3.6 \(\int (A+C \cos ^2(c+d x)) \sec ^3(c+d x) \, dx\)

Optimal. Leaf size=40 \[ \frac{(A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A \tan (c+d x) \sec (c+d x)}{2 d} \]

[Out]

((A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*d)

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Rubi [A]  time = 0.0373045, antiderivative size = 40, 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, 3770} \[ \frac{(A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A \tan (c+d x) \sec (c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]

[Out]

((A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*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 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx &=\frac{A \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} (A+2 C) \int \sec (c+d x) \, dx\\ &=\frac{(A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A \sec (c+d x) \tan (c+d x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.0282361, size = 48, normalized size = 1.2 \[ \frac{A \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A \tan (c+d x) \sec (c+d x)}{2 d}+\frac{C \tanh ^{-1}(\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]

[Out]

(A*ArcTanh[Sin[c + d*x]])/(2*d) + (C*ArcTanh[Sin[c + d*x]])/d + (A*Sec[c + d*x]*Tan[c + d*x])/(2*d)

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Maple [A]  time = 0.064, size = 59, normalized size = 1.5 \begin{align*}{\frac{A\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{A\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \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)^3,x)

[Out]

1/2*A*sec(d*x+c)*tan(d*x+c)/d+1/2/d*A*ln(sec(d*x+c)+tan(d*x+c))+1/d*C*ln(sec(d*x+c)+tan(d*x+c))

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Maxima [A]  time = 1.07052, size = 78, normalized size = 1.95 \begin{align*} \frac{{\left (A + 2 \, C\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (A + 2 \, C\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac{2 \, A \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1}}{4 \, 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)^3,x, algorithm="maxima")

[Out]

1/4*((A + 2*C)*log(sin(d*x + c) + 1) - (A + 2*C)*log(sin(d*x + c) - 1) - 2*A*sin(d*x + c)/(sin(d*x + c)^2 - 1)
)/d

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Fricas [A]  time = 1.70487, size = 192, normalized size = 4.8 \begin{align*} \frac{{\left (A + 2 \, C\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (A + 2 \, C\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, A \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \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)^3,x, algorithm="fricas")

[Out]

1/4*((A + 2*C)*cos(d*x + c)^2*log(sin(d*x + c) + 1) - (A + 2*C)*cos(d*x + c)^2*log(-sin(d*x + c) + 1) + 2*A*si
n(d*x + c))/(d*cos(d*x + c)^2)

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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((A+C*cos(d*x+c)**2)*sec(d*x+c)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.18859, size = 81, normalized size = 2.02 \begin{align*} \frac{{\left (A + 2 \, C\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) -{\left (A + 2 \, C\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac{2 \, A \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1}}{4 \, 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)^3,x, algorithm="giac")

[Out]

1/4*((A + 2*C)*log(abs(sin(d*x + c) + 1)) - (A + 2*C)*log(abs(sin(d*x + c) - 1)) - 2*A*sin(d*x + c)/(sin(d*x +
c)^2 - 1))/d