Optimal. Leaf size=79 \[ \frac{b \cos (e+f x)}{a f (a+b) \sqrt{a-b \cos ^2(e+f x)+b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right )}{a^{3/2} f} \]
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Rubi [A] time = 0.0938341, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3186, 382, 377, 206} \[ \frac{b \cos (e+f x)}{a f (a+b) \sqrt{a-b \cos ^2(e+f x)+b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right )}{a^{3/2} f} \]
Antiderivative was successfully verified.
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Rule 3186
Rule 382
Rule 377
Rule 206
Rubi steps
\begin{align*} \int \frac{\csc (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \left (a+b-b x^2\right )^{3/2}} \, dx,x,\cos (e+f x)\right )}{f}\\ &=\frac{b \cos (e+f x)}{a (a+b) f \sqrt{a+b-b \cos ^2(e+f x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{a+b-b x^2}} \, dx,x,\cos (e+f x)\right )}{a f}\\ &=\frac{b \cos (e+f x)}{a (a+b) f \sqrt{a+b-b \cos ^2(e+f x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{\cos (e+f x)}{\sqrt{a+b-b \cos ^2(e+f x)}}\right )}{a f}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a+b-b \cos ^2(e+f x)}}\right )}{a^{3/2} f}+\frac{b \cos (e+f x)}{a (a+b) f \sqrt{a+b-b \cos ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.338754, size = 93, normalized size = 1.18 \[ \frac{\frac{\sqrt{2} \sqrt{a} b \cos (e+f x)}{(a+b) \sqrt{2 a-b \cos (2 (e+f x))+b}}-\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \cos (e+f x)}{\sqrt{2 a-b \cos (2 (e+f x))+b}}\right )}{a^{3/2} f} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.155, size = 165, normalized size = 2.1 \begin{align*}{\frac{1}{f\cos \left ( fx+e \right ) }\sqrt{- \left ( -b \left ( \sin \left ( fx+e \right ) \right ) ^{2}-a \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}} \left ( -{\frac{1}{2}\ln \left ({\frac{1}{ \left ( \sin \left ( fx+e \right ) \right ) ^{2}} \left ( 2\,a+ \left ( -a+b \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{2}+2\,\sqrt{a}\sqrt{- \left ( -b \left ( \sin \left ( fx+e \right ) \right ) ^{2}-a \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}} \right ) } \right ){a}^{-{\frac{3}{2}}}}+{\frac{b \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{a \left ( a+b \right ) }{\frac{1}{\sqrt{- \left ( -b \left ( \sin \left ( fx+e \right ) \right ) ^{2}-a \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}}}}} \right ){\frac{1}{\sqrt{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{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 [B] time = 2.28678, size = 1008, normalized size = 12.76 \begin{align*} \left [-\frac{4 \, \sqrt{-b \cos \left (f x + e\right )^{2} + a + b} a b \cos \left (f x + e\right ) -{\left ({\left (a b + b^{2}\right )} \cos \left (f x + e\right )^{2} - a^{2} - 2 \, a b - b^{2}\right )} \sqrt{a} \log \left (\frac{2 \,{\left ({\left (a^{2} - 6 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \,{\left (3 \, a^{2} + 2 \, a b - b^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \,{\left ({\left (a - b\right )} \cos \left (f x + e\right )^{3} +{\left (a + b\right )} \cos \left (f x + e\right )\right )} \sqrt{-b \cos \left (f x + e\right )^{2} + a + b} \sqrt{a} + a^{2} + 2 \, a b + b^{2}\right )}}{\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1}\right )}{4 \,{\left ({\left (a^{3} b + a^{2} b^{2}\right )} f \cos \left (f x + e\right )^{2} -{\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} f\right )}}, -\frac{2 \, \sqrt{-b \cos \left (f x + e\right )^{2} + a + b} a b \cos \left (f x + e\right ) -{\left ({\left (a b + b^{2}\right )} \cos \left (f x + e\right )^{2} - a^{2} - 2 \, a b - b^{2}\right )} \sqrt{-a} \arctan \left (-\frac{{\left ({\left (a - b\right )} \cos \left (f x + e\right )^{2} + a + b\right )} \sqrt{-b \cos \left (f x + e\right )^{2} + a + b} \sqrt{-a}}{2 \,{\left (a b \cos \left (f x + e\right )^{3} -{\left (a^{2} + a b\right )} \cos \left (f x + e\right )\right )}}\right )}{2 \,{\left ({\left (a^{3} b + a^{2} b^{2}\right )} f \cos \left (f x + e\right )^{2} -{\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} f\right )}}\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{\left (e + f x \right )}}{\left (a + b \sin ^{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 )}{{\left (b \sin \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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