3.8 \(\int \sec (c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=17 \[ -\frac{a \log (1-\sin (c+d x))}{d} \]

[Out]

-((a*Log[1 - Sin[c + d*x]])/d)

________________________________________________________________________________________

Rubi [A]  time = 0.0199603, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2667, 31} \[ -\frac{a \log (1-\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*Log[1 - Sin[c + d*x]])/d)

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0140097, size = 26, normalized size = 1.53 \[ \frac{a \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.029, size = 16, normalized size = 0.9 \begin{align*} -{\frac{a\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.945267, size = 20, normalized size = 1.18 \begin{align*} -\frac{a \log \left (\sin \left (d x + c\right ) - 1\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*log(sin(d*x + c) - 1)/d

________________________________________________________________________________________

Fricas [A]  time = 1.7122, size = 39, normalized size = 2.29 \begin{align*} -\frac{a \log \left (-\sin \left (d x + c\right ) + 1\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*log(-sin(d*x + c) + 1)/d

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*(Integral(sin(c + d*x)*sec(c + d*x), x) + Integral(sec(c + d*x), x))

________________________________________________________________________________________

Giac [B]  time = 1.20579, size = 50, normalized size = 2.94 \begin{align*} \frac{a \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) - 2 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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