3.75 \(\int \frac{\tan (c+d x)}{(a+a \sec (c+d x))^2} \, dx\)

Optimal. Leaf size=36 \[ -\frac{1}{a^2 d (\cos (c+d x)+1)}-\frac{\log (\cos (c+d x)+1)}{a^2 d} \]

[Out]

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

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Rubi [A]  time = 0.0347561, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3879, 43} \[ -\frac{1}{a^2 d (\cos (c+d x)+1)}-\frac{\log (\cos (c+d x)+1)}{a^2 d} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]/(a + a*Sec[c + d*x])^2,x]

[Out]

-(1/(a^2*d*(1 + Cos[c + d*x]))) - Log[1 + Cos[c + d*x]]/(a^2*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 (c+d x)}{(a+a \sec (c+d x))^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x}{(a+a x)^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{a^2 (1+x)^2}+\frac{1}{a^2 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{1}{a^2 d (1+\cos (c+d x))}-\frac{\log (1+\cos (c+d x))}{a^2 d}\\ \end{align*}

Mathematica [A]  time = 0.132192, size = 56, normalized size = 1.56 \[ -\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) \left (2 \cos (c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+1\right )}{2 a^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]/(a + a*Sec[c + d*x])^2,x]

[Out]

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

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Maple [A]  time = 0.024, size = 50, normalized size = 1.4 \begin{align*}{\frac{1}{d{a}^{2} \left ( 1+\sec \left ( dx+c \right ) \right ) }}-{\frac{\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{d{a}^{2}}}+{\frac{\ln \left ( \sec \left ( dx+c \right ) \right ) }{d{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)/(a+a*sec(d*x+c))^2,x)

[Out]

1/d/a^2/(1+sec(d*x+c))-1/d/a^2*ln(1+sec(d*x+c))+1/d/a^2*ln(sec(d*x+c))

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Maxima [A]  time = 1.13091, size = 47, normalized size = 1.31 \begin{align*} -\frac{\frac{1}{a^{2} \cos \left (d x + c\right ) + a^{2}} + \frac{\log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2}}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)/(a+a*sec(d*x+c))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.12163, size = 113, normalized size = 3.14 \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 ) + 1}{a^{2} d \cos \left (d x + c\right ) + a^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)/(a+a*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 29.8006, size = 177, normalized size = 4.92 \begin{align*} \begin{cases} \frac{\log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \sec{\left (c + d x \right )}}{2 a^{2} d \sec{\left (c + d x \right )} + 2 a^{2} d} + \frac{\log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a^{2} d \sec{\left (c + d x \right )} + 2 a^{2} d} - \frac{2 \log{\left (\sec{\left (c + d x \right )} + 1 \right )} \sec{\left (c + d x \right )}}{2 a^{2} d \sec{\left (c + d x \right )} + 2 a^{2} d} - \frac{2 \log{\left (\sec{\left (c + d x \right )} + 1 \right )}}{2 a^{2} d \sec{\left (c + d x \right )} + 2 a^{2} d} + \frac{2}{2 a^{2} d \sec{\left (c + d x \right )} + 2 a^{2} d} & \text{for}\: d \neq 0 \\\frac{x \tan{\left (c \right )}}{\left (a \sec{\left (c \right )} + a\right )^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)/(a+a*sec(d*x+c))**2,x)

[Out]

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

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Giac [A]  time = 1.26897, size = 77, normalized size = 2.14 \begin{align*} \frac{\frac{2 \, \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{2}} + \frac{\cos \left (d x + c\right ) - 1}{a^{2}{\left (\cos \left (d x + c\right ) + 1\right )}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)/(a+a*sec(d*x+c))^2,x, algorithm="giac")

[Out]

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