\(\int \tan ^4(a+b x) \, dx\) [85]

Optimal result

Integrand size = 8, antiderivative size = 28 \[ \int \tan ^4(a+b x) \, dx=x-\frac {\tan (a+b x)}{b}+\frac {\tan ^3(a+b x)}{3 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3554, 8} \[ \int \tan ^4(a+b x) \, dx=\frac {\tan ^3(a+b x)}{3 b}-\frac {\tan (a+b x)}{b}+x \]

[In]

Int[Tan[a + b*x]^4,x]

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \tan ^4(a+b x) \, dx=\frac {\arctan (\tan (a+b x))}{b}-\frac {\tan (a+b x)}{b}+\frac {\tan ^3(a+b x)}{3 b} \]

[In]

Integrate[Tan[a + b*x]^4,x]

[Out]

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

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(\frac {\frac {\left (\tan ^{3}\left (b x +a \right )\right )}{3}-\tan \left (b x +a \right )+b x +a}{b}\)	\(28\)
default	\(\frac {\frac {\left (\tan ^{3}\left (b x +a \right )\right )}{3}-\tan \left (b x +a \right )+b x +a}{b}\)	\(28\)
risch	\(x -\frac {4 i \left (3 \,{\mathrm e}^{4 i \left (b x +a \right )}+3 \,{\mathrm e}^{2 i \left (b x +a \right )}+2\right )}{3 b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{3}}\)	\(46\)
norman	\(\frac {x \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-x +\frac {2 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}-\frac {20 \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}+\frac {2 \left (\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}+3 x \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-3 x \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{\left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{3}}\)	\(108\)
parallelrisch	\(\frac {3 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) x b -9 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) x b +6 \left (\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+9 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) x b -20 \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-3 b x +6 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{3 b \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{3} \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )^{3}}\)	\(119\)

[In]

int(sec(b*x+a)^4*sin(b*x+a)^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \tan ^4(a+b x) \, dx=\frac {3 \, b x \cos \left (b x + a\right )^{3} - {\left (4 \, \cos \left (b x + a\right )^{2} - 1\right )} \sin \left (b x + a\right )}{3 \, b \cos \left (b x + a\right )^{3}} \]

[In]

integrate(sec(b*x+a)^4*sin(b*x+a)^4,x, algorithm="fricas")

[Out]

1/3*(3*b*x*cos(b*x + a)^3 - (4*cos(b*x + a)^2 - 1)*sin(b*x + a))/(b*cos(b*x + a)^3)

Sympy [F]

\[ \int \tan ^4(a+b x) \, dx=\int \sin ^{4}{\left (a + b x \right )} \sec ^{4}{\left (a + b x \right )}\, dx \]

[In]

integrate(sec(b*x+a)**4*sin(b*x+a)**4,x)

[Out]

Integral(sin(a + b*x)**4*sec(a + b*x)**4, x)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \tan ^4(a+b x) \, dx=\frac {\tan \left (b x + a\right )^{3} + 3 \, b x + 3 \, a - 3 \, \tan \left (b x + a\right )}{3 \, b} \]

[In]

integrate(sec(b*x+a)^4*sin(b*x+a)^4,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \tan ^4(a+b x) \, dx=\frac {\tan \left (b x + a\right )^{3} + 3 \, b x + 3 \, a - 3 \, \tan \left (b x + a\right )}{3 \, b} \]

[In]

integrate(sec(b*x+a)^4*sin(b*x+a)^4,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \tan ^4(a+b x) \, dx=x-\frac {\mathrm {tan}\left (a+b\,x\right )-\frac {{\mathrm {tan}\left (a+b\,x\right )}^3}{3}}{b} \]

[In]

int(sin(a + b*x)^4/cos(a + b*x)^4,x)

[Out]

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