3.643 \(\int \frac{a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)}{(b \sec (c+d x)+a \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=26 \[ -\frac{1}{2 d (a \sin (c+d x)+b \sec (c+d x))^2} \]

[Out]

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

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Rubi [A]  time = 0.045177, antiderivative size = 26, 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 = {4385} \[ -\frac{1}{2 d (a \sin (c+d x)+b \sec (c+d x))^2} \]

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])^3,x]

[Out]

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

Rule 4385

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[ActivateTrig[y], ActivateTrig[u], x]}, Simp[(q*A
ctivateTrig[y^(m + 1)])/(m + 1), x] /;  !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1] &&  !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))^3} \, dx &=-\frac{1}{2 d (b \sec (c+d x)+a \sin (c+d x))^2}\\ \end{align*}

Mathematica [A]  time = 0.727144, size = 29, normalized size = 1.12 \[ -\frac{2 \cos ^2(c+d x)}{d (a \sin (2 (c+d x))+2 b)^2} \]

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])^3,x]

[Out]

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

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Maple [A]  time = 0.171, size = 25, normalized size = 1. \begin{align*} -{\frac{1}{2\,d \left ( b\sec \left ( dx+c \right ) +a\sin \left ( dx+c \right ) \right ) ^{2}}} \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))^3,x)

[Out]

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

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Maxima [A]  time = 0.982554, size = 32, normalized size = 1.23 \begin{align*} -\frac{1}{2 \,{\left (b \sec \left (d x + c\right ) + a \sin \left (d x + c\right )\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))^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.98695, size = 149, normalized size = 5.73 \begin{align*} \frac{\cos \left (d x + c\right )^{2}}{2 \,{\left (a^{2} d \cos \left (d x + c\right )^{4} - a^{2} d \cos \left (d x + c\right )^{2} - 2 \, a b d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - b^{2} d\right )}} \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))^3,x, algorithm="fricas")

[Out]

1/2*cos(d*x + c)^2/(a^2*d*cos(d*x + c)^4 - a^2*d*cos(d*x + c)^2 - 2*a*b*d*cos(d*x + c)*sin(d*x + c) - b^2*d)

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Sympy [A]  time = 62.2404, size = 80, normalized size = 3.08 \begin{align*} \begin{cases} - \frac{1}{2 a^{2} d \sin ^{2}{\left (c + d x \right )} + 4 a b d \sin{\left (c + d x \right )} \sec{\left (c + d x \right )} + 2 b^{2} d \sec ^{2}{\left (c + d x \right )}} & \text{for}\: d \neq 0 \\\frac{x \left (a \cos{\left (c \right )} + b \tan{\left (c \right )} \sec{\left (c \right )}\right )}{\left (a \sin{\left (c \right )} + b \sec{\left (c \right )}\right )^{3}} & \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))**3,x)

[Out]

Piecewise((-1/(2*a**2*d*sin(c + d*x)**2 + 4*a*b*d*sin(c + d*x)*sec(c + d*x) + 2*b**2*d*sec(c + d*x)**2), Ne(d,
0)), (x*(a*cos(c) + b*tan(c)*sec(c))/(a*sin(c) + b*sec(c))**3, True))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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))^3,x, algorithm="giac")

[Out]

Exception raised: TypeError