Optimal. Leaf size=146 \[ -\frac{2 b \sec ^2(e+f x) \left (2 \left (a^2-2 b^2\right )+a b \tan (e+f x)\right )}{3 f (d \sec (e+f x))^{3/2}}+\frac{2 a \left (a^2+6 b^2\right ) \sec ^2(e+f x)^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right )}{3 f (d \sec (e+f x))^{3/2}}-\frac{2 (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{3 f (d \sec (e+f x))^{3/2}} \]
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Rubi [A] time = 0.124466, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3512, 739, 780, 231} \[ -\frac{2 b \sec ^2(e+f x) \left (2 \left (a^2-2 b^2\right )+a b \tan (e+f x)\right )}{3 f (d \sec (e+f x))^{3/2}}+\frac{2 a \left (a^2+6 b^2\right ) \sec ^2(e+f x)^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right )}{3 f (d \sec (e+f x))^{3/2}}-\frac{2 (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{3 f (d \sec (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3512
Rule 739
Rule 780
Rule 231
Rubi steps
\begin{align*} \int \frac{(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{3/2}} \, dx &=\frac{\sec ^2(e+f x)^{3/4} \operatorname{Subst}\left (\int \frac{(a+x)^3}{\left (1+\frac{x^2}{b^2}\right )^{7/4}} \, dx,x,b \tan (e+f x)\right )}{b f (d \sec (e+f x))^{3/2}}\\ &=-\frac{2 (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{3 f (d \sec (e+f x))^{3/2}}+\frac{\left (2 b \sec ^2(e+f x)^{3/4}\right ) \operatorname{Subst}\left (\int \frac{(a+x) \left (\frac{1}{2} \left (4+\frac{a^2}{b^2}\right )-\frac{3 a x}{2 b^2}\right )}{\left (1+\frac{x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{3 f (d \sec (e+f x))^{3/2}}\\ &=-\frac{2 (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{3 f (d \sec (e+f x))^{3/2}}-\frac{2 b \sec ^2(e+f x) \left (2 \left (a^2-2 b^2\right )+a b \tan (e+f x)\right )}{3 f (d \sec (e+f x))^{3/2}}+\frac{\left (a \left (6+\frac{a^2}{b^2}\right ) b \sec ^2(e+f x)^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{3 f (d \sec (e+f x))^{3/2}}\\ &=\frac{2 a \left (a^2+6 b^2\right ) F\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sec ^2(e+f x)^{3/4}}{3 f (d \sec (e+f x))^{3/2}}-\frac{2 (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{3 f (d \sec (e+f x))^{3/2}}-\frac{2 b \sec ^2(e+f x) \left (2 \left (a^2-2 b^2\right )+a b \tan (e+f x)\right )}{3 f (d \sec (e+f x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 1.30836, size = 117, normalized size = 0.8 \[ \frac{\sec ^2(e+f x) \left (\left (b^3-3 a^2 b\right ) \cos (2 (e+f x))+2 a \left (a^2+6 b^2\right ) \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right )-3 a^2 b+a^3 \sin (2 (e+f x))-3 a b^2 \sin (2 (e+f x))+7 b^3\right )}{3 f (d \sec (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.303, size = 342, normalized size = 2.3 \begin{align*}{\frac{2}{3\,f \left ( \cos \left ( fx+e \right ) \right ) ^{2}} \left ( 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}^{3}+6\,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{b}^{2}+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 ){a}^{3}+6\,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 ) a{b}^{2}-3\,{a}^{2} \left ( \cos \left ( fx+e \right ) \right ) ^{2}b+{b}^{3} \left ( \cos \left ( fx+e \right ) \right ) ^{2}+\cos \left ( fx+e \right ){a}^{3}\sin \left ( fx+e \right ) -3\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) a{b}^{2}+3\,{b}^{3} \right ) \left ({\frac{d}{\cos \left ( fx+e \right ) }} \right ) ^{-{\frac{3}{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{{\left (b \tan \left (f x + e\right ) + a\right )}^{3}}{\left (d \sec \left (f x + e\right )\right )^{\frac{3}{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{{\left (b^{3} \tan \left (f x + e\right )^{3} + 3 \, a b^{2} \tan \left (f x + e\right )^{2} + 3 \, a^{2} b \tan \left (f x + e\right ) + a^{3}\right )} \sqrt{d \sec \left (f x + e\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{\left (a + b \tan{\left (e + f x \right )}\right )^{3}}{\left (d \sec{\left (e + f x \right )}\right )^{\frac{3}{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{{\left (b \tan \left (f x + e\right ) + a\right )}^{3}}{\left (d \sec \left (f x + e\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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