3.640 \(\int (b \sec (c+d x)+a \sin (c+d x)) (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0280635, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.024, Rules used = {4385} \[ \frac{(a \sin (c+d x)+b \sec (c+d x))^2}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 (b \sec (c+d x)+a \sin (c+d x)) (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx &=\frac{(b \sec (c+d x)+a \sin (c+d x))^2}{2 d}\\ \end{align*}

Mathematica [B]  time = 0.0368757, size = 67, normalized size = 2.58 \[ -\frac{a^2 \cos ^2(c+d x)}{2 d}-\frac{a b \tan ^{-1}(\tan (c+d x))}{d}+\frac{a b \tan (c+d x)}{d}+a b x+\frac{b^2 \sec ^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.119, size = 57, normalized size = 2.2 \begin{align*}{\frac{1}{d} \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}{a}^{2}}{2}}+ab \left ( \tan \left ( dx+c \right ) -dx-c \right ) +ab \left ( dx+c \right ) +{\frac{{b}^{2}}{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.96683, size = 32, normalized size = 1.23 \begin{align*} \frac{{\left (b \sec \left (d x + c\right ) + a \sin \left (d x + c\right )\right )}^{2}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c)+a*sin(d*x+c))*(a*cos(d*x+c)+b*sec(d*x+c)*tan(d*x+c)),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 = 2.39611, size = 150, normalized size = 5.77 \begin{align*} -\frac{2 \, a^{2} \cos \left (d x + c\right )^{4} - a^{2} \cos \left (d x + c\right )^{2} - 4 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, b^{2}}{4 \, d \cos \left (d x + c\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 5.01278, size = 73, normalized size = 2.81 \begin{align*} \begin{cases} \frac{a^{2} \sin ^{2}{\left (c + d x \right )}}{2 d} + \frac{a b \sin{\left (c + d x \right )} \sec{\left (c + d x \right )}}{d} + \frac{b^{2} \sec ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + b \sec{\left (c \right )}\right ) \left (a \cos{\left (c \right )} + b \tan{\left (c \right )} \sec{\left (c \right )}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18359, size = 61, normalized size = 2.35 \begin{align*} \frac{b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) - \frac{a^{2}}{\tan \left (d x + c\right )^{2} + 1}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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