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

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

[Out]

A*x + (C*Tan[c + d*x])/d

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Rubi [A]  time = 0.0130266, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3767, 8} \[ A x+\frac{C \tan (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[A + C*Sec[c + d*x]^2,x]

[Out]

A*x + (C*Tan[c + d*x])/d

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

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

Rubi steps

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

Mathematica [A]  time = 0.0026345, size = 15, normalized size = 1. \[ A x+\frac{C \tan (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[A + C*Sec[c + d*x]^2,x]

[Out]

A*x + (C*Tan[c + d*x])/d

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Maple [A]  time = 0.015, size = 16, normalized size = 1.1 \begin{align*} Ax+{\frac{C\tan \left ( dx+c \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.920524, size = 20, normalized size = 1.33 \begin{align*} A x + \frac{C \tan \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*x + C*tan(d*x + c)/d

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Fricas [B]  time = 0.465923, size = 76, normalized size = 5.07 \begin{align*} \frac{A d x \cos \left (d x + c\right ) + C \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*sec(d*x+c)^2,x, algorithm="fricas")

[Out]

(A*d*x*cos(d*x + c) + C*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 \sec ^{2}{\left (c + d x \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(A + C*sec(c + d*x)**2, x)

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Giac [A]  time = 1.14451, size = 20, normalized size = 1.33 \begin{align*} A x + \frac{C \tan \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*x + C*tan(d*x + c)/d