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

Optimal. Leaf size=35 \[ \frac{2}{a^3 d (\cos (c+d x)+1)}+\frac{\log (\cos (c+d x)+1)}{a^3 d} \]

[Out]

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

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Rubi [A]  time = 0.0514823, antiderivative size = 35, 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, 43} \[ \frac{2}{a^3 d (\cos (c+d x)+1)}+\frac{\log (\cos (c+d x)+1)}{a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2/(a^3*d*(1 + Cos[c + d*x])) + Log[1 + Cos[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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.062097, size = 33, normalized size = 0.94 \[ \frac{\tan ^2\left (\frac{1}{2} (c+d x)\right )+2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.092, size = 51, normalized size = 1.5 \begin{align*} -2\,{\frac{1}{d{a}^{3} \left ( 1+\sec \left ( dx+c \right ) \right ) }}+{\frac{\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{d{a}^{3}}}-{\frac{\ln \left ( \sec \left ( dx+c \right ) \right ) }{d{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/d/a^3/(1+sec(d*x+c))+1/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 = 1.06252, size = 49, normalized size = 1.4 \begin{align*} \frac{\frac{2}{a^{3} \cos \left (d x + c\right ) + a^{3}} + \frac{\log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.11642, size = 112, normalized size = 3.2 \begin{align*} \frac{{\left (\cos \left (d x + c\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 2}{a^{3} d \cos \left (d x + c\right ) + a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 27.4519, size = 457, normalized size = 13.06 \begin{align*} \begin{cases} - \frac{\log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \sec ^{2}{\left (c + d x \right )}}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec{\left (c + d x \right )} + 2 a^{3} d} - \frac{2 \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \sec{\left (c + d x \right )}}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec{\left (c + d x \right )} + 2 a^{3} d} - \frac{\log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec{\left (c + d x \right )} + 2 a^{3} d} + \frac{2 \log{\left (\sec{\left (c + d x \right )} + 1 \right )} \sec ^{2}{\left (c + d x \right )}}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec{\left (c + d x \right )} + 2 a^{3} d} + \frac{4 \log{\left (\sec{\left (c + d x \right )} + 1 \right )} \sec{\left (c + d x \right )}}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec{\left (c + d x \right )} + 2 a^{3} d} + \frac{2 \log{\left (\sec{\left (c + d x \right )} + 1 \right )}}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec{\left (c + d x \right )} + 2 a^{3} d} + \frac{\tan ^{2}{\left (c + d x \right )}}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec{\left (c + d x \right )} + 2 a^{3} d} - \frac{2 \sec{\left (c + d x \right )}}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec{\left (c + d x \right )} + 2 a^{3} d} - \frac{2}{2 a^{3} d \sec ^{2}{\left (c + d x \right )} + 4 a^{3} d \sec{\left (c + d x \right )} + 2 a^{3} d} & \text{for}\: d \neq 0 \\\frac{x \tan ^{3}{\left (c \right )}}{\left (a \sec{\left (c \right )} + a\right )^{3}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-log(tan(c + d*x)**2 + 1)*sec(c + d*x)**2/(2*a**3*d*sec(c + d*x)**2 + 4*a**3*d*sec(c + d*x) + 2*a**
3*d) - 2*log(tan(c + d*x)**2 + 1)*sec(c + d*x)/(2*a**3*d*sec(c + d*x)**2 + 4*a**3*d*sec(c + d*x) + 2*a**3*d) -
log(tan(c + d*x)**2 + 1)/(2*a**3*d*sec(c + d*x)**2 + 4*a**3*d*sec(c + d*x) + 2*a**3*d) + 2*log(sec(c + d*x) +
1)*sec(c + d*x)**2/(2*a**3*d*sec(c + d*x)**2 + 4*a**3*d*sec(c + d*x) + 2*a**3*d) + 4*log(sec(c + d*x) + 1)*se
c(c + d*x)/(2*a**3*d*sec(c + d*x)**2 + 4*a**3*d*sec(c + d*x) + 2*a**3*d) + 2*log(sec(c + d*x) + 1)/(2*a**3*d*s
ec(c + d*x)**2 + 4*a**3*d*sec(c + d*x) + 2*a**3*d) + tan(c + d*x)**2/(2*a**3*d*sec(c + d*x)**2 + 4*a**3*d*sec(
c + d*x) + 2*a**3*d) - 2*sec(c + d*x)/(2*a**3*d*sec(c + d*x)**2 + 4*a**3*d*sec(c + d*x) + 2*a**3*d) - 2/(2*a**
3*d*sec(c + d*x)**2 + 4*a**3*d*sec(c + d*x) + 2*a**3*d), Ne(d, 0)), (x*tan(c)**3/(a*sec(c) + a)**3, True))

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Giac [A]  time = 1.84627, size = 76, normalized size = 2.17 \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{\cos \left (d x + c\right ) - 1}{a^{3}{\left (\cos \left (d x + c\right ) + 1\right )}}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^3/(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 + (cos(d*x + c) - 1)/(a^3*(cos(d*x + c) + 1)))/d