Optimal. Leaf size=136 \[ -\frac{b (5 a-3 b) \sec (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right )}{a^{5/2} f}-\frac{b \sec (e+f x)}{3 a f (a-b) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}} \]
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Rubi [A] time = 0.145504, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3664, 414, 527, 12, 377, 207} \[ -\frac{b (5 a-3 b) \sec (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right )}{a^{5/2} f}-\frac{b \sec (e+f x)}{3 a f (a-b) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3664
Rule 414
Rule 527
Rule 12
Rule 377
Rule 207
Rubi steps
\begin{align*} \int \frac{\csc (e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (-1+x^2\right ) \left (a-b+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac{b \sec (e+f x)}{3 a (a-b) f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{3 a-b-2 b x^2}{\left (-1+x^2\right ) \left (a-b+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{3 a (a-b) f}\\ &=-\frac{b \sec (e+f x)}{3 a (a-b) f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac{(5 a-3 b) b \sec (e+f x)}{3 a^2 (a-b)^2 f \sqrt{a-b+b \sec ^2(e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{3 (a-b)^2}{\left (-1+x^2\right ) \sqrt{a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{3 a^2 (a-b)^2 f}\\ &=-\frac{b \sec (e+f x)}{3 a (a-b) f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac{(5 a-3 b) b \sec (e+f x)}{3 a^2 (a-b)^2 f \sqrt{a-b+b \sec ^2(e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{\left (-1+x^2\right ) \sqrt{a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{a^2 f}\\ &=-\frac{b \sec (e+f x)}{3 a (a-b) f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac{(5 a-3 b) b \sec (e+f x)}{3 a^2 (a-b)^2 f \sqrt{a-b+b \sec ^2(e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1+a x^2} \, dx,x,\frac{\sec (e+f x)}{\sqrt{a-b+b \sec ^2(e+f x)}}\right )}{a^2 f}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a-b+b \sec ^2(e+f x)}}\right )}{a^{5/2} f}-\frac{b \sec (e+f x)}{3 a (a-b) f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac{(5 a-3 b) b \sec (e+f x)}{3 a^2 (a-b)^2 f \sqrt{a-b+b \sec ^2(e+f x)}}\\ \end{align*}
Mathematica [B] time = 4.70739, size = 305, normalized size = 2.24 \[ \frac{\cos (e+f x) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left (-\frac{2 \sqrt{2} \sqrt{a} b \left (3 \left (2 a^2-3 a b+b^2\right ) \cos (2 (e+f x))+6 a^2+a b-3 b^2\right )}{(a-b)^2 ((a-b) \cos (2 (e+f x))+a+b)^2}-\frac{3 \sec ^2\left (\frac{1}{2} (e+f x)\right ) \left (\tanh ^{-1}\left (\frac{a-(a-2 b) \tan ^2\left (\frac{1}{2} (e+f x)\right )}{\sqrt{a} \sqrt{a \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )-1\right )^2+4 b \tan ^2\left (\frac{1}{2} (e+f x)\right )}}\right )+\tanh ^{-1}\left (\frac{a \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )-1\right )+2 b}{\sqrt{a} \sqrt{a \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )-1\right )^2+4 b \tan ^2\left (\frac{1}{2} (e+f x)\right )}}\right )\right )}{\sqrt{\sec ^4\left (\frac{1}{2} (e+f x)\right ) ((a-b) \cos (2 (e+f x))+a+b)}}\right )}{6 a^{5/2} f} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.415, size = 27448, normalized size = 201.8 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (f x + e\right )}{{\left (b \tan \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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Fricas [B] time = 3.21182, size = 1561, normalized size = 11.48 \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 )}{{\left (b \tan \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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