\(\int (a+b \sec ^2(e+f x)) \tan ^4(e+f x) \, dx\) [318]

Optimal result

Integrand size = 21, antiderivative size = 48 \[ \int \left (a+b \sec ^2(e+f x)\right ) \tan ^4(e+f x) \, dx=a x-\frac {a \tan (e+f x)}{f}+\frac {a \tan ^3(e+f x)}{3 f}+\frac {b \tan ^5(e+f x)}{5 f} \]

[Out]

a*x-a*tan(f*x+e)/f+1/3*a*tan(f*x+e)^3/f+1/5*b*tan(f*x+e)^5/f

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4226, 1816, 209} \[ \int \left (a+b \sec ^2(e+f x)\right ) \tan ^4(e+f x) \, dx=\frac {a \tan ^3(e+f x)}{3 f}-\frac {a \tan (e+f x)}{f}+a x+\frac {b \tan ^5(e+f x)}{5 f} \]

[In]

Int[(a + b*Sec[e + f*x]^2)*Tan[e + f*x]^4,x]

[Out]

a*x - (a*Tan[e + f*x])/f + (a*Tan[e + f*x]^3)/(3*f) + (b*Tan[e + f*x]^5)/(5*f)

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 1816

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 4226

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]}, Dist[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] && IntegerQ[n/2] && (IntegerQ[m/2] ||
EqQ[n, 2])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^4 \left (a+b \left (1+x^2\right )\right )}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \left (-a+a x^2+b x^4+\frac {a}{1+x^2}\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {a \tan (e+f x)}{f}+\frac {a \tan ^3(e+f x)}{3 f}+\frac {b \tan ^5(e+f x)}{5 f}+\frac {a \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = a x-\frac {a \tan (e+f x)}{f}+\frac {a \tan ^3(e+f x)}{3 f}+\frac {b \tan ^5(e+f x)}{5 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.19 \[ \int \left (a+b \sec ^2(e+f x)\right ) \tan ^4(e+f x) \, dx=\frac {a \arctan (\tan (e+f x))}{f}-\frac {a \tan (e+f x)}{f}+\frac {a \tan ^3(e+f x)}{3 f}+\frac {b \tan ^5(e+f x)}{5 f} \]

[In]

Integrate[(a + b*Sec[e + f*x]^2)*Tan[e + f*x]^4,x]

[Out]

(a*ArcTan[Tan[e + f*x]])/f - (a*Tan[e + f*x])/f + (a*Tan[e + f*x]^3)/(3*f) + (b*Tan[e + f*x]^5)/(5*f)

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.98

method	result	size
parts	\(\frac {a \left (\frac {\tan \left (f x +e \right )^{3}}{3}-\tan \left (f x +e \right )+\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {b \tan \left (f x +e \right )^{5}}{5 f}\)	\(47\)
derivativedivides	\(\frac {a \left (\frac {\tan \left (f x +e \right )^{3}}{3}-\tan \left (f x +e \right )+f x +e \right )+\frac {b \sin \left (f x +e \right )^{5}}{5 \cos \left (f x +e \right )^{5}}}{f}\)	\(50\)
default	\(\frac {a \left (\frac {\tan \left (f x +e \right )^{3}}{3}-\tan \left (f x +e \right )+f x +e \right )+\frac {b \sin \left (f x +e \right )^{5}}{5 \cos \left (f x +e \right )^{5}}}{f}\)	\(50\)
risch	\(a x +\frac {2 i \left (-30 a \,{\mathrm e}^{8 i \left (f x +e \right )}+15 b \,{\mathrm e}^{8 i \left (f x +e \right )}-90 a \,{\mathrm e}^{6 i \left (f x +e \right )}-110 a \,{\mathrm e}^{4 i \left (f x +e \right )}+30 b \,{\mathrm e}^{4 i \left (f x +e \right )}-70 a \,{\mathrm e}^{2 i \left (f x +e \right )}-20 a +3 b \right )}{15 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{5}}\)	\(103\)

[In]

int((a+b*sec(f*x+e)^2)*tan(f*x+e)^4,x,method=_RETURNVERBOSE)

[Out]

a/f*(1/3*tan(f*x+e)^3-tan(f*x+e)+arctan(tan(f*x+e)))+1/5*b*tan(f*x+e)^5/f

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.50 \[ \int \left (a+b \sec ^2(e+f x)\right ) \tan ^4(e+f x) \, dx=\frac {15 \, a f x \cos \left (f x + e\right )^{5} - {\left ({\left (20 \, a - 3 \, b\right )} \cos \left (f x + e\right )^{4} - {\left (5 \, a - 6 \, b\right )} \cos \left (f x + e\right )^{2} - 3 \, b\right )} \sin \left (f x + e\right )}{15 \, f \cos \left (f x + e\right )^{5}} \]

[In]

integrate((a+b*sec(f*x+e)^2)*tan(f*x+e)^4,x, algorithm="fricas")

[Out]

1/15*(15*a*f*x*cos(f*x + e)^5 - ((20*a - 3*b)*cos(f*x + e)^4 - (5*a - 6*b)*cos(f*x + e)^2 - 3*b)*sin(f*x + e))
/(f*cos(f*x + e)^5)

Sympy [A] (verification not implemented)

Time = 0.83 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.12 \[ \int \left (a+b \sec ^2(e+f x)\right ) \tan ^4(e+f x) \, dx=a \left (\begin {cases} x + \frac {\tan ^{3}{\left (e + f x \right )}}{3 f} - \frac {\tan {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \tan ^{4}{\left (e \right )} & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} x \tan ^{4}{\left (e \right )} \sec ^{2}{\left (e \right )} & \text {for}\: f = 0 \\\frac {\tan ^{5}{\left (e + f x \right )}}{5 f} & \text {otherwise} \end {cases}\right ) \]

[In]

integrate((a+b*sec(f*x+e)**2)*tan(f*x+e)**4,x)

[Out]

a*Piecewise((x + tan(e + f*x)**3/(3*f) - tan(e + f*x)/f, Ne(f, 0)), (x*tan(e)**4, True)) + b*Piecewise((x*tan(
e)**4*sec(e)**2, Eq(f, 0)), (tan(e + f*x)**5/(5*f), True))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.94 \[ \int \left (a+b \sec ^2(e+f x)\right ) \tan ^4(e+f x) \, dx=\frac {3 \, b \tan \left (f x + e\right )^{5} + 5 \, a \tan \left (f x + e\right )^{3} + 15 \, {\left (f x + e\right )} a - 15 \, a \tan \left (f x + e\right )}{15 \, f} \]

[In]

integrate((a+b*sec(f*x+e)^2)*tan(f*x+e)^4,x, algorithm="maxima")

[Out]

1/15*(3*b*tan(f*x + e)^5 + 5*a*tan(f*x + e)^3 + 15*(f*x + e)*a - 15*a*tan(f*x + e))/f

Giac [A] (verification not implemented)

none

Time = 0.94 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.94 \[ \int \left (a+b \sec ^2(e+f x)\right ) \tan ^4(e+f x) \, dx=\frac {3 \, b \tan \left (f x + e\right )^{5} + 5 \, a \tan \left (f x + e\right )^{3} + 15 \, {\left (f x + e\right )} a - 15 \, a \tan \left (f x + e\right )}{15 \, f} \]

[In]

integrate((a+b*sec(f*x+e)^2)*tan(f*x+e)^4,x, algorithm="giac")

[Out]

1/15*(3*b*tan(f*x + e)^5 + 5*a*tan(f*x + e)^3 + 15*(f*x + e)*a - 15*a*tan(f*x + e))/f

Mupad [B] (verification not implemented)

Time = 19.46 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.83 \[ \int \left (a+b \sec ^2(e+f x)\right ) \tan ^4(e+f x) \, dx=\frac {\frac {b\,{\mathrm {tan}\left (e+f\,x\right )}^5}{5}+\frac {a\,{\mathrm {tan}\left (e+f\,x\right )}^3}{3}-a\,\mathrm {tan}\left (e+f\,x\right )+a\,f\,x}{f} \]

[In]

int(tan(e + f*x)^4*(a + b/cos(e + f*x)^2),x)

[Out]

((a*tan(e + f*x)^3)/3 - a*tan(e + f*x) + (b*tan(e + f*x)^5)/5 + a*f*x)/f