3.641 \(\int \frac{a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)}{b \sec (c+d x)+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=22 \[ \frac{\log (a \sin (c+d x)+b \sec (c+d x))}{d} \]

[Out]

Log[b*Sec[c + d*x] + a*Sin[c + d*x]]/d

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Rubi [A]  time = 0.0484212, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.023, Rules used = {4383} \[ \frac{\log (a \sin (c+d x)+b \sec (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cos[c + d*x] + b*Sec[c + d*x]*Tan[c + d*x])/(b*Sec[c + d*x] + a*Sin[c + d*x]),x]

[Out]

Log[b*Sec[c + d*x] + a*Sin[c + d*x]]/d

Rule 4383

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[ActivateTrig[y], ActivateTrig[u], x]}, Simp[q*Log[Remo
veContent[ActivateTrig[y], x]], x] /;  !FalseQ[q]] /;  !InertTrigFreeQ[u]

Rubi steps

\begin{align*} \int \frac{a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)}{b \sec (c+d x)+a \sin (c+d x)} \, dx &=\frac{\log (b \sec (c+d x)+a \sin (c+d x))}{d}\\ \end{align*}

Mathematica [A]  time = 0.462226, size = 29, normalized size = 1.32 \[ \frac{\log (a \sin (2 (c+d x))+2 b)-\log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cos[c + d*x] + b*Sec[c + d*x]*Tan[c + d*x])/(b*Sec[c + d*x] + a*Sin[c + d*x]),x]

[Out]

(-Log[Cos[c + d*x]] + Log[2*b + a*Sin[2*(c + d*x)]])/d

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Maple [A]  time = 0.09, size = 23, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( b\sec \left ( dx+c \right ) +a\sin \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.965781, size = 30, normalized size = 1.36 \begin{align*} \frac{\log \left (b \sec \left (d x + c\right ) + a \sin \left (d x + c\right )\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*sec(d*x + c) + a*sin(d*x + c))/d

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Fricas [A]  time = 2.76279, size = 85, normalized size = 3.86 \begin{align*} \frac{\log \left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b\right ) - \log \left (-\cos \left (d x + c\right )\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 7.52807, size = 63, normalized size = 2.86 \begin{align*} \begin{cases} x \tan{\left (c \right )} & \text{for}\: a = 0 \wedge d = 0 \\\frac{\log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} & \text{for}\: a = 0 \\\frac{x \left (a \cos{\left (c \right )} + b \tan{\left (c \right )} \sec{\left (c \right )}\right )}{a \sin{\left (c \right )} + b \sec{\left (c \right )}} & \text{for}\: d = 0 \\\frac{\log{\left (\sin{\left (c + d x \right )} + \frac{b \sec{\left (c + d x \right )}}{a} \right )}}{d} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cos(d*x+c)+b*sec(d*x+c)*tan(d*x+c))/(b*sec(d*x+c)+a*sin(d*x+c)),x)

[Out]

Piecewise((x*tan(c), Eq(a, 0) & Eq(d, 0)), (log(tan(c + d*x)**2 + 1)/(2*d), Eq(a, 0)), (x*(a*cos(c) + b*tan(c)
*sec(c))/(a*sin(c) + b*sec(c)), Eq(d, 0)), (log(sin(c + d*x) + b*sec(c + d*x)/a)/d, True))

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Giac [A]  time = 1.30741, size = 57, normalized size = 2.59 \begin{align*} \frac{2 \, \log \left (b \tan \left (d x + c\right )^{2} + a \tan \left (d x + c\right ) + b\right ) - \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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