Optimal. Leaf size=110 \[ -\frac{(2 a-b) \sqrt{a+b \sin ^2(e+f x)}}{2 a f}+\frac{(2 a-b) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right )}{2 \sqrt{a} f}-\frac{\csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{2 a f} \]
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Rubi [A] time = 0.102533, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3194, 78, 50, 63, 208} \[ -\frac{(2 a-b) \sqrt{a+b \sin ^2(e+f x)}}{2 a f}+\frac{(2 a-b) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right )}{2 \sqrt{a} f}-\frac{\csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{2 a f} \]
Antiderivative was successfully verified.
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Rule 3194
Rule 78
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \cot ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(1-x) \sqrt{a+b x}}{x^2} \, dx,x,\sin ^2(e+f x)\right )}{2 f}\\ &=-\frac{\csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{2 a f}-\frac{(2 a-b) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\sin ^2(e+f x)\right )}{4 a f}\\ &=-\frac{(2 a-b) \sqrt{a+b \sin ^2(e+f x)}}{2 a f}-\frac{\csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{2 a f}-\frac{(2 a-b) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{4 f}\\ &=-\frac{(2 a-b) \sqrt{a+b \sin ^2(e+f x)}}{2 a f}-\frac{\csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{2 a f}-\frac{(2 a-b) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sin ^2(e+f x)}\right )}{2 b f}\\ &=\frac{(2 a-b) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right )}{2 \sqrt{a} f}-\frac{(2 a-b) \sqrt{a+b \sin ^2(e+f x)}}{2 a f}-\frac{\csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{2 a f}\\ \end{align*}
Mathematica [A] time = 0.198836, size = 77, normalized size = 0.7 \[ \frac{(2 a-b) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right )-\sqrt{a} \left (\csc ^2(e+f x)+2\right ) \sqrt{a+b \sin ^2(e+f x)}}{2 \sqrt{a} f} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.22, size = 130, normalized size = 1.2 \begin{align*} -{\frac{1}{f}\sqrt{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}}+{\frac{1}{f}\sqrt{a}\ln \left ({\frac{1}{\sin \left ( fx+e \right ) } \left ( 2\,a+2\,\sqrt{a}\sqrt{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}} \right ) } \right ) }-{\frac{b}{2\,f}\ln \left ({\frac{1}{\sin \left ( fx+e \right ) } \left ( 2\,a+2\,\sqrt{a}\sqrt{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}} \right ) } \right ){\frac{1}{\sqrt{a}}}}-{\frac{1}{2\,f \left ( \sin \left ( fx+e \right ) \right ) ^{2}}\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 [A] time = 8.42464, size = 595, normalized size = 5.41 \begin{align*} \left [-\frac{{\left ({\left (2 \, a - b\right )} \cos \left (f x + e\right )^{2} - 2 \, a + b\right )} \sqrt{a} \log \left (\frac{2 \,{\left (b \cos \left (f x + e\right )^{2} + 2 \, \sqrt{-b \cos \left (f x + e\right )^{2} + a + b} \sqrt{a} - 2 \, a - b\right )}}{\cos \left (f x + e\right )^{2} - 1}\right ) + 2 \,{\left (2 \, a \cos \left (f x + e\right )^{2} - 3 \, a\right )} \sqrt{-b \cos \left (f x + e\right )^{2} + a + b}}{4 \,{\left (a f \cos \left (f x + e\right )^{2} - a f\right )}}, -\frac{{\left ({\left (2 \, a - b\right )} \cos \left (f x + e\right )^{2} - 2 \, a + b\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-b \cos \left (f x + e\right )^{2} + a + b} \sqrt{-a}}{a}\right ) +{\left (2 \, a \cos \left (f x + e\right )^{2} - 3 \, a\right )} \sqrt{-b \cos \left (f x + e\right )^{2} + a + b}}{2 \,{\left (a f \cos \left (f x + e\right )^{2} - a 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 \sqrt{a + b \sin ^{2}{\left (e + f x \right )}} \cot ^{3}{\left (e + f x \right )}\, 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 \sqrt{b \sin \left (f x + e\right )^{2} + a} \cot \left (f x + e\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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