3.5 \(\int \tan ^5(c+d x) \, dx\)

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

[Out]

-(Log[Cos[c + d*x]]/d) - Tan[c + d*x]^2/(2*d) + Tan[c + d*x]^4/(4*d)

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Rubi [A]  time = 0.0197208, 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} \[ \frac{\tan ^4(c+d x)}{4 d}-\frac{\tan ^2(c+d x)}{2 d}-\frac{\log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^5,x]

[Out]

-(Log[Cos[c + d*x]]/d) - Tan[c + d*x]^2/(2*d) + Tan[c + d*x]^4/(4*d)

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

Mathematica [A]  time = 0.0457365, size = 37, normalized size = 0.86 \[ -\frac{-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))}{4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^5,x]

[Out]

-(4*Log[Cos[c + d*x]] + 2*Tan[c + d*x]^2 - Tan[c + d*x]^4)/(4*d)

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Maple [A]  time = 0.003, size = 44, normalized size = 1. \begin{align*}{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^5,x)

[Out]

1/4*tan(d*x+c)^4/d-1/2*tan(d*x+c)^2/d+1/2/d*ln(1+tan(d*x+c)^2)

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Maxima [A]  time = 1.57118, size = 73, normalized size = 1.7 \begin{align*} \frac{\frac{4 \, \sin \left (d x + c\right )^{2} - 3}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 2 \, \log \left (\sin \left (d x + c\right )^{2} - 1\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^5,x, algorithm="maxima")

[Out]

1/4*((4*sin(d*x + c)^2 - 3)/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 2*log(sin(d*x + c)^2 - 1))/d

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Fricas [A]  time = 1.753, size = 101, normalized size = 2.35 \begin{align*} \frac{\tan \left (d x + c\right )^{4} - 2 \, \tan \left (d x + c\right )^{2} - 2 \, \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^5,x, algorithm="fricas")

[Out]

1/4*(tan(d*x + c)^4 - 2*tan(d*x + c)^2 - 2*log(1/(tan(d*x + c)^2 + 1)))/d

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Sympy [A]  time = 0.441391, size = 44, normalized size = 1.02 \begin{align*} \begin{cases} \frac{\log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{\tan ^{4}{\left (c + d x \right )}}{4 d} - \frac{\tan ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \tan ^{5}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**5,x)

[Out]

Piecewise((log(tan(c + d*x)**2 + 1)/(2*d) + tan(c + d*x)**4/(4*d) - tan(c + d*x)**2/(2*d), Ne(d, 0)), (x*tan(c
)**5, True))

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Giac [B]  time = 2.63925, size = 691, normalized size = 16.07 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^5,x, algorithm="giac")

[Out]

-1/4*(2*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2
*tan(d*x)*tan(c) + 1))*tan(d*x)^4*tan(c)^4 + 3*tan(d*x)^4*tan(c)^4 - 8*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)
^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^3*tan(c)^3 + 2*
tan(d*x)^4*tan(c)^2 - 8*tan(d*x)^3*tan(c)^3 + 2*tan(d*x)^2*tan(c)^4 + 12*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(
c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^2*tan(c)^2 -
tan(d*x)^4 - 8*tan(d*x)^3*tan(c) + 4*tan(d*x)^2*tan(c)^2 - 8*tan(d*x)*tan(c)^3 - tan(c)^4 - 8*log(4*(tan(c)^2
+ 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*t
an(d*x)*tan(c) + 2*tan(d*x)^2 - 8*tan(d*x)*tan(c) + 2*tan(c)^2 + 2*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 -
2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)) + 3)/(d*tan(d*x)^4*tan(c)^4
- 4*d*tan(d*x)^3*tan(c)^3 + 6*d*tan(d*x)^2*tan(c)^2 - 4*d*tan(d*x)*tan(c) + d)