3.1446 \(\int \sec (c+d x) (a+b \sin (c+d x)) \tan (c+d x) \, dx\)

Optimal. Leaf size=27 \[ \frac{a \sec (c+d x)}{d}+\frac{b \tan (c+d x)}{d}-b x \]

[Out]

-(b*x) + (a*Sec[c + d*x])/d + (b*Tan[c + d*x])/d

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Rubi [A]  time = 0.0491544, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2838, 2606, 8, 3473} \[ \frac{a \sec (c+d x)}{d}+\frac{b \tan (c+d x)}{d}-b x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*x) + (a*Sec[c + d*x])/d + (b*Tan[c + d*x])/d

Rule 2838

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)
*(x_)]), x_Symbol] :> Dist[a, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[(g*Cos[e + f*x
])^p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rubi steps

\begin{align*} \int \sec (c+d x) (a+b \sin (c+d x)) \tan (c+d x) \, dx &=a \int \sec (c+d x) \tan (c+d x) \, dx+b \int \tan ^2(c+d x) \, dx\\ &=\frac{b \tan (c+d x)}{d}-b \int 1 \, dx+\frac{a \operatorname{Subst}(\int 1 \, dx,x,\sec (c+d x))}{d}\\ &=-b x+\frac{a \sec (c+d x)}{d}+\frac{b \tan (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.0155202, size = 36, normalized size = 1.33 \[ \frac{a \sec (c+d x)}{d}-\frac{b \tan ^{-1}(\tan (c+d x))}{d}+\frac{b \tan (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.031, size = 32, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({\frac{a}{\cos \left ( dx+c \right ) }}+b \left ( \tan \left ( dx+c \right ) -dx-c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.51906, size = 43, normalized size = 1.59 \begin{align*} -\frac{{\left (d x + c - \tan \left (d x + c\right )\right )} b - \frac{a}{\cos \left (d x + c\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.5509, size = 82, normalized size = 3.04 \begin{align*} -\frac{b d x \cos \left (d x + c\right ) - b \sin \left (d x + c\right ) - a}{d \cos \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \sin{\left (c + d x \right )}\right ) \sin{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**2*sin(d*x+c)*(a+b*sin(d*x+c)),x)

[Out]

Integral((a + b*sin(c + d*x))*sin(c + d*x)*sec(c + d*x)**2, x)

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Giac [A]  time = 1.19254, size = 58, normalized size = 2.15 \begin{align*} -\frac{{\left (d x + c\right )} b + \frac{2 \,{\left (b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a\right )}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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