3.89 \(\int \frac{\tan ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx\)

Optimal. Leaf size=46 \[ \frac{\sec (c+d x)}{a^3 d}+\frac{3 \log (\cos (c+d x))}{a^3 d}-\frac{4 \log (\cos (c+d x)+1)}{a^3 d} \]

[Out]

(3*Log[Cos[c + d*x]])/(a^3*d) - (4*Log[1 + Cos[c + d*x]])/(a^3*d) + Sec[c + d*x]/(a^3*d)

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Rubi [A]  time = 0.0541435, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3879, 88} \[ \frac{\sec (c+d x)}{a^3 d}+\frac{3 \log (\cos (c+d x))}{a^3 d}-\frac{4 \log (\cos (c+d x)+1)}{a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^5/(a + a*Sec[c + d*x])^3,x]

[Out]

(3*Log[Cos[c + d*x]])/(a^3*d) - (4*Log[1 + Cos[c + d*x]])/(a^3*d) + Sec[c + d*x]/(a^3*d)

Rule 3879

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[1/(a^(m - n
- 1)*b^n*d), Subst[Int[((a - b*x)^((m - 1)/2)*(a + b*x)^((m - 1)/2 + n))/x^(m + n), x], x, Sin[c + d*x]], x] /
; FreeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{\tan ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(a-a x)^2}{x^2 (a+a x)} \, dx,x,\cos (c+d x)\right )}{a^4 d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{a}{x^2}-\frac{3 a}{x}+\frac{4 a}{1+x}\right ) \, dx,x,\cos (c+d x)\right )}{a^4 d}\\ &=\frac{3 \log (\cos (c+d x))}{a^3 d}-\frac{4 \log (1+\cos (c+d x))}{a^3 d}+\frac{\sec (c+d x)}{a^3 d}\\ \end{align*}

Mathematica [A]  time = 0.113691, size = 36, normalized size = 0.78 \[ \frac{\sec (c+d x)-8 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+3 \log (\cos (c+d x))}{a^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^5/(a + a*Sec[c + d*x])^3,x]

[Out]

(-8*Log[Cos[(c + d*x)/2]] + 3*Log[Cos[c + d*x]] + Sec[c + d*x])/(a^3*d)

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Maple [A]  time = 0.071, size = 46, normalized size = 1. \begin{align*}{\frac{\sec \left ( dx+c \right ) }{{a}^{3}d}}-4\,{\frac{\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{{a}^{3}d}}+{\frac{\ln \left ( \sec \left ( dx+c \right ) \right ) }{{a}^{3}d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^5/(a+a*sec(d*x+c))^3,x)

[Out]

sec(d*x+c)/a^3/d-4/d/a^3*ln(1+sec(d*x+c))+1/d/a^3*ln(sec(d*x+c))

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Maxima [A]  time = 1.16601, size = 61, normalized size = 1.33 \begin{align*} -\frac{\frac{4 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}} - \frac{3 \, \log \left (\cos \left (d x + c\right )\right )}{a^{3}} - \frac{1}{a^{3} \cos \left (d x + c\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^5/(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

-(4*log(cos(d*x + c) + 1)/a^3 - 3*log(cos(d*x + c))/a^3 - 1/(a^3*cos(d*x + c)))/d

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Fricas [A]  time = 1.22972, size = 144, normalized size = 3.13 \begin{align*} \frac{3 \, \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) - 4 \, \cos \left (d x + c\right ) \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 1}{a^{3} d \cos \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^5/(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\tan ^{5}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec{\left (c + d x \right )} + 1}\, dx}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**5/(a+a*sec(d*x+c))**3,x)

[Out]

Integral(tan(c + d*x)**5/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x)/a**3

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Giac [B]  time = 3.67744, size = 151, normalized size = 3.28 \begin{align*} \frac{\frac{\log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{3}} + \frac{3 \, \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{a^{3}} - \frac{\frac{3 \,{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + 1}{a^{3}{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^5/(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

(log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1))/a^3 + 3*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) -
1))/a^3 - (3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)/(a^3*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)))/d