Optimal. Leaf size=294 \[ \frac {\left (3 a^2-56 a b+48 b^2\right ) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{192 b f}-\frac {\left (3 a^3+8 a^2 b-80 a b^2+64 b^3\right ) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{128 b^2 f}+\frac {\left (3 a^4+8 a^3 b+48 a^2 b^2-192 a b^3+128 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{128 b^{5/2} f}-\frac {(a-b)^{3/2} \tan ^{-1}\left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f}+\frac {b \tan ^7(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{8 f}+\frac {(9 a-8 b) \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{48 f} \]
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Rubi [A] time = 0.45, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, 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 203
Rule 206
Rule 217
Rule 377
Rule 477
Rule 523
Rule 582
Rule 3670
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*}
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Mathematica [C] time = 6.55, size = 908, normalized size = 3.09 \[ \frac {-\frac {b \left (3 a^4+8 b a^3-16 b^2 a^2-64 b^3 a+64 b^4\right ) \sqrt {\frac {a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \sqrt {-\frac {a \cot ^2(e+f x)}{b}} \sqrt {-\frac {a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) F\left (\left .\sin ^{-1}\left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right )\right |1\right ) \sin ^4(e+f x)}{a (a+b+(a-b) \cos (2 (e+f x)))}-\frac {4 b \left (-64 b^4+128 a b^3-64 a^2 b^2\right ) \sqrt {\cos (2 (e+f x))+1} \sqrt {\frac {a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left (\frac {\sqrt {-\frac {a \cot ^2(e+f x)}{b}} \sqrt {-\frac {a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) F\left (\left .\sin ^{-1}\left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right )\right |1\right ) \sin ^4(e+f x)}{4 a \sqrt {\cos (2 (e+f x))+1} \sqrt {a+b+(a-b) \cos (2 (e+f x))}}-\frac {\sqrt {-\frac {a \cot ^2(e+f x)}{b}} \sqrt {-\frac {a (\cos (2 (e+f x))+1) \csc ^2(e+f x)}{b}} \sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \Pi \left (-\frac {b}{a-b};\left .\sin ^{-1}\left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right )\right |1\right ) \sin ^4(e+f x)}{2 (a-b) \sqrt {\cos (2 (e+f x))+1} \sqrt {a+b+(a-b) \cos (2 (e+f x))}}\right )}{\sqrt {a+b+(a-b) \cos (2 (e+f x))}}}{64 b^2 f}+\frac {\sqrt {\frac {\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left (\frac {1}{8} b \tan (e+f x) \sec ^6(e+f x)+\frac {1}{48} (9 a \sin (e+f x)-26 b \sin (e+f x)) \sec ^5(e+f x)+\frac {\left (3 \sin (e+f x) a^2-128 b \sin (e+f x) a+184 b^2 \sin (e+f x)\right ) \sec ^3(e+f x)}{192 b}+\frac {\left (-9 \sin (e+f x) a^3-30 b \sin (e+f x) a^2+424 b^2 \sin (e+f x) a-400 b^3 \sin (e+f x)\right ) \sec (e+f x)}{384 b^2}\right )}{f} \]
Antiderivative was successfully verified.
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fricas [A] time = 4.94, size = 1059, normalized size = 3.60 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \tan \left (f x + e\right )^{6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.28, size = 669, normalized size = 2.28 \[ \frac {\left (\tan ^{3}\left (f x +e \right )\right ) \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{\frac {5}{2}}}{8 f b}-\frac {a \tan \left (f x +e \right ) \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{\frac {5}{2}}}{16 f \,b^{2}}+\frac {a^{2} \tan \left (f x +e \right ) \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}{64 f \,b^{2}}+\frac {3 a^{3} \tan \left (f x +e \right ) \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{128 f \,b^{2}}+\frac {3 a^{4} \ln \left (\tan \left (f x +e \right ) \sqrt {b}+\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}\right )}{128 f \,b^{\frac {5}{2}}}-\frac {\tan \left (f x +e \right ) \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{\frac {5}{2}}}{6 f b}+\frac {a \tan \left (f x +e \right ) \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}{24 f b}+\frac {a^{2} \tan \left (f x +e \right ) \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{16 f b}+\frac {a^{3} \ln \left (\tan \left (f x +e \right ) \sqrt {b}+\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}\right )}{16 f \,b^{\frac {3}{2}}}+\frac {\tan \left (f x +e \right ) \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}{4 f}+\frac {3 a \tan \left (f x +e \right ) \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{8 f}+\frac {3 a^{2} \ln \left (\tan \left (f x +e \right ) \sqrt {b}+\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}\right )}{8 f \sqrt {b}}-\frac {b \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}\, \tan \left (f x +e \right )}{2 f}-\frac {3 \sqrt {b}\, a \ln \left (\tan \left (f x +e \right ) \sqrt {b}+\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}\right )}{2 f}+\frac {b^{\frac {3}{2}} \ln \left (\tan \left (f x +e \right ) \sqrt {b}+\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}\right )}{f}-\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {\left (a -b \right ) b^{2} \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}\right )}{f \left (a -b \right )}+\frac {2 a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {\left (a -b \right ) b^{2} \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}\right )}{f b \left (a -b \right )}-\frac {a^{2} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {\left (a -b \right ) b^{2} \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}\right )}{f \,b^{2} \left (a -b \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \tan \left (f x + e\right )^{6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {tan}\left (e+f\,x\right )}^6\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}} \tan ^{6}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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