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

Optimal. Leaf size=28 \[ \frac{\sec (c+d x)}{a d}+\frac{\log (\cos (c+d x))}{a d} \]

[Out]

Log[Cos[c + d*x]]/(a*d) + Sec[c + d*x]/(a*d)

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Rubi [A]  time = 0.0481519, antiderivative size = 28, 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} \[ \frac{\sec (c+d x)}{a d}+\frac{\log (\cos (c+d x))}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[Cos[c + d*x]]/(a*d) + Sec[c + d*x]/(a*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)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{a-a x}{x^2} \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{a}{x^2}-\frac{a}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=\frac{\log (\cos (c+d x))}{a d}+\frac{\sec (c+d x)}{a d}\\ \end{align*}

Mathematica [A]  time = 0.065714, size = 21, normalized size = 0.75 \[ \frac{\sec (c+d x)+\log (\cos (c+d x))}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Log[Cos[c + d*x]] + Sec[c + d*x])/(a*d)

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Maple [A]  time = 0.046, size = 30, normalized size = 1.1 \begin{align*}{\frac{\sec \left ( dx+c \right ) }{da}}-{\frac{\ln \left ( \sec \left ( dx+c \right ) \right ) }{da}} \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)),x)

[Out]

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

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Maxima [A]  time = 1.14619, size = 38, normalized size = 1.36 \begin{align*} \frac{\frac{\log \left (\cos \left (d x + c\right )\right )}{a} + \frac{1}{a \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)^3/(a+a*sec(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.15497, size = 78, normalized size = 2.79 \begin{align*} \frac{\cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) + 1}{a 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)^3/(a+a*sec(d*x+c)),x, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [B]  time = 1.97942, size = 150, normalized size = 5.36 \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} - \frac{\log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{a} + \frac{\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1}{a{\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)^3/(a+a*sec(d*x+c)),x, algorithm="giac")

[Out]

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