Optimal. Leaf size=63 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a+b}}\right )}{f (a+b)^{3/2}}-\frac {1}{f (a+b) \sqrt {a+b \sin ^2(e+f x)}} \]
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Rubi [A] time = 0.07, antiderivative size = 63, 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} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a+b}}\right )}{f (a+b)^{3/2}}-\frac {1}{f (a+b) \sqrt {a+b \sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 3194
Rubi steps
\begin {align*} \int \frac {\tan (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{(1-x) (a+b x)^{3/2}} \, dx,x,\sin ^2(e+f x)\right )}{2 f}\\ &=-\frac {1}{(a+b) f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{(1-x) \sqrt {a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{2 (a+b) f}\\ &=-\frac {1}{(a+b) f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sin ^2(e+f x)}\right )}{b (a+b) f}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a+b}}\right )}{(a+b)^{3/2} f}-\frac {1}{(a+b) f \sqrt {a+b \sin ^2(e+f x)}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 54, normalized size = 0.86 \[ -\frac {\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1-\frac {b \cos ^2(e+f x)}{a+b}\right )}{f (a+b) \sqrt {a-b \cos ^2(e+f x)+b}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 281, normalized size = 4.46 \[ \left [\frac {{\left (b \cos \left (f x + e\right )^{2} - a - b\right )} \sqrt {a + b} \log \left (\frac {b \cos \left (f x + e\right )^{2} - 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a + b} - 2 \, a - 2 \, b}{\cos \left (f x + e\right )^{2}}\right ) + 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} {\left (a + b\right )}}{2 \, {\left ({\left (a^{2} b + 2 \, a b^{2} + b^{3}\right )} f \cos \left (f x + e\right )^{2} - {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} f\right )}}, -\frac {{\left (b \cos \left (f x + e\right )^{2} - a - b\right )} \sqrt {-a - b} \arctan \left (\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-a - b}}{a + b}\right ) - \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} {\left (a + b\right )}}{{\left (a^{2} b + 2 \, a b^{2} + b^{3}\right )} f \cos \left (f x + e\right )^{2} - {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.89, size = 250, normalized size = 3.97 \[ -\frac {\frac {\frac {{\left (a^{2} b + 2 \, a b^{2} + b^{3}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}{a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}} + \frac {a^{2} b + 2 \, a b^{2} + b^{3}}{a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}}}{\sqrt {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 2 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 4 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}} + \frac {2 \, \arctan \left (-\frac {\sqrt {a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 2 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 4 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a} - \sqrt {a}}{2 \, \sqrt {-a - b}}\right )}{{\left (a + b\right )} \sqrt {-a - b}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 8.49, size = 1317, normalized size = 20.90 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.66, size = 143, normalized size = 2.27 \[ -\frac {\frac {\operatorname {arsinh}\left (\frac {b \sin \left (f x + e\right )}{\sqrt {a b} {\left (\sin \left (f x + e\right ) + 1\right )}} - \frac {a}{\sqrt {a b} {\left (\sin \left (f x + e\right ) + 1\right )}}\right )}{{\left (a + b\right )}^{\frac {3}{2}}} - \frac {\operatorname {arsinh}\left (-\frac {b \sin \left (f x + e\right )}{\sqrt {a b} {\left (\sin \left (f x + e\right ) - 1\right )}} - \frac {a}{\sqrt {a b} {\left (\sin \left (f x + e\right ) - 1\right )}}\right )}{{\left (a + b\right )}^{\frac {3}{2}}} + \frac {2}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a + \sqrt {b \sin \left (f x + e\right )^{2} + a} b}}{2 \, f} \]
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 \[ \int \frac {\mathrm {tan}\left (e+f\,x\right )}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\tan {\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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