3.639 \(\int (b \sec (c+d x)+a \sin (c+d x))^2 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx\)

Optimal. Leaf size=26 \[ \frac{(a \sin (c+d x)+b \sec (c+d x))^3}{3 d} \]

[Out]

(b*Sec[c + d*x] + a*Sin[c + d*x])^3/(3*d)

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Rubi [A]  time = 0.0429013, 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} \[ \frac{(a \sin (c+d x)+b \sec (c+d x))^3}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[(b*Sec[c + d*x] + a*Sin[c + d*x])^2*(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])^3/(3*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))^2 (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))^3}{3 d}\\ \end{align*}

Mathematica [A]  time = 1.27789, size = 31, normalized size = 1.19 \[ \frac{\sec ^3(c+d x) (a \sin (2 (c+d x))+2 b)^3}{24 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Sec[c + d*x] + a*Sin[c + d*x])^2*(a*Cos[c + d*x] + b*Sec[c + d*x]*Tan[c + d*x]),x]

[Out]

(Sec[c + d*x]^3*(2*b + a*Sin[2*(c + d*x)])^3)/(24*d)

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Maple [B]  time = 0.172, size = 118, normalized size = 4.5 \begin{align*}{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{{a}^{2}b \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d\cos \left ( dx+c \right ) }}+{\frac{{a}^{2}b \left ( \sin \left ( dx+c \right ) \right ) ^{2}\cos \left ( dx+c \right ) }{d}}+{\frac{a{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{a{b}^{2}\sin \left ( dx+c \right ) }{d}}+{\frac{{b}^{3}}{3\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*sec(d*x+c)+a*sin(d*x+c))^2*(a*cos(d*x+c)+b*sec(d*x+c)*tan(d*x+c)),x)

[Out]

1/3/d*a^3*sin(d*x+c)^3+1/d*a^2*b*sin(d*x+c)^4/cos(d*x+c)+1/d*a^2*b*sin(d*x+c)^2*cos(d*x+c)+1/d*a*b^2*sin(d*x+c
)^3/cos(d*x+c)^2+1/d*a*b^2*sin(d*x+c)+1/3/d*b^3/cos(d*x+c)^3

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Maxima [A]  time = 0.96345, 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 )}^{3}}{3 \, 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))^2*(a*cos(d*x+c)+b*sec(d*x+c)*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/3*(b*sec(d*x + c) + a*sin(d*x + c))^3/d

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Fricas [B]  time = 2.60107, size = 217, normalized size = 8.35 \begin{align*} -\frac{3 \, a^{2} b \cos \left (d x + c\right )^{4} - 3 \, a^{2} b \cos \left (d x + c\right )^{2} - b^{3} +{\left (a^{3} \cos \left (d x + c\right )^{5} - a^{3} \cos \left (d x + c\right )^{3} - 3 \, a b^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3 \, d \cos \left (d x + c\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c)+a*sin(d*x+c))^2*(a*cos(d*x+c)+b*sec(d*x+c)*tan(d*x+c)),x, algorithm="fricas")

[Out]

-1/3*(3*a^2*b*cos(d*x + c)^4 - 3*a^2*b*cos(d*x + c)^2 - b^3 + (a^3*cos(d*x + c)^5 - a^3*cos(d*x + c)^3 - 3*a*b
^2*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^3)

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Sympy [A]  time = 18.2675, size = 100, normalized size = 3.85 \begin{align*} \begin{cases} \frac{a^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{a^{2} b \sin ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{d} + \frac{a b^{2} \sin{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{d} + \frac{b^{3} \sec ^{3}{\left (c + d x \right )}}{3 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + b \sec{\left (c \right )}\right )^{2} \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))**2*(a*cos(d*x+c)+b*sec(d*x+c)*tan(d*x+c)),x)

[Out]

Piecewise((a**3*sin(c + d*x)**3/(3*d) + a**2*b*sin(c + d*x)**2*sec(c + d*x)/d + a*b**2*sin(c + d*x)*sec(c + d*
x)**2/d + b**3*sec(c + d*x)**3/(3*d), Ne(d, 0)), (x*(a*sin(c) + b*sec(c))**2*(a*cos(c) + b*tan(c)*sec(c)), Tru
e))

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c)+a*sin(d*x+c))^2*(a*cos(d*x+c)+b*sec(d*x+c)*tan(d*x+c)),x, algorithm="giac")

[Out]

Timed out