Integrand size = 21, antiderivative size = 64 \[ \int \left (a+b \sec ^2(e+f x)\right ) \tan ^6(e+f x) \, dx=-a x+\frac {a \tan (e+f x)}{f}-\frac {a \tan ^3(e+f x)}{3 f}+\frac {a \tan ^5(e+f x)}{5 f}+\frac {b \tan ^7(e+f x)}{7 f} \] Output:
-a*x+a*tan(f*x+e)/f-1/3*a*tan(f*x+e)^3/f+1/5*a*tan(f*x+e)^5/f+1/7*b*tan(f* x+e)^7/f
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.14 \[ \int \left (a+b \sec ^2(e+f x)\right ) \tan ^6(e+f x) \, dx=-\frac {a \arctan (\tan (e+f x))}{f}+\frac {a \tan (e+f x)}{f}-\frac {a \tan ^3(e+f x)}{3 f}+\frac {a \tan ^5(e+f x)}{5 f}+\frac {b \tan ^7(e+f x)}{7 f} \] Input:
Integrate[(a + b*Sec[e + f*x]^2)*Tan[e + f*x]^6,x]
Output:
-((a*ArcTan[Tan[e + f*x]])/f) + (a*Tan[e + f*x])/f - (a*Tan[e + f*x]^3)/(3 *f) + (a*Tan[e + f*x]^5)/(5*f) + (b*Tan[e + f*x]^7)/(7*f)
Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 4629, 2075, 363, 254, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \tan ^6(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \tan (e+f x)^6 \left (a+b \sec (e+f x)^2\right )dx\) |
\(\Big \downarrow \) 4629 |
\(\displaystyle \frac {\int \frac {\tan ^6(e+f x) \left (a+b \left (\tan ^2(e+f x)+1\right )\right )}{\tan ^2(e+f x)+1}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 2075 |
\(\displaystyle \frac {\int \frac {\tan ^6(e+f x) \left (b \tan ^2(e+f x)+a+b\right )}{\tan ^2(e+f x)+1}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 363 |
\(\displaystyle \frac {a \int \frac {\tan ^6(e+f x)}{\tan ^2(e+f x)+1}d\tan (e+f x)+\frac {1}{7} b \tan ^7(e+f x)}{f}\) |
\(\Big \downarrow \) 254 |
\(\displaystyle \frac {a \int \left (\tan ^4(e+f x)-\tan ^2(e+f x)-\frac {1}{\tan ^2(e+f x)+1}+1\right )d\tan (e+f x)+\frac {1}{7} b \tan ^7(e+f x)}{f}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a \left (-\arctan (\tan (e+f x))+\frac {1}{5} \tan ^5(e+f x)-\frac {1}{3} \tan ^3(e+f x)+\tan (e+f x)\right )+\frac {1}{7} b \tan ^7(e+f x)}{f}\) |
Input:
Int[(a + b*Sec[e + f*x]^2)*Tan[e + f*x]^6,x]
Output:
((b*Tan[e + f*x]^7)/7 + a*(-ArcTan[Tan[e + f*x]] + Tan[e + f*x] - Tan[e + f*x]^3/3 + Tan[e + f*x]^5/5))/f
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*Expa ndToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{e, m, p, q}, x] && Binomi alQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0] && ! BinomialMatchQ[{u, v}, x]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*((d_.)*tan[(e_.) + (f _.)*(x_)])^(m_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim p[ff/f Subst[Int[(d*ff*x)^m*((a + b*(1 + ff^2*x^2)^(n/2))^p/(1 + ff^2*x^2 )), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && Inte gerQ[n/2] && (IntegerQ[m/2] || EqQ[n, 2])
Time = 1.54 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.89
method | result | size |
parts | \(\frac {a \left (\frac {\tan \left (f x +e \right )^{5}}{5}-\frac {\tan \left (f x +e \right )^{3}}{3}+\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {b \tan \left (f x +e \right )^{7}}{7 f}\) | \(57\) |
derivativedivides | \(\frac {a \left (\frac {\tan \left (f x +e \right )^{5}}{5}-\frac {\tan \left (f x +e \right )^{3}}{3}+\tan \left (f x +e \right )-f x -e \right )+\frac {b \sin \left (f x +e \right )^{7}}{7 \cos \left (f x +e \right )^{7}}}{f}\) | \(61\) |
default | \(\frac {a \left (\frac {\tan \left (f x +e \right )^{5}}{5}-\frac {\tan \left (f x +e \right )^{3}}{3}+\tan \left (f x +e \right )-f x -e \right )+\frac {b \sin \left (f x +e \right )^{7}}{7 \cos \left (f x +e \right )^{7}}}{f}\) | \(61\) |
risch | \(-a x -\frac {2 i \left (-315 a \,{\mathrm e}^{12 i \left (f x +e \right )}+105 b \,{\mathrm e}^{12 i \left (f x +e \right )}-1260 a \,{\mathrm e}^{10 i \left (f x +e \right )}-2555 a \,{\mathrm e}^{8 i \left (f x +e \right )}+525 b \,{\mathrm e}^{8 i \left (f x +e \right )}-3080 a \,{\mathrm e}^{6 i \left (f x +e \right )}-2121 a \,{\mathrm e}^{4 i \left (f x +e \right )}+315 b \,{\mathrm e}^{4 i \left (f x +e \right )}-812 a \,{\mathrm e}^{2 i \left (f x +e \right )}-161 a +15 b \right )}{105 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{7}}\) | \(140\) |
Input:
int((a+b*sec(f*x+e)^2)*tan(f*x+e)^6,x,method=_RETURNVERBOSE)
Output:
a/f*(1/5*tan(f*x+e)^5-1/3*tan(f*x+e)^3+tan(f*x+e)-arctan(tan(f*x+e)))+1/7* b*tan(f*x+e)^7/f
Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.39 \[ \int \left (a+b \sec ^2(e+f x)\right ) \tan ^6(e+f x) \, dx=-\frac {105 \, a f x \cos \left (f x + e\right )^{7} - {\left ({\left (161 \, a - 15 \, b\right )} \cos \left (f x + e\right )^{6} - {\left (77 \, a - 45 \, b\right )} \cos \left (f x + e\right )^{4} + 3 \, {\left (7 \, a - 15 \, b\right )} \cos \left (f x + e\right )^{2} + 15 \, b\right )} \sin \left (f x + e\right )}{105 \, f \cos \left (f x + e\right )^{7}} \] Input:
integrate((a+b*sec(f*x+e)^2)*tan(f*x+e)^6,x, algorithm="fricas")
Output:
-1/105*(105*a*f*x*cos(f*x + e)^7 - ((161*a - 15*b)*cos(f*x + e)^6 - (77*a - 45*b)*cos(f*x + e)^4 + 3*(7*a - 15*b)*cos(f*x + e)^2 + 15*b)*sin(f*x + e ))/(f*cos(f*x + e)^7)
Time = 1.29 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.03 \[ \int \left (a+b \sec ^2(e+f x)\right ) \tan ^6(e+f x) \, dx=a \left (\begin {cases} - x + \frac {\tan ^{5}{\left (e + f x \right )}}{5 f} - \frac {\tan ^{3}{\left (e + f x \right )}}{3 f} + \frac {\tan {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \tan ^{6}{\left (e \right )} & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} x \tan ^{6}{\left (e \right )} \sec ^{2}{\left (e \right )} & \text {for}\: f = 0 \\\frac {\tan ^{7}{\left (e + f x \right )}}{7 f} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((a+b*sec(f*x+e)**2)*tan(f*x+e)**6,x)
Output:
a*Piecewise((-x + tan(e + f*x)**5/(5*f) - tan(e + f*x)**3/(3*f) + tan(e + f*x)/f, Ne(f, 0)), (x*tan(e)**6, True)) + b*Piecewise((x*tan(e)**6*sec(e)* *2, Eq(f, 0)), (tan(e + f*x)**7/(7*f), True))
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.88 \[ \int \left (a+b \sec ^2(e+f x)\right ) \tan ^6(e+f x) \, dx=\frac {15 \, b \tan \left (f x + e\right )^{7} + 21 \, a \tan \left (f x + e\right )^{5} - 35 \, a \tan \left (f x + e\right )^{3} - 105 \, {\left (f x + e\right )} a + 105 \, a \tan \left (f x + e\right )}{105 \, f} \] Input:
integrate((a+b*sec(f*x+e)^2)*tan(f*x+e)^6,x, algorithm="maxima")
Output:
1/105*(15*b*tan(f*x + e)^7 + 21*a*tan(f*x + e)^5 - 35*a*tan(f*x + e)^3 - 1 05*(f*x + e)*a + 105*a*tan(f*x + e))/f
Time = 0.38 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.12 \[ \int \left (a+b \sec ^2(e+f x)\right ) \tan ^6(e+f x) \, dx=-\frac {{\left (f x + e\right )} a}{f} + \frac {15 \, b f^{6} \tan \left (f x + e\right )^{7} + 21 \, a f^{6} \tan \left (f x + e\right )^{5} - 35 \, a f^{6} \tan \left (f x + e\right )^{3} + 105 \, a f^{6} \tan \left (f x + e\right )}{105 \, f^{7}} \] Input:
integrate((a+b*sec(f*x+e)^2)*tan(f*x+e)^6,x, algorithm="giac")
Output:
-(f*x + e)*a/f + 1/105*(15*b*f^6*tan(f*x + e)^7 + 21*a*f^6*tan(f*x + e)^5 - 35*a*f^6*tan(f*x + e)^3 + 105*a*f^6*tan(f*x + e))/f^7
Time = 15.31 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.80 \[ \int \left (a+b \sec ^2(e+f x)\right ) \tan ^6(e+f x) \, dx=\frac {\frac {b\,{\mathrm {tan}\left (e+f\,x\right )}^7}{7}+\frac {a\,{\mathrm {tan}\left (e+f\,x\right )}^5}{5}-\frac {a\,{\mathrm {tan}\left (e+f\,x\right )}^3}{3}+a\,\mathrm {tan}\left (e+f\,x\right )-a\,f\,x}{f} \] Input:
int(tan(e + f*x)^6*(a + b/cos(e + f*x)^2),x)
Output:
(a*tan(e + f*x) - (a*tan(e + f*x)^3)/3 + (a*tan(e + f*x)^5)/5 + (b*tan(e + f*x)^7)/7 - a*f*x)/f
Time = 0.15 (sec) , antiderivative size = 393, normalized size of antiderivative = 6.14 \[ \int \left (a+b \sec ^2(e+f x)\right ) \tan ^6(e+f x) \, dx=\frac {21 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{6} \tan \left (f x +e \right )^{5} a -35 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{6} \tan \left (f x +e \right )^{3} a +105 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{6} \tan \left (f x +e \right ) a -105 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{6} a f x -63 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} \tan \left (f x +e \right )^{5} a +105 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} \tan \left (f x +e \right )^{3} a -315 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} \tan \left (f x +e \right ) a +315 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} a f x +63 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} \tan \left (f x +e \right )^{5} a -105 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} \tan \left (f x +e \right )^{3} a +315 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} \tan \left (f x +e \right ) a -315 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a f x -21 \cos \left (f x +e \right ) \tan \left (f x +e \right )^{5} a +35 \cos \left (f x +e \right ) \tan \left (f x +e \right )^{3} a -105 \cos \left (f x +e \right ) \tan \left (f x +e \right ) a +105 \cos \left (f x +e \right ) a f x -15 \sin \left (f x +e \right )^{7} b}{105 \cos \left (f x +e \right ) f \left (\sin \left (f x +e \right )^{6}-3 \sin \left (f x +e \right )^{4}+3 \sin \left (f x +e \right )^{2}-1\right )} \] Input:
int((a+b*sec(f*x+e)^2)*tan(f*x+e)^6,x)
Output:
(21*cos(e + f*x)*sin(e + f*x)**6*tan(e + f*x)**5*a - 35*cos(e + f*x)*sin(e + f*x)**6*tan(e + f*x)**3*a + 105*cos(e + f*x)*sin(e + f*x)**6*tan(e + f* x)*a - 105*cos(e + f*x)*sin(e + f*x)**6*a*f*x - 63*cos(e + f*x)*sin(e + f* x)**4*tan(e + f*x)**5*a + 105*cos(e + f*x)*sin(e + f*x)**4*tan(e + f*x)**3 *a - 315*cos(e + f*x)*sin(e + f*x)**4*tan(e + f*x)*a + 315*cos(e + f*x)*si n(e + f*x)**4*a*f*x + 63*cos(e + f*x)*sin(e + f*x)**2*tan(e + f*x)**5*a - 105*cos(e + f*x)*sin(e + f*x)**2*tan(e + f*x)**3*a + 315*cos(e + f*x)*sin( e + f*x)**2*tan(e + f*x)*a - 315*cos(e + f*x)*sin(e + f*x)**2*a*f*x - 21*c os(e + f*x)*tan(e + f*x)**5*a + 35*cos(e + f*x)*tan(e + f*x)**3*a - 105*co s(e + f*x)*tan(e + f*x)*a + 105*cos(e + f*x)*a*f*x - 15*sin(e + f*x)**7*b) /(105*cos(e + f*x)*f*(sin(e + f*x)**6 - 3*sin(e + f*x)**4 + 3*sin(e + f*x) **2 - 1))