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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 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])

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]

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 = 0.02, size = 25, normalized size = 1.00 \[ \frac {a \sec (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d} \]

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

________________________________________________________________________________________

fricas [A]  time = 0.72, size = 34, normalized size = 1.36 \[ -\frac {a \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) - a}{d \cos \left (d x + c\right )} \]

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))

________________________________________________________________________________________

giac [B]  time = 0.50, size = 106, normalized size = 4.24 \[ \frac {a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {3 \, a + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1}}{d} \]

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

________________________________________________________________________________________

maple [A]  time = 0.28, size = 25, normalized size = 1.00 \[ \frac {a \sec \left (d x +c \right )}{d}+\frac {a \ln \left (\sec \left (d x +c \right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.79, size = 26, normalized size = 1.04 \[ -\frac {a \log \left (\cos \left (d x + c\right )\right ) - \frac {a}{\cos \left (d x + c\right )}}{d} \]

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

________________________________________________________________________________________

mupad [B]  time = 1.17, size = 40, normalized size = 1.60 \[ \frac {2\,a\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d}-\frac {2\,a}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)*(a + a/cos(c + d*x)),x)

[Out]

(2*a*atanh(tan(c/2 + (d*x)/2)^2))/d - (2*a)/(d*(tan(c/2 + (d*x)/2)^2 - 1))

________________________________________________________________________________________

sympy [A]  time = 0.26, size = 37, normalized size = 1.48 \[ \begin {cases} \frac {a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a \sec {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \sec {\relax (c )} + a\right ) \tan {\relax (c )} & \text {otherwise} \end {cases} \]

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))

________________________________________________________________________________________