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

Optimal. Leaf size=30 \[ \frac{(a \sin (c+d x)+b \sec (c+d x))^{n+1}}{d (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0592225, antiderivative size = 30, 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} \[ \frac{(a \sin (c+d x)+b \sec (c+d x))^{n+1}}{d (n+1)} \]

Antiderivative was successfully verified.

[In]

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

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))^n (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))^{1+n}}{d (1+n)}\\ \end{align*}

Mathematica [A]  time = 1.12582, size = 51, normalized size = 1.7 \[ \frac{\sec (c+d x) (a \sin (2 (c+d x))+2 b) (a \sin (c+d x)+b \sec (c+d x))^n}{2 d (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.078, size = 31, normalized size = 1. \begin{align*}{\frac{ \left ( b\sec \left ( dx+c \right ) +a\sin \left ( dx+c \right ) \right ) ^{1+n}}{d \left ( 1+n \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))^n*(a*cos(d*x+c)+b*sec(d*x+c)*tan(d*x+c)),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.09156, size = 150, normalized size = 5. \begin{align*} \frac{{\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b\right )} \left (\frac{a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b}{\cos \left (d x + c\right )}\right )^{n}}{{\left (d n + d\right )} \cos \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*cos(d*x + c)*sin(d*x + c) + b)*((a*cos(d*x + c)*sin(d*x + c) + b)/cos(d*x + c))^n/((d*n + d)*cos(d*x + c))

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Sympy [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))**n*(a*cos(d*x+c)+b*sec(d*x+c)*tan(d*x+c)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sec \left (d x + c\right ) \tan \left (d x + c\right ) + a \cos \left (d x + c\right )\right )}{\left (b \sec \left (d x + c\right ) + a \sin \left (d x + c\right )\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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