Optimal. Leaf size=129 \[ \frac{b (5 a+3 b) \cos (e+f x)}{3 a^2 f (a+b)^2 \sqrt{a-b \cos ^2(e+f x)+b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right )}{a^{5/2} f}+\frac{b \cos (e+f x)}{3 a f (a+b) \left (a-b \cos ^2(e+f x)+b\right )^{3/2}} \]
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Rubi [A] time = 0.152596, antiderivative size = 129, 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 = {3186, 414, 527, 12, 377, 206} \[ \frac{b (5 a+3 b) \cos (e+f x)}{3 a^2 f (a+b)^2 \sqrt{a-b \cos ^2(e+f x)+b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right )}{a^{5/2} f}+\frac{b \cos (e+f x)}{3 a f (a+b) \left (a-b \cos ^2(e+f x)+b\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3186
Rule 414
Rule 527
Rule 12
Rule 377
Rule 206
Rubi steps
\begin{align*} \int \frac{\csc (e+f x)}{\left (a+b \sin ^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,\cos (e+f x)\right )}{f}\\ &=\frac{b \cos (e+f x)}{3 a (a+b) f \left (a+b-b \cos ^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,\cos (e+f x)\right )}{3 a (a+b) f}\\ &=\frac{b \cos (e+f x)}{3 a (a+b) f \left (a+b-b \cos ^2(e+f x)\right )^{3/2}}+\frac{b (5 a+3 b) \cos (e+f x)}{3 a^2 (a+b)^2 f \sqrt{a+b-b \cos ^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,\cos (e+f x)\right )}{3 a^2 (a+b)^2 f}\\ &=\frac{b \cos (e+f x)}{3 a (a+b) f \left (a+b-b \cos ^2(e+f x)\right )^{3/2}}+\frac{b (5 a+3 b) \cos (e+f x)}{3 a^2 (a+b)^2 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^2 f}\\ &=\frac{b \cos (e+f x)}{3 a (a+b) f \left (a+b-b \cos ^2(e+f x)\right )^{3/2}}+\frac{b (5 a+3 b) \cos (e+f x)}{3 a^2 (a+b)^2 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^2 f}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a+b-b \cos ^2(e+f x)}}\right )}{a^{5/2} f}+\frac{b \cos (e+f x)}{3 a (a+b) f \left (a+b-b \cos ^2(e+f x)\right )^{3/2}}+\frac{b (5 a+3 b) \cos (e+f x)}{3 a^2 (a+b)^2 f \sqrt{a+b-b \cos ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.590195, size = 127, normalized size = 0.98 \[ \frac{\frac{\sqrt{2} b \cos (e+f x) \left (12 a^2-b (5 a+3 b) \cos (2 (e+f x))+13 a b+3 b^2\right )}{3 a^2 (a+b)^2 (2 a-b \cos (2 (e+f x))+b)^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \cos (e+f x)}{\sqrt{2 a-b \cos (2 (e+f x))+b}}\right )}{a^{5/2}}}{f} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.726, size = 249, normalized size = 1.9 \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{5}{2}}}}+{\frac{b \left ( 2\,b \left ( \sin \left ( fx+e \right ) \right ) ^{2}+3\,a+b \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{3\,a \left ( a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \left ({a}^{2}+2\,ab+{b}^{2} \right ) }{\frac{1}{\sqrt{- \left ( -b \left ( \sin \left ( fx+e \right ) \right ) ^{2}-a \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}}}}}+{\frac{b \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{{a}^{2} \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 = 4.16613, size = 1723, normalized size = 13.36 \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 \sin \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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