Optimal. Leaf size=94 \[ \frac{6 a E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{5 d^2 f \sqrt{\cos (e+f x)} \sqrt{d \sec (e+f x)}}+\frac{2 a \sin (e+f x)}{5 d f (d \sec (e+f x))^{3/2}}-\frac{2 b}{5 f (d \sec (e+f x))^{5/2}} \]
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Rubi [A] time = 0.0704951, antiderivative size = 94, normalized size of antiderivative = 1., 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 = {3486, 3769, 3771, 2639} \[ \frac{6 a E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{5 d^2 f \sqrt{\cos (e+f x)} \sqrt{d \sec (e+f x)}}+\frac{2 a \sin (e+f x)}{5 d f (d \sec (e+f x))^{3/2}}-\frac{2 b}{5 f (d \sec (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3486
Rule 3769
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{a+b \tan (e+f x)}{(d \sec (e+f x))^{5/2}} \, dx &=-\frac{2 b}{5 f (d \sec (e+f x))^{5/2}}+a \int \frac{1}{(d \sec (e+f x))^{5/2}} \, dx\\ &=-\frac{2 b}{5 f (d \sec (e+f x))^{5/2}}+\frac{2 a \sin (e+f x)}{5 d f (d \sec (e+f x))^{3/2}}+\frac{(3 a) \int \frac{1}{\sqrt{d \sec (e+f x)}} \, dx}{5 d^2}\\ &=-\frac{2 b}{5 f (d \sec (e+f x))^{5/2}}+\frac{2 a \sin (e+f x)}{5 d f (d \sec (e+f x))^{3/2}}+\frac{(3 a) \int \sqrt{\cos (e+f x)} \, dx}{5 d^2 \sqrt{\cos (e+f x)} \sqrt{d \sec (e+f x)}}\\ &=-\frac{2 b}{5 f (d \sec (e+f x))^{5/2}}+\frac{6 a E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{5 d^2 f \sqrt{\cos (e+f x)} \sqrt{d \sec (e+f x)}}+\frac{2 a \sin (e+f x)}{5 d f (d \sec (e+f x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.605078, size = 74, normalized size = 0.79 \[ \frac{2 \sqrt{d \sec (e+f x)} \left (\cos ^2(e+f x) (a \sin (e+f x)-b \cos (e+f x))+3 a \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right )\right )}{5 d^3 f} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.204, size = 345, normalized size = 3.7 \begin{align*} -{\frac{2}{5\,f\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{3}} \left ( 3\,i\sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}{\it EllipticE} \left ({\frac{i \left ( \cos \left ( fx+e \right ) -1 \right ) }{\sin \left ( fx+e \right ) }},i \right ) \cos \left ( fx+e \right ) a\sin \left ( fx+e \right ) -3\,i\sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( fx+e \right ) -1 \right ) }{\sin \left ( fx+e \right ) }},i \right ) \cos \left ( fx+e \right ) a\sin \left ( fx+e \right ) +3\,i\sin \left ( fx+e \right ){\it EllipticE} \left ({\frac{i \left ( \cos \left ( fx+e \right ) -1 \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}a-3\,i\sin \left ( fx+e \right ){\it EllipticF} \left ({\frac{i \left ( \cos \left ( fx+e \right ) -1 \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}a+\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{3}b+ \left ( \cos \left ( fx+e \right ) \right ) ^{4}a+2\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}a-3\,a\cos \left ( fx+e \right ) \right ) \left ({\frac{d}{\cos \left ( fx+e \right ) }} \right ) ^{-{\frac{5}{2}}}} \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}{2}}}\,{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{\sqrt{d \sec \left (f x + e\right )}{\left (b \tan \left (f x + e\right ) + a\right )}}{d^{3} \sec \left (f x + e\right )^{3}}, 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}{2}}}\, 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}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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