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

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Rubi [A]  time = 0.06, antiderivative size = 30, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 1.22, size = 51, normalized size = 1.70 \[ \frac {\sec (c+d x) (a \sin (2 (c+d x))+2 b) (a \sin (c+d x)+b \sec (c+d x))^n}{2 d (n+1)} \]

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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fricas [A]  time = 3.05, size = 59, normalized size = 1.97 \[ \frac {{\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b\right )} \left (\frac {a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b}{\cos \left (d x + c\right )}\right )^{n}}{{\left (d n + d\right )} \cos \left (d x + c\right )} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sec \left (d x + c\right ) \tan \left (d x + c\right ) + a \cos \left (d x + c\right )\right )} {\left (b \sec \left (d x + c\right ) + a \sin \left (d x + c\right )\right )}^{n}\,{d x} \]

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)

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maple [A]  time = 0.46, size = 31, normalized size = 1.03 \[ \frac {\left (b \sec \left (d x +c \right )+a \sin \left (d x +c \right )\right )^{n +1}}{d \left (n +1\right )} \]

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))^(n+1)/d/(n+1)

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maxima [A]  time = 0.31, size = 30, normalized size = 1.00 \[ \frac {{\left (b \sec \left (d x + c\right ) + a \sin \left (d x + c\right )\right )}^{n + 1}}{d {\left (n + 1\right )}} \]

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]

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

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mupad [B]  time = 5.54, size = 63, normalized size = 2.10 \[ \left \{\begin {array}{cl} \frac {\ln \left (a\,\sin \left (c+d\,x\right )+\frac {b}{\cos \left (c+d\,x\right )}\right )}{d} & \text {\ if\ \ }n=-1\\ \frac {{\left (a\,\sin \left (c+d\,x\right )+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{n+1}}{d\,\left (n+1\right )} & \text {\ if\ \ }n\neq -1 \end {array}\right . \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

piecewise(n == -1, log(a*sin(c + d*x) + b/cos(c + d*x))/d, n ~= -1, (a*sin(c + d*x) + b/cos(c + d*x))^(n + 1)/
(d*(n + 1)))

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sympy [A]  time = 74.07, size = 138, normalized size = 4.60 \[ \begin {cases} \frac {x \left (a \cos {\relax (c )} + b \tan {\relax (c )} \sec {\relax (c )}\right )}{a \sin {\relax (c )} + b \sec {\relax (c )}} & \text {for}\: d = 0 \wedge n = -1 \\x \left (a \sin {\relax (c )} + b \sec {\relax (c )}\right )^{n} \left (a \cos {\relax (c )} + b \tan {\relax (c )} \sec {\relax (c )}\right ) & \text {for}\: d = 0 \\\frac {\log {\left (\sin {\left (c + d x \right )} + \frac {b \sec {\left (c + d x \right )}}{a} \right )}}{d} & \text {for}\: n = -1 \\\frac {a \left (a \sin {\left (c + d x \right )} + b \sec {\left (c + d x \right )}\right )^{n} \sin {\left (c + d x \right )}}{d n + d} + \frac {b \left (a \sin {\left (c + d x \right )} + b \sec {\left (c + d x \right )}\right )^{n} \sec {\left (c + d x \right )}}{d n + d} & \text {otherwise} \end {cases} \]

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]

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

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