Optimal. Leaf size=165 \[ \frac{\left (8 a^2-8 a b-b^2\right ) \sqrt{a+b \sin ^2(e+f x)}}{8 a^2 f}-\frac{\left (8 a^2-8 a b-b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right )}{8 a^{3/2} f}+\frac{(8 a+b) \csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{8 a^2 f}-\frac{\csc ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 a f} \]
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Rubi [A] time = 0.154806, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {3194, 89, 78, 50, 63, 208} \[ \frac{\left (8 a^2-8 a b-b^2\right ) \sqrt{a+b \sin ^2(e+f x)}}{8 a^2 f}-\frac{\left (8 a^2-8 a b-b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right )}{8 a^{3/2} f}+\frac{(8 a+b) \csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{8 a^2 f}-\frac{\csc ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 a f} \]
Antiderivative was successfully verified.
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Rule 3194
Rule 89
Rule 78
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \cot ^5(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(1-x)^2 \sqrt{a+b x}}{x^3} \, dx,x,\sin ^2(e+f x)\right )}{2 f}\\ &=-\frac{\csc ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 a f}+\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{1}{2} (-8 a-b)+2 a x\right ) \sqrt{a+b x}}{x^2} \, dx,x,\sin ^2(e+f x)\right )}{4 a f}\\ &=\frac{(8 a+b) \csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{8 a^2 f}-\frac{\csc ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 a f}+\frac{\left (8 a^2-8 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\sin ^2(e+f x)\right )}{16 a^2 f}\\ &=\frac{\left (8 a^2-8 a b-b^2\right ) \sqrt{a+b \sin ^2(e+f x)}}{8 a^2 f}+\frac{(8 a+b) \csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{8 a^2 f}-\frac{\csc ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 a f}+\frac{\left (8 a^2-8 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{16 a f}\\ &=\frac{\left (8 a^2-8 a b-b^2\right ) \sqrt{a+b \sin ^2(e+f x)}}{8 a^2 f}+\frac{(8 a+b) \csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{8 a^2 f}-\frac{\csc ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 a f}+\frac{\left (8 a^2-8 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sin ^2(e+f x)}\right )}{8 a b f}\\ &=-\frac{\left (8 a^2-8 a b-b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right )}{8 a^{3/2} f}+\frac{\left (8 a^2-8 a b-b^2\right ) \sqrt{a+b \sin ^2(e+f x)}}{8 a^2 f}+\frac{(8 a+b) \csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{8 a^2 f}-\frac{\csc ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 a f}\\ \end{align*}
Mathematica [A] time = 0.589232, size = 103, normalized size = 0.62 \[ \frac{\left (-8 a^2+8 a b+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right )+\sqrt{a} \sqrt{a+b \sin ^2(e+f x)} \left ((8 a-b) \csc ^2(e+f x)-2 a \csc ^4(e+f x)+8 a\right )}{8 a^{3/2} f} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.502, size = 230, normalized size = 1.4 \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}{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{b}{8\,af \left ( \sin \left ( fx+e \right ) \right ) ^{2}}\sqrt{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}}+{\frac{{b}^{2}}{8\,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 ){a}^{-{\frac{3}{2}}}}+{\frac{1}{f \left ( \sin \left ( fx+e \right ) \right ) ^{2}}\sqrt{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}}-{\frac{1}{4\,f \left ( \sin \left ( fx+e \right ) \right ) ^{4}}\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 = 11.5458, size = 965, normalized size = 5.85 \begin{align*} \left [-\frac{{\left ({\left (8 \, a^{2} - 8 \, a b - b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \,{\left (8 \, a^{2} - 8 \, a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + 8 \, a^{2} - 8 \, a b - b^{2}\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 (8 \, a^{2} \cos \left (f x + e\right )^{4} -{\left (24 \, a^{2} - a b\right )} \cos \left (f x + e\right )^{2} + 14 \, a^{2} - a b\right )} \sqrt{-b \cos \left (f x + e\right )^{2} + a + b}}{16 \,{\left (a^{2} f \cos \left (f x + e\right )^{4} - 2 \, a^{2} f \cos \left (f x + e\right )^{2} + a^{2} f\right )}}, \frac{{\left ({\left (8 \, a^{2} - 8 \, a b - b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \,{\left (8 \, a^{2} - 8 \, a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + 8 \, a^{2} - 8 \, a b - b^{2}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-b \cos \left (f x + e\right )^{2} + a + b} \sqrt{-a}}{a}\right ) +{\left (8 \, a^{2} \cos \left (f x + e\right )^{4} -{\left (24 \, a^{2} - a b\right )} \cos \left (f x + e\right )^{2} + 14 \, a^{2} - a b\right )} \sqrt{-b \cos \left (f x + e\right )^{2} + a + b}}{8 \,{\left (a^{2} f \cos \left (f x + e\right )^{4} - 2 \, a^{2} f \cos \left (f x + e\right )^{2} + a^{2} f\right )}}\right ] \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 \sqrt{b \sin \left (f x + e\right )^{2} + a} \cot \left (f x + e\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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