Optimal. Leaf size=294 \[ \frac{\left (3 a^2-56 a b+48 b^2\right ) \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{192 b f}-\frac{\left (8 a^2 b+3 a^3-80 a b^2+64 b^3\right ) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{128 b^2 f}+\frac{\left (48 a^2 b^2+8 a^3 b+3 a^4-192 a b^3+128 b^4\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{128 b^{5/2} f}+\frac{b \tan ^7(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}+\frac{(9 a-8 b) \tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{48 f}-\frac{(a-b)^{3/2} \tan ^{-1}\left (\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{f} \]
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Rubi [A] time = 0.447992, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {3670, 477, 582, 523, 217, 206, 377, 203} \[ \frac{\left (3 a^2-56 a b+48 b^2\right ) \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{192 b f}-\frac{\left (8 a^2 b+3 a^3-80 a b^2+64 b^3\right ) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{128 b^2 f}+\frac{\left (48 a^2 b^2+8 a^3 b+3 a^4-192 a b^3+128 b^4\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{128 b^{5/2} f}+\frac{b \tan ^7(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}+\frac{(9 a-8 b) \tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{48 f}-\frac{(a-b)^{3/2} \tan ^{-1}\left (\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 477
Rule 582
Rule 523
Rule 217
Rule 206
Rule 377
Rule 203
Rubi steps
\begin{align*} \int \tan ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^6 \left (a+b x^2\right )^{3/2}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{b \tan ^7(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}+\frac{\operatorname{Subst}\left (\int \frac{x^6 \left (a (8 a-7 b)+(9 a-8 b) b x^2\right )}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{8 f}\\ &=\frac{(9 a-8 b) \tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{48 f}+\frac{b \tan ^7(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}-\frac{\operatorname{Subst}\left (\int \frac{x^4 \left (5 a (9 a-8 b) b-b \left (3 a^2-56 a b+48 b^2\right ) x^2\right )}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{48 b f}\\ &=\frac{\left (3 a^2-56 a b+48 b^2\right ) \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{192 b f}+\frac{(9 a-8 b) \tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{48 f}+\frac{b \tan ^7(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}+\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (-3 a b \left (3 a^2-56 a b+48 b^2\right )-3 b \left (3 a^3+8 a^2 b-80 a b^2+64 b^3\right ) x^2\right )}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{192 b^2 f}\\ &=-\frac{\left (3 a^3+8 a^2 b-80 a b^2+64 b^3\right ) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{128 b^2 f}+\frac{\left (3 a^2-56 a b+48 b^2\right ) \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{192 b f}+\frac{(9 a-8 b) \tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{48 f}+\frac{b \tan ^7(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}-\frac{\operatorname{Subst}\left (\int \frac{-3 a b \left (3 a^3+8 a^2 b-80 a b^2+64 b^3\right )-3 b \left (3 a^4+8 a^3 b+48 a^2 b^2-192 a b^3+128 b^4\right ) x^2}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{384 b^3 f}\\ &=-\frac{\left (3 a^3+8 a^2 b-80 a b^2+64 b^3\right ) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{128 b^2 f}+\frac{\left (3 a^2-56 a b+48 b^2\right ) \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{192 b f}+\frac{(9 a-8 b) \tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{48 f}+\frac{b \tan ^7(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}-\frac{(a-b)^2 \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{f}+\frac{\left (3 a^4+8 a^3 b+48 a^2 b^2-192 a b^3+128 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{128 b^2 f}\\ &=-\frac{\left (3 a^3+8 a^2 b-80 a b^2+64 b^3\right ) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{128 b^2 f}+\frac{\left (3 a^2-56 a b+48 b^2\right ) \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{192 b f}+\frac{(9 a-8 b) \tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{48 f}+\frac{b \tan ^7(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}-\frac{(a-b)^2 \operatorname{Subst}\left (\int \frac{1}{1-(-a+b) x^2} \, dx,x,\frac{\tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{f}+\frac{\left (3 a^4+8 a^3 b+48 a^2 b^2-192 a b^3+128 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{128 b^2 f}\\ &=-\frac{(a-b)^{3/2} \tan ^{-1}\left (\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{f}+\frac{\left (3 a^4+8 a^3 b+48 a^2 b^2-192 a b^3+128 b^4\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{128 b^{5/2} f}-\frac{\left (3 a^3+8 a^2 b-80 a b^2+64 b^3\right ) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{128 b^2 f}+\frac{\left (3 a^2-56 a b+48 b^2\right ) \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{192 b f}+\frac{(9 a-8 b) \tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{48 f}+\frac{b \tan ^7(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}\\ \end{align*}
Mathematica [C] time = 6.48861, size = 908, normalized size = 3.09 \[ \frac{-\frac{b \left (3 a^4+8 b a^3-16 b^2 a^2-64 b^3 a+64 b^4\right ) \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right ),1\right ) \sin ^4(e+f x)}{a (a+b+(a-b) \cos (2 (e+f x)))}-\frac{4 b \left (-64 b^4+128 a b^3-64 a^2 b^2\right ) \sqrt{\cos (2 (e+f x))+1} \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left (\frac{\sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right ),1\right ) \sin ^4(e+f x)}{4 a \sqrt{\cos (2 (e+f x))+1} \sqrt{a+b+(a-b) \cos (2 (e+f x))}}-\frac{\sqrt{-\frac{a \cot ^2(e+f x)}{b}} \sqrt{-\frac{a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \Pi \left (-\frac{b}{a-b};\left .\sin ^{-1}\left (\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right )\right |1\right ) \sin ^4(e+f x)}{2 (a-b) \sqrt{\cos (2 (e+f x))+1} \sqrt{a+b+(a-b) \cos (2 (e+f x))}}\right )}{\sqrt{a+b+(a-b) \cos (2 (e+f x))}}}{64 b^2 f}+\frac{\sqrt{\frac{\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left (\frac{1}{8} b \tan (e+f x) \sec ^6(e+f x)+\frac{1}{48} (9 a \sin (e+f x)-26 b \sin (e+f x)) \sec ^5(e+f x)+\frac{\left (3 \sin (e+f x) a^2-128 b \sin (e+f x) a+184 b^2 \sin (e+f x)\right ) \sec ^3(e+f x)}{192 b}+\frac{\left (-9 \sin (e+f x) a^3-30 b \sin (e+f x) a^2+424 b^2 \sin (e+f x) a-400 b^3 \sin (e+f x)\right ) \sec (e+f x)}{384 b^2}\right )}{f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.024, size = 669, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 36.2956, size = 2572, normalized size = 8.75 \begin{align*} \text{result too large to display} \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 \left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac{3}{2}} \tan ^{6}{\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{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} \tan \left (f x + e\right )^{6}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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