Optimal. Leaf size=78 \[ -\frac{3 a d \sin (e+f x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{4}{3},\frac{7}{3},\cos ^2(e+f x)\right )}{8 f \sqrt{\sin ^2(e+f x)} (d \sec (e+f x))^{8/3}}-\frac{3 b}{5 f (d \sec (e+f x))^{5/3}} \]
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Rubi [A] time = 0.0639063, antiderivative size = 78, 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} \[ -\frac{3 a d \sin (e+f x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{4}{3},\frac{7}{3},\cos ^2(e+f x)\right )}{8 f \sqrt{\sin ^2(e+f x)} (d \sec (e+f x))^{8/3}}-\frac{3 b}{5 f (d \sec (e+f x))^{5/3}} \]
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)}{(d \sec (e+f x))^{5/3}} \, dx &=-\frac{3 b}{5 f (d \sec (e+f x))^{5/3}}+a \int \frac{1}{(d \sec (e+f x))^{5/3}} \, dx\\ &=-\frac{3 b}{5 f (d \sec (e+f x))^{5/3}}+\left (a \sqrt [3]{\frac{\cos (e+f x)}{d}} \sqrt [3]{d \sec (e+f x)}\right ) \int \left (\frac{\cos (e+f x)}{d}\right )^{5/3} \, dx\\ &=-\frac{3 b}{5 f (d \sec (e+f x))^{5/3}}-\frac{3 a \cos ^3(e+f x) \, _2F_1\left (\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \sin (e+f x)}{8 d^2 f \sqrt{\sin ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.433942, size = 94, normalized size = 1.21 \[ \frac{2 a \sin (e+f x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{2}{3},\frac{3}{2},\sin ^2(e+f x)\right )+3 \sqrt [3]{\cos ^2(e+f x)} (a \sin (e+f x)-b \cos (e+f x))}{5 d f \sqrt [3]{\cos ^2(e+f x)} (d \sec (e+f x))^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.094, size = 0, normalized size = 0. \begin{align*} \int{(a+b\tan \left ( fx+e \right ) ) \left ( d\sec \left ( fx+e \right ) \right ) ^{-{\frac{5}{3}}}}\, 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{5}{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{1}{3}}{\left (b \tan \left (f x + e\right ) + a\right )}}{d^{2} \sec \left (f x + e\right )^{2}}, 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 )}}{\left (d \sec{\left (e + f x \right )}\right )^{\frac{5}{3}}}\, 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{5}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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