Optimal. Leaf size=57 \[ \frac {1}{a f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{a^{3/2} f} \]
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Rubi [A] time = 0.08, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3194, 51, 63, 208} \[ \frac {1}{a f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{a^{3/2} f} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 3194
Rubi steps
\begin {align*} \int \frac {\cot (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\sin ^2(e+f x)\right )}{2 f}\\ &=\frac {1}{a 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 f}\\ &=\frac {1}{a 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 b f}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{a^{3/2} f}+\frac {1}{a f \sqrt {a+b \sin ^2(e+f x)}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 46, normalized size = 0.81 \[ \frac {\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b \sin ^2(e+f x)}{a}+1\right )}{a f \sqrt {a+b \sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 225, normalized size = 3.95 \[ \left [\frac {{\left (b \cos \left (f x + e\right )^{2} - a - b\right )} \sqrt {a} \log \left (\frac {2 \, {\left (b \cos \left (f x + e\right )^{2} + 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a} - 2 \, a - b\right )}}{\cos \left (f x + e\right )^{2} - 1}\right ) - 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} a}{2 \, {\left (a^{2} b f \cos \left (f x + e\right )^{2} - {\left (a^{3} + a^{2} b\right )} f\right )}}, \frac {{\left (b \cos \left (f x + e\right )^{2} - a - b\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-a}}{a}\right ) - \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} a}{a^{2} b f \cos \left (f x + e\right )^{2} - {\left (a^{3} + a^{2} b\right )} f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 57, normalized size = 1.00 \[ \frac {\arctan \left (\frac {\sqrt {b \sin \left (f x + e\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a f} + \frac {1}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.58, size = 64, normalized size = 1.12 \[ \frac {1}{a f \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{f \,a^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 46, normalized size = 0.81 \[ -\frac {\frac {\operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right )}{a^{\frac {3}{2}}} - \frac {1}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {cot}\left (e+f\,x\right )}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot {\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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