3.12 \(\int (A+C \cos ^2(c+d x)) \sec ^2(c+d x) \, dx\)

Optimal. Leaf size=15 \[ \frac{A \tan (c+d x)}{d}+C x \]

[Out]

C*x + (A*Tan[c + d*x])/d

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Rubi [A]  time = 0.0244259, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3012, 8} \[ \frac{A \tan (c+d x)}{d}+C x \]

Antiderivative was successfully verified.

[In]

Int[(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]

[Out]

C*x + (A*Tan[c + d*x])/d

Rule 3012

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*Cos[e
+ f*x]*(b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Dist[(A*(m + 2) + C*(m + 1))/(b^2*(m + 1)), Int[(b*Sin[e
+ f*x])^(m + 2), x], x] /; FreeQ[{b, e, f, A, C}, x] && LtQ[m, -1]

Rule 8

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

Rubi steps

\begin{align*} \int \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx &=\frac{A \tan (c+d x)}{d}+C \int 1 \, dx\\ &=C x+\frac{A \tan (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.0086647, size = 15, normalized size = 1. \[ \frac{A \tan (c+d x)}{d}+C x \]

Antiderivative was successfully verified.

[In]

Integrate[(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]

[Out]

C*x + (A*Tan[c + d*x])/d

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Maple [A]  time = 0.064, size = 21, normalized size = 1.4 \begin{align*}{\frac{A\tan \left ( dx+c \right ) +C \left ( dx+c \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+C*cos(d*x+c)^2)*sec(d*x+c)^2,x)

[Out]

1/d*(A*tan(d*x+c)+C*(d*x+c))

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Maxima [A]  time = 1.49507, size = 27, normalized size = 1.8 \begin{align*} \frac{{\left (d x + c\right )} C + A \tan \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^2,x, algorithm="maxima")

[Out]

((d*x + c)*C + A*tan(d*x + c))/d

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Fricas [B]  time = 1.59324, size = 76, normalized size = 5.07 \begin{align*} \frac{C d x \cos \left (d x + c\right ) + A \sin \left (d x + c\right )}{d \cos \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^2,x, algorithm="fricas")

[Out]

(C*d*x*cos(d*x + c) + A*sin(d*x + c))/(d*cos(d*x + c))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+C*cos(d*x+c)**2)*sec(d*x+c)**2,x)

[Out]

Integral((A + C*cos(c + d*x)**2)*sec(c + d*x)**2, x)

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Giac [A]  time = 1.16327, size = 27, normalized size = 1.8 \begin{align*} \frac{{\left (d x + c\right )} C + A \tan \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^2,x, algorithm="giac")

[Out]

((d*x + c)*C + A*tan(d*x + c))/d