Optimal. Leaf size=171 \[ -\frac{b (13 a-2 b) \sec (e+f x)}{6 a f (a+b)^3 \sqrt{a+b \sec ^2(e+f x)}}-\frac{5 b \sec (e+f x)}{6 f (a+b)^2 \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac{(a-4 b) \tanh ^{-1}\left (\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right )}{2 f (a+b)^{7/2}}-\frac{\cot (e+f x) \csc (e+f x)}{2 f (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}} \]
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Rubi [A] time = 0.206449, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {4134, 471, 527, 12, 377, 207} \[ -\frac{b (13 a-2 b) \sec (e+f x)}{6 a f (a+b)^3 \sqrt{a+b \sec ^2(e+f x)}}-\frac{5 b \sec (e+f x)}{6 f (a+b)^2 \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac{(a-4 b) \tanh ^{-1}\left (\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right )}{2 f (a+b)^{7/2}}-\frac{\cot (e+f x) \csc (e+f x)}{2 f (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4134
Rule 471
Rule 527
Rule 12
Rule 377
Rule 207
Rubi steps
\begin{align*} \int \frac{\csc ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{\left (-1+x^2\right )^2 \left (a+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac{\cot (e+f x) \csc (e+f x)}{2 (a+b) f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{a-4 b x^2}{\left (-1+x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{2 (a+b) f}\\ &=-\frac{\cot (e+f x) \csc (e+f x)}{2 (a+b) f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac{5 b \sec (e+f x)}{6 (a+b)^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{a (3 a-2 b)-10 a b x^2}{\left (-1+x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{6 a (a+b)^2 f}\\ &=-\frac{\cot (e+f x) \csc (e+f x)}{2 (a+b) f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac{5 b \sec (e+f x)}{6 (a+b)^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac{(13 a-2 b) b \sec (e+f x)}{6 a (a+b)^3 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{3 a^2 (a-4 b)}{\left (-1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\sec (e+f x)\right )}{6 a^2 (a+b)^3 f}\\ &=-\frac{\cot (e+f x) \csc (e+f x)}{2 (a+b) f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac{5 b \sec (e+f x)}{6 (a+b)^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac{(13 a-2 b) b \sec (e+f x)}{6 a (a+b)^3 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{(a-4 b) \operatorname{Subst}\left (\int \frac{1}{\left (-1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\sec (e+f x)\right )}{2 (a+b)^3 f}\\ &=-\frac{\cot (e+f x) \csc (e+f x)}{2 (a+b) f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac{5 b \sec (e+f x)}{6 (a+b)^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac{(13 a-2 b) b \sec (e+f x)}{6 a (a+b)^3 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{(a-4 b) \operatorname{Subst}\left (\int \frac{1}{-1-(-a-b) x^2} \, dx,x,\frac{\sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right )}{2 (a+b)^3 f}\\ &=-\frac{(a-4 b) \tanh ^{-1}\left (\frac{\sqrt{a+b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}}\right )}{2 (a+b)^{7/2} f}-\frac{\cot (e+f x) \csc (e+f x)}{2 (a+b) f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac{5 b \sec (e+f x)}{6 (a+b)^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac{(13 a-2 b) b \sec (e+f x)}{6 a (a+b)^3 f \sqrt{a+b \sec ^2(e+f x)}}\\ \end{align*}
Mathematica [C] time = 1.54846, size = 151, normalized size = 0.88 \[ -\frac{\sec ^5(e+f x) (a \cos (2 (e+f x))+a+2 b) \left ((a+b) \csc ^2(e+f x) \left (\left (3 a^2+2 b^2\right ) \cos (2 (e+f x))+3 a^2+6 a b-2 b^2\right )-3 a (a-4 b) (a \cos (2 (e+f x))+a+2 b) \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},1-\frac{a \sin ^2(e+f x)}{a+b}\right )\right )}{24 a f (a+b)^3 \left (a+b \sec ^2(e+f x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.021, size = 11110, normalized size = 65. \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [B] time = 1.31295, size = 2086, normalized size = 12.2 \begin{align*} \text{result too large to display} \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 )^{3}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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