Optimal. Leaf size=83 \[ \frac{1}{a^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right )}{a^{5/2} f}+\frac{1}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}} \]
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Rubi [A] time = 0.0876517, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3194, 51, 63, 208} \[ \frac{1}{a^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right )}{a^{5/2} f}+\frac{1}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3194
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\cot (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{5/2}} \, dx,x,\sin ^2(e+f x)\right )}{2 f}\\ &=\frac{1}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{3/2}} \, dx,x,\sin ^2(e+f x)\right )}{2 a f}\\ &=\frac{1}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac{1}{a^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{2 a^2 f}\\ &=\frac{1}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac{1}{a^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sin ^2(e+f x)}\right )}{a^2 b f}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right )}{a^{5/2} f}+\frac{1}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac{1}{a^2 f \sqrt{a+b \sin ^2(e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.0643929, size = 49, normalized size = 0.59 \[ \frac{\, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{b \sin ^2(e+f x)}{a}+1\right )}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.888, size = 271, normalized size = 3.3 \begin{align*} -{\frac{7}{12\,{a}^{2}f}\sqrt{-b \left ( \cos \left ( fx+e \right ) \right ) ^{2}+{\frac{ab+{b}^{2}}{b}}}{\frac{1}{\sqrt{-ab}}} \left ( \sin \left ( fx+e \right ) +{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}}+{\frac{7}{12\,{a}^{2}f}\sqrt{-b \left ( \cos \left ( fx+e \right ) \right ) ^{2}+{\frac{ab+{b}^{2}}{b}}}{\frac{1}{\sqrt{-ab}}} \left ( \sin \left ( fx+e \right ) -{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}}-{\frac{1}{12\,{a}^{2}fb}\sqrt{-b \left ( \cos \left ( fx+e \right ) \right ) ^{2}+{\frac{ab+{b}^{2}}{b}}} \left ( \sin \left ( fx+e \right ) +{\frac{1}{b}\sqrt{-ab}} \right ) ^{-2}}-{\frac{1}{12\,{a}^{2}fb}\sqrt{-b \left ( \cos \left ( fx+e \right ) \right ) ^{2}+{\frac{ab+{b}^{2}}{b}}} \left ( \sin \left ( fx+e \right ) -{\frac{1}{b}\sqrt{-ab}} \right ) ^{-2}}-{\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 ){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 [B] time = 2.57794, size = 911, normalized size = 10.98 \begin{align*} \left [\frac{3 \,{\left (b^{2} \cos \left (f x + e\right )^{4} - 2 \,{\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 (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 (3 \, a b \cos \left (f x + e\right )^{2} - 4 \, a^{2} - 3 \, a b\right )} \sqrt{-b \cos \left (f x + e\right )^{2} + a + b}}{6 \,{\left (a^{3} b^{2} f \cos \left (f x + e\right )^{4} - 2 \,{\left (a^{4} b + a^{3} b^{2}\right )} f \cos \left (f x + e\right )^{2} +{\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} f\right )}}, \frac{3 \,{\left (b^{2} \cos \left (f x + e\right )^{4} - 2 \,{\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{\sqrt{-b \cos \left (f x + e\right )^{2} + a + b} \sqrt{-a}}{a}\right ) -{\left (3 \, a b \cos \left (f x + e\right )^{2} - 4 \, a^{2} - 3 \, a b\right )} \sqrt{-b \cos \left (f x + e\right )^{2} + a + b}}{3 \,{\left (a^{3} b^{2} f \cos \left (f x + e\right )^{4} - 2 \,{\left (a^{4} b + a^{3} b^{2}\right )} f \cos \left (f x + e\right )^{2} +{\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} 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 [A] time = 1.10856, size = 100, normalized size = 1.2 \begin{align*} \frac{\arctan \left (\frac{\sqrt{b \sin \left (f x + e\right )^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2} f} + \frac{3 \, b \sin \left (f x + e\right )^{2} + 4 \, a}{3 \,{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} a^{2} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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