) ^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{3}} \left ({\frac{b+a \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{ \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 = 2.21597, size = 428, normalized size = 3.48 \begin{align*} -\frac{{\left (2 \,{\left (a^{2} - 3 \, a b\right )} \cos \left (f x + e\right )^{5} -{\left (3 \, a^{2} - 10 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{3} - 2 \,{\left (3 \, a b - 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}}}}{3 \,{\left ({\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} f \cos \left (f x + e\right )^{4} -{\left (a^{4} + 2 \, a^{3} b - 2 \, a b^{3} - b^{4}\right )} f \cos \left (f x + e\right )^{2} -{\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\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 )^{4}}{{\left (b \sec \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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