Optimal. Leaf size=79 \[ \frac {2 \cos ^2(e+f x)^{3/4} (d \tan (e+f x))^{3/2} (b \csc (e+f x))^m \, _2F_1\left (\frac {3}{4},\frac {1}{4} (3-2 m);\frac {1}{4} (7-2 m);\sin ^2(e+f x)\right )}{d f (3-2 m)} \]
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Rubi [A] time = 0.15, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2618, 2602, 2577} \[ \frac {2 \cos ^2(e+f x)^{3/4} (d \tan (e+f x))^{3/2} (b \csc (e+f x))^m \, _2F_1\left (\frac {3}{4},\frac {1}{4} (3-2 m);\frac {1}{4} (7-2 m);\sin ^2(e+f x)\right )}{d f (3-2 m)} \]
Antiderivative was successfully verified.
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Rule 2577
Rule 2602
Rule 2618
Rubi steps
\begin {align*} \int (b \csc (e+f x))^m \sqrt {d \tan (e+f x)} \, dx &=\left ((b \csc (e+f x))^m \left (\frac {\sin (e+f x)}{b}\right )^m\right ) \int \left (\frac {\sin (e+f x)}{b}\right )^{-m} \sqrt {d \tan (e+f x)} \, dx\\ &=\frac {\left (\cos ^{\frac {3}{2}}(e+f x) (b \csc (e+f x))^{2+m} \left (\frac {\sin (e+f x)}{b}\right )^{\frac {1}{2}+m} (d \tan (e+f x))^{3/2}\right ) \int \frac {\left (\frac {\sin (e+f x)}{b}\right )^{\frac {1}{2}-m}}{\sqrt {\cos (e+f x)}} \, dx}{b d}\\ &=\frac {2 \cos ^2(e+f x)^{3/4} (b \csc (e+f x))^{2+m} \, _2F_1\left (\frac {3}{4},\frac {1}{4} (3-2 m);\frac {1}{4} (7-2 m);\sin ^2(e+f x)\right ) \sin ^2(e+f x) (d \tan (e+f x))^{3/2}}{b^2 d f (3-2 m)}\\ \end {align*}
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Mathematica [A] time = 3.20, size = 87, normalized size = 1.10 \[ -\frac {2 (d \tan (e+f x))^{3/2} \sec ^2(e+f x)^{-m/2} (b \csc (e+f x))^m \, _2F_1\left (\frac {1}{4} (3-2 m),1-\frac {m}{2};\frac {1}{4} (7-2 m);-\tan ^2(e+f x)\right )}{d f (2 m-3)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {d \tan \left (f x + e\right )} \left (b \csc \left (f x + e\right )\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {d \tan \left (f x + e\right )} \left (b \csc \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.57, size = 0, normalized size = 0.00 \[ \int \left (b \csc \left (f x +e \right )\right )^{m} \sqrt {d \tan \left (f x +e \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {d \tan \left (f x + e\right )} \left (b \csc \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^m \,d x \]
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 \[ \int \left (b \csc {\left (e + f x \right )}\right )^{m} \sqrt {d \tan {\left (e + f x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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