Optimal. Leaf size=83 \[ \frac {(a+b)^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{a b^{5/2} f}-\frac {(a+2 b) \tan (e+f x)}{b^2 f}-\frac {x}{a}+\frac {\tan ^3(e+f x)}{3 b f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.27, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {4141, 1975, 479, 582, 522, 203, 205} \[ \frac {(a+b)^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{a b^{5/2} f}-\frac {(a+2 b) \tan (e+f x)}{b^2 f}-\frac {x}{a}+\frac {\tan ^3(e+f x)}{3 b f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 205
Rule 479
Rule 522
Rule 582
Rule 1975
Rule 4141
Rubi steps
\begin {align*} \int \frac {\tan ^6(e+f x)}{a+b \sec ^2(e+f x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^6}{\left (1+x^2\right ) \left (a+b \left (1+x^2\right )\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^6}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\tan ^3(e+f x)}{3 b f}-\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (3 (a+b)+3 (a+2 b) x^2\right )}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{3 b f}\\ &=-\frac {(a+2 b) \tan (e+f x)}{b^2 f}+\frac {\tan ^3(e+f x)}{3 b f}+\frac {\operatorname {Subst}\left (\int \frac {3 (a+b) (a+2 b)+3 \left (a^2+3 a b+3 b^2\right ) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{3 b^2 f}\\ &=-\frac {(a+2 b) \tan (e+f x)}{b^2 f}+\frac {\tan ^3(e+f x)}{3 b f}-\frac {\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{a f}+\frac {(a+b)^3 \operatorname {Subst}\left (\int \frac {1}{a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{a b^2 f}\\ &=-\frac {x}{a}+\frac {(a+b)^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{a b^{5/2} f}-\frac {(a+2 b) \tan (e+f x)}{b^2 f}+\frac {\tan ^3(e+f x)}{3 b f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 2.79, size = 229, normalized size = 2.76 \[ \frac {\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left (-\frac {(3 a+7 b) \sec (e) \sin (f x) \sec (e+f x)}{b^2 f}-\frac {3 (a+b)^{5/2} (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left (\frac {(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt {a+b} \sqrt {b (\cos (e)-i \sin (e))^4}}\right )}{a b^2 f \sqrt {b (\cos (e)-i \sin (e))^4}}-\frac {3 x}{a}+\frac {\sec (e) \sin (f x) \sec ^3(e+f x)}{b f}+\frac {\tan (e) \sec ^2(e+f x)}{b f}\right )}{6 \left (a+b \sec ^2(e+f x)\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.54, size = 373, normalized size = 4.49 \[ \left [-\frac {12 \, b^{2} f x \cos \left (f x + e\right )^{3} - 3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {-\frac {a + b}{b}} \cos \left (f x + e\right )^{3} \log \left (\frac {{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, {\left ({\left (a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{3} - b^{2} \cos \left (f x + e\right )\right )} \sqrt {-\frac {a + b}{b}} \sin \left (f x + e\right ) + b^{2}}{a^{2} \cos \left (f x + e\right )^{4} + 2 \, a b \cos \left (f x + e\right )^{2} + b^{2}}\right ) + 4 \, {\left ({\left (3 \, a^{2} + 7 \, a b\right )} \cos \left (f x + e\right )^{2} - a b\right )} \sin \left (f x + e\right )}{12 \, a b^{2} f \cos \left (f x + e\right )^{3}}, -\frac {6 \, b^{2} f x \cos \left (f x + e\right )^{3} + 3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {\frac {a + b}{b}} \arctan \left (\frac {{\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt {\frac {a + b}{b}}}{2 \, {\left (a + b\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) \cos \left (f x + e\right )^{3} + 2 \, {\left ({\left (3 \, a^{2} + 7 \, a b\right )} \cos \left (f x + e\right )^{2} - a b\right )} \sin \left (f x + e\right )}{6 \, a b^{2} f \cos \left (f x + e\right )^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 6.31, size = 132, normalized size = 1.59 \[ -\frac {\frac {3 \, {\left (f x + e\right )}}{a} - \frac {3 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (b) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b + b^{2}}}\right )\right )}}{\sqrt {a b + b^{2}} a b^{2}} - \frac {b^{2} \tan \left (f x + e\right )^{3} - 3 \, a b \tan \left (f x + e\right ) - 6 \, b^{2} \tan \left (f x + e\right )}{b^{3}}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.84, size = 186, normalized size = 2.24 \[ \frac {\tan ^{3}\left (f x +e \right )}{3 b f}-\frac {a \tan \left (f x +e \right )}{f \,b^{2}}-\frac {2 \tan \left (f x +e \right )}{b f}+\frac {a^{2} \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {\left (a +b \right ) b}}\right )}{f \,b^{2} \sqrt {\left (a +b \right ) b}}+\frac {3 a \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {\left (a +b \right ) b}}\right )}{f b \sqrt {\left (a +b \right ) b}}+\frac {3 \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {\left (a +b \right ) b}}\right )}{f \sqrt {\left (a +b \right ) b}}+\frac {b \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {\left (a +b \right ) b}}\right )}{f a \sqrt {\left (a +b \right ) b}}-\frac {\arctan \left (\tan \left (f x +e \right )\right )}{f a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.44, size = 95, normalized size = 1.14 \[ -\frac {\frac {3 \, {\left (f x + e\right )}}{a} - \frac {b \tan \left (f x + e\right )^{3} - 3 \, {\left (a + 2 \, b\right )} \tan \left (f x + e\right )}{b^{2}} - \frac {3 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{\sqrt {{\left (a + b\right )} b} a b^{2}}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.92, size = 1109, normalized size = 13.36 \[ \frac {{\mathrm {tan}\left (e+f\,x\right )}^3}{3\,b\,f}-\frac {\mathrm {atan}\left (\frac {40\,a^2\,\mathrm {tan}\left (e+f\,x\right )}{30\,a\,b+40\,a^2+10\,b^2+\frac {30\,a^3}{b}+\frac {12\,a^4}{b^2}+\frac {2\,a^5}{b^3}}+\frac {30\,a^3\,\mathrm {tan}\left (e+f\,x\right )}{30\,a\,b^2+40\,a^2\,b+30\,a^3+10\,b^3+\frac {12\,a^4}{b}+\frac {2\,a^5}{b^2}}+\frac {12\,a^4\,\mathrm {tan}\left (e+f\,x\right )}{30\,a\,b^3+30\,a^3\,b+12\,a^4+10\,b^4+40\,a^2\,b^2+\frac {2\,a^5}{b}}+\frac {2\,a^5\,\mathrm {tan}\left (e+f\,x\right )}{2\,a^5+12\,a^4\,b+30\,a^3\,b^2+40\,a^2\,b^3+30\,a\,b^4+10\,b^5}+\frac {10\,b^2\,\mathrm {tan}\left (e+f\,x\right )}{30\,a\,b+40\,a^2+10\,b^2+\frac {30\,a^3}{b}+\frac {12\,a^4}{b^2}+\frac {2\,a^5}{b^3}}+\frac {30\,a\,b\,\mathrm {tan}\left (e+f\,x\right )}{30\,a\,b+40\,a^2+10\,b^2+\frac {30\,a^3}{b}+\frac {12\,a^4}{b^2}+\frac {2\,a^5}{b^3}}\right )}{a\,f}-\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (a+2\,b\right )}{b^2\,f}-\frac {\mathrm {atan}\left (\frac {\frac {\sqrt {-b^5\,{\left (a+b\right )}^5}\,\left (\frac {2\,\mathrm {tan}\left (e+f\,x\right )\,\left (a^6+6\,a^5\,b+15\,a^4\,b^2+20\,a^3\,b^3+15\,a^2\,b^4+6\,a\,b^5+2\,b^6\right )}{b^3}+\frac {\sqrt {-b^5\,{\left (a+b\right )}^5}\,\left (\frac {4\,a^4\,b^3+12\,a^3\,b^4+8\,a^2\,b^5}{b^3}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (4\,a^3\,b^5+8\,a^2\,b^6\right )\,\sqrt {-b^5\,{\left (a+b\right )}^5}}{a\,b^8}\right )}{2\,a\,b^5}\right )\,1{}\mathrm {i}}{2\,a\,b^5}+\frac {\sqrt {-b^5\,{\left (a+b\right )}^5}\,\left (\frac {2\,\mathrm {tan}\left (e+f\,x\right )\,\left (a^6+6\,a^5\,b+15\,a^4\,b^2+20\,a^3\,b^3+15\,a^2\,b^4+6\,a\,b^5+2\,b^6\right )}{b^3}-\frac {\sqrt {-b^5\,{\left (a+b\right )}^5}\,\left (\frac {4\,a^4\,b^3+12\,a^3\,b^4+8\,a^2\,b^5}{b^3}-\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (4\,a^3\,b^5+8\,a^2\,b^6\right )\,\sqrt {-b^5\,{\left (a+b\right )}^5}}{a\,b^8}\right )}{2\,a\,b^5}\right )\,1{}\mathrm {i}}{2\,a\,b^5}}{\frac {2\,\left (a^5+6\,a^4\,b+15\,a^3\,b^2+19\,a^2\,b^3+12\,a\,b^4+3\,b^5\right )}{b^3}-\frac {\sqrt {-b^5\,{\left (a+b\right )}^5}\,\left (\frac {2\,\mathrm {tan}\left (e+f\,x\right )\,\left (a^6+6\,a^5\,b+15\,a^4\,b^2+20\,a^3\,b^3+15\,a^2\,b^4+6\,a\,b^5+2\,b^6\right )}{b^3}+\frac {\sqrt {-b^5\,{\left (a+b\right )}^5}\,\left (\frac {4\,a^4\,b^3+12\,a^3\,b^4+8\,a^2\,b^5}{b^3}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (4\,a^3\,b^5+8\,a^2\,b^6\right )\,\sqrt {-b^5\,{\left (a+b\right )}^5}}{a\,b^8}\right )}{2\,a\,b^5}\right )}{2\,a\,b^5}+\frac {\sqrt {-b^5\,{\left (a+b\right )}^5}\,\left (\frac {2\,\mathrm {tan}\left (e+f\,x\right )\,\left (a^6+6\,a^5\,b+15\,a^4\,b^2+20\,a^3\,b^3+15\,a^2\,b^4+6\,a\,b^5+2\,b^6\right )}{b^3}-\frac {\sqrt {-b^5\,{\left (a+b\right )}^5}\,\left (\frac {4\,a^4\,b^3+12\,a^3\,b^4+8\,a^2\,b^5}{b^3}-\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (4\,a^3\,b^5+8\,a^2\,b^6\right )\,\sqrt {-b^5\,{\left (a+b\right )}^5}}{a\,b^8}\right )}{2\,a\,b^5}\right )}{2\,a\,b^5}}\right )\,\sqrt {-b^5\,{\left (a+b\right )}^5}\,1{}\mathrm {i}}{a\,b^5\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{6}{\left (e + f x \right )}}{a + b \sec ^{2}{\left (e + f x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________