3.5 \(\int (a+a \sec (c+d x)) \tan (c+d x) \, dx\)

Optimal. Leaf size=25 \[ \frac{a \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.0192775, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3879, 43} \[ \frac{a \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0178748, size = 25, normalized size = 1. \[ \frac{a \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.014, size = 25, normalized size = 1. \begin{align*}{\frac{a\ln \left ( \sec \left ( dx+c \right ) \right ) }{d}}+{\frac{a\sec \left ( dx+c \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.14846, size = 35, normalized size = 1.4 \begin{align*} -\frac{a \log \left (\cos \left (d x + c\right )\right ) - \frac{a}{\cos \left (d x + c\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.697311, size = 80, normalized size = 3.2 \begin{align*} -\frac{a \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) - a}{d \cos \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.423997, size = 37, normalized size = 1.48 \begin{align*} \begin{cases} \frac{a \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{a \sec{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a \sec{\left (c \right )} + a\right ) \tan{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.41025, size = 143, normalized size = 5.72 \begin{align*} \frac{a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{3 \, a + \frac{a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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