3.109 \(\int \tan ^5(a+b x) \, dx\)

Optimal. Leaf size=43 \[ \frac{\tan ^4(a+b x)}{4 b}-\frac{\tan ^2(a+b x)}{2 b}-\frac{\log (\cos (a+b x))}{b} \]

[Out]

-(Log[Cos[a + b*x]]/b) - Tan[a + b*x]^2/(2*b) + Tan[a + b*x]^4/(4*b)

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Rubi [A]  time = 0.0210946, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3473, 3475} \[ \frac{\tan ^4(a+b x)}{4 b}-\frac{\tan ^2(a+b x)}{2 b}-\frac{\log (\cos (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Tan[a + b*x]^5,x]

[Out]

-(Log[Cos[a + b*x]]/b) - Tan[a + b*x]^2/(2*b) + Tan[a + b*x]^4/(4*b)

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]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \tan ^5(a+b x) \, dx &=\frac{\tan ^4(a+b x)}{4 b}-\int \tan ^3(a+b x) \, dx\\ &=-\frac{\tan ^2(a+b x)}{2 b}+\frac{\tan ^4(a+b x)}{4 b}+\int \tan (a+b x) \, dx\\ &=-\frac{\log (\cos (a+b x))}{b}-\frac{\tan ^2(a+b x)}{2 b}+\frac{\tan ^4(a+b x)}{4 b}\\ \end{align*}

Mathematica [A]  time = 0.051624, size = 37, normalized size = 0.86 \[ -\frac{-\tan ^4(a+b x)+2 \tan ^2(a+b x)+4 \log (\cos (a+b x))}{4 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[a + b*x]^5,x]

[Out]

-(4*Log[Cos[a + b*x]] + 2*Tan[a + b*x]^2 - Tan[a + b*x]^4)/(4*b)

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Maple [A]  time = 0.026, size = 40, normalized size = 0.9 \begin{align*} -{\frac{\ln \left ( \cos \left ( bx+a \right ) \right ) }{b}}-{\frac{ \left ( \tan \left ( bx+a \right ) \right ) ^{2}}{2\,b}}+{\frac{ \left ( \tan \left ( bx+a \right ) \right ) ^{4}}{4\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(b*x+a)^5*sin(b*x+a)^5,x)

[Out]

-ln(cos(b*x+a))/b-1/2*tan(b*x+a)^2/b+1/4*tan(b*x+a)^4/b

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Maxima [A]  time = 0.965203, size = 73, normalized size = 1.7 \begin{align*} \frac{\frac{4 \, \sin \left (b x + a\right )^{2} - 3}{\sin \left (b x + a\right )^{4} - 2 \, \sin \left (b x + a\right )^{2} + 1} - 2 \, \log \left (\sin \left (b x + a\right )^{2} - 1\right )}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^5*sin(b*x+a)^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.674, size = 116, normalized size = 2.7 \begin{align*} -\frac{4 \, \cos \left (b x + a\right )^{4} \log \left (-\cos \left (b x + a\right )\right ) + 4 \, \cos \left (b x + a\right )^{2} - 1}{4 \, b \cos \left (b x + a\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^5*sin(b*x+a)^5,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)**5*sin(b*x+a)**5,x)

[Out]

Timed out

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Giac [B]  time = 1.22642, size = 305, normalized size = 7.09 \begin{align*} \frac{\frac{3 \,{\left (\frac{\cos \left (b x + a\right ) + 1}{\cos \left (b x + a\right ) - 1} + \frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1}\right )}^{2} + \frac{20 \,{\left (\cos \left (b x + a\right ) + 1\right )}}{\cos \left (b x + a\right ) - 1} + \frac{20 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + 44}{{\left (\frac{\cos \left (b x + a\right ) + 1}{\cos \left (b x + a\right ) - 1} + \frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 2\right )}^{2}} + 2 \, \log \left ({\left | -\frac{\cos \left (b x + a\right ) + 1}{\cos \left (b x + a\right ) - 1} - \frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 2 \right |}\right ) - 2 \, \log \left ({\left | -\frac{\cos \left (b x + a\right ) + 1}{\cos \left (b x + a\right ) - 1} - \frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 2 \right |}\right )}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^5*sin(b*x+a)^5,x, algorithm="giac")

[Out]

1/4*((3*((cos(b*x + a) + 1)/(cos(b*x + a) - 1) + (cos(b*x + a) - 1)/(cos(b*x + a) + 1))^2 + 20*(cos(b*x + a) +
1)/(cos(b*x + a) - 1) + 20*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 44)/((cos(b*x + a) + 1)/(cos(b*x + a) - 1)
+ (cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 2)^2 + 2*log(abs(-(cos(b*x + a) + 1)/(cos(b*x + a) - 1) - (cos(b*x
+ a) - 1)/(cos(b*x + a) + 1) + 2)) - 2*log(abs(-(cos(b*x + a) + 1)/(cos(b*x + a) - 1) - (cos(b*x + a) - 1)/(co
s(b*x + a) + 1) - 2)))/b