3.85 \(\int \tan ^4(a+b x) \, dx\)

Optimal. Leaf size=28 \[ \frac {\tan ^3(a+b x)}{3 b}-\frac {\tan (a+b x)}{b}+x \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3473, 8} \[ \frac {\tan ^3(a+b x)}{3 b}-\frac {\tan (a+b x)}{b}+x \]

Antiderivative was successfully verified.

[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 3473

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*} \int \tan ^4(a+b x) \, dx &=\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*}

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Mathematica [A]  time = 0.01, size = 38, normalized size = 1.36 \[ \frac {\tan ^{-1}(\tan (a+b x))}{b}+\frac {\tan ^3(a+b x)}{3 b}-\frac {\tan (a+b x)}{b} \]

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.45, size = 46, normalized size = 1.64 \[ \frac {3 \, b x \cos \left (b x + a\right )^{3} - {\left (4 \, \cos \left (b x + a\right )^{2} - 1\right )} \sin \left (b x + a\right )}{3 \, b \cos \left (b x + a\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.25, size = 29, normalized size = 1.04 \[ \frac {\tan \left (b x + a\right )^{3} + 3 \, b x + 3 \, a - 3 \, \tan \left (b x + a\right )}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.03, size = 28, normalized size = 1.00 \[ \frac {\frac {\left (\tan ^{3}\left (b x +a \right )\right )}{3}-\tan \left (b x +a \right )+b x +a}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.42, size = 29, normalized size = 1.04 \[ \frac {\tan \left (b x + a\right )^{3} + 3 \, b x + 3 \, a - 3 \, \tan \left (b x + a\right )}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 0.40, size = 24, normalized size = 0.86 \[ x-\frac {\mathrm {tan}\left (a+b\,x\right )-\frac {{\mathrm {tan}\left (a+b\,x\right )}^3}{3}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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