Optimal. Leaf size=76 \[ -\frac{3 a d \sin (e+f x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{2}{3},\frac{5}{3},\cos ^2(e+f x)\right )}{4 f \sqrt{\sin ^2(e+f x)} (d \sec (e+f x))^{4/3}}-\frac{3 b}{f \sqrt [3]{d \sec (e+f x)}} \]
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Rubi [A] time = 0.0598214, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3486, 3772, 2643} \[ -\frac{3 a d \sin (e+f x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{2}{3},\frac{5}{3},\cos ^2(e+f x)\right )}{4 f \sqrt{\sin ^2(e+f x)} (d \sec (e+f x))^{4/3}}-\frac{3 b}{f \sqrt [3]{d \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3486
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \frac{a+b \tan (e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx &=-\frac{3 b}{f \sqrt [3]{d \sec (e+f x)}}+a \int \frac{1}{\sqrt [3]{d \sec (e+f x)}} \, dx\\ &=-\frac{3 b}{f \sqrt [3]{d \sec (e+f x)}}+\left (a \left (\frac{\cos (e+f x)}{d}\right )^{2/3} (d \sec (e+f x))^{2/3}\right ) \int \sqrt [3]{\frac{\cos (e+f x)}{d}} \, dx\\ &=-\frac{3 b}{f \sqrt [3]{d \sec (e+f x)}}-\frac{3 a \cos ^2(e+f x) \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(e+f x)\right ) (d \sec (e+f x))^{2/3} \sin (e+f x)}{4 d f \sqrt{\sin ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.929204, size = 119, normalized size = 1.57 \[ -\frac{3 (a \cot (e+f x)+b) \left (a \sqrt{\sin ^2(e+f x)} \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \text{Hypergeometric2F1}\left (-\frac{1}{6},\frac{1}{2},\frac{5}{6},\sec ^2(e+f x)\right )+b \sin (e+f x)\right )}{f \sqrt [3]{d \sec (e+f x)} \left (a \sqrt{\sin ^2(e+f x)} \cot (e+f x)+b \sin (e+f x)\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.138, size = 0, normalized size = 0. \begin{align*} \int{(a+b\tan \left ( fx+e \right ) ){\frac{1}{\sqrt [3]{d\sec \left ( fx+e \right ) }}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \tan \left (f x + e\right ) + a}{\left (d \sec \left (f x + e\right )\right )^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (d \sec \left (f x + e\right )\right )^{\frac{2}{3}}{\left (b \tan \left (f x + e\right ) + a\right )}}{d \sec \left (f x + e\right )}, x\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{a + b \tan{\left (e + f x \right )}}{\sqrt [3]{d \sec{\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{b \tan \left (f x + e\right ) + a}{\left (d \sec \left (f x + e\right )\right )^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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