Optimal. Leaf size=126 \[ -\frac{\left (8 a^2+8 a b+3 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right )}{8 a^{5/2} f}+\frac{(8 a+3 b) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{8 a^2 f}-\frac{\csc ^4(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{4 a f} \]
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Rubi [A] time = 0.128517, antiderivative size = 126, 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, 89, 78, 63, 208} \[ -\frac{\left (8 a^2+8 a b+3 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right )}{8 a^{5/2} f}+\frac{(8 a+3 b) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{8 a^2 f}-\frac{\csc ^4(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{4 a f} \]
Antiderivative was successfully verified.
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Rule 3194
Rule 89
Rule 78
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\cot ^5(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(1-x)^2}{x^3 \sqrt{a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{2 f}\\ &=-\frac{\csc ^4(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{4 a f}+\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{2} (-8 a-3 b)+2 a x}{x^2 \sqrt{a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{4 a f}\\ &=\frac{(8 a+3 b) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{8 a^2 f}-\frac{\csc ^4(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{4 a f}+\frac{\left (8 a^2+8 a b+3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{16 a^2 f}\\ &=\frac{(8 a+3 b) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{8 a^2 f}-\frac{\csc ^4(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{4 a f}+\frac{\left (8 a^2+8 a b+3 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^2 b f}\\ &=-\frac{\left (8 a^2+8 a b+3 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right )}{8 a^{5/2} f}+\frac{(8 a+3 b) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{8 a^2 f}-\frac{\csc ^4(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{4 a f}\\ \end{align*}
Mathematica [A] time = 0.36684, size = 101, normalized size = 0.8 \[ \frac{\sqrt{a} \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)} \left (-2 a \csc ^2(e+f x)+8 a+3 b\right )-\left (8 a^2+8 a b+3 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right )}{8 a^{5/2} f} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.662, size = 219, normalized size = 1.7 \begin{align*} -{\frac{1}{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}{af \left ( \sin \left ( fx+e \right ) \right ) ^{2}}\sqrt{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}}-{\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 ){a}^{-{\frac{3}{2}}}}-{\frac{1}{4\,af \left ( \sin \left ( fx+e \right ) \right ) ^{4}}\sqrt{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}}+{\frac{3\,b}{8\,{a}^{2}f \left ( \sin \left ( fx+e \right ) \right ) ^{2}}\sqrt{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}}-{\frac{3\,{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{5}{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 = 3.08619, size = 923, normalized size = 7.33 \begin{align*} \left [\frac{{\left ({\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \,{\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 8 \, a^{2} + 8 \, a b + 3 \, 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 ({\left (8 \, a^{2} + 3 \, a b\right )} \cos \left (f x + e\right )^{2} - 6 \, a^{2} - 3 \, a b\right )} \sqrt{-b \cos \left (f x + e\right )^{2} + a + b}}{16 \,{\left (a^{3} f \cos \left (f x + e\right )^{4} - 2 \, a^{3} f \cos \left (f x + e\right )^{2} + a^{3} f\right )}}, \frac{{\left ({\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \,{\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-b \cos \left (f x + e\right )^{2} + a + b} \sqrt{-a}}{a}\right ) -{\left ({\left (8 \, a^{2} + 3 \, a b\right )} \cos \left (f x + e\right )^{2} - 6 \, a^{2} - 3 \, a b\right )} \sqrt{-b \cos \left (f x + e\right )^{2} + a + b}}{8 \,{\left (a^{3} f \cos \left (f x + e\right )^{4} - 2 \, a^{3} f \cos \left (f x + e\right )^{2} + a^{3} 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{\cot ^{5}{\left (e + f x \right )}}{\sqrt{a + b \sin ^{2}{\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 \frac{\cot \left (f x + e\right )^{5}}{\sqrt{b \sin \left (f x + e\right )^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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