3.526 \(\int \sec (c+d x) (a+b \tan (c+d x))^2 \, dx\)

Optimal. Leaf size=65 \[ \frac{\left (2 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{3 a b \sec (c+d x)}{2 d}+\frac{b \sec (c+d x) (a+b \tan (c+d x))}{2 d} \]

[Out]

((2*a^2 - b^2)*ArcTanh[Sin[c + d*x]])/(2*d) + (3*a*b*Sec[c + d*x])/(2*d) + (b*Sec[c + d*x]*(a + b*Tan[c + d*x]
))/(2*d)

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Rubi [A]  time = 0.051964, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {3508, 3486, 3770} \[ \frac{\left (2 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{3 a b \sec (c+d x)}{2 d}+\frac{b \sec (c+d x) (a+b \tan (c+d x))}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]*(a + b*Tan[c + d*x])^2,x]

[Out]

((2*a^2 - b^2)*ArcTanh[Sin[c + d*x]])/(2*d) + (3*a*b*Sec[c + d*x])/(2*d) + (b*Sec[c + d*x]*(a + b*Tan[c + d*x]
))/(2*d)

Rule 3508

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*(d*Se
c[e + f*x])^m*(a + b*Tan[e + f*x]))/(f*(m + 1)), x] + Dist[1/(m + 1), Int[(d*Sec[e + f*x])^m*(a^2*(m + 1) - b^
2 + a*b*(m + 2)*Tan[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 + b^2, 0] && NeQ[m, -1]

Rule 3486

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*(d*Sec[
e + f*x])^m)/(f*m), x] + Dist[a, Int[(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2
*m] || NeQ[a^2 + b^2, 0])

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

Mathematica [A]  time = 0.038608, size = 67, normalized size = 1.03 \[ \frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a b \sec (c+d x)}{d}-\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^2 \tan (c+d x) \sec (c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]*(a + b*Tan[c + d*x])^2,x]

[Out]

(a^2*ArcTanh[Sin[c + d*x]])/d - (b^2*ArcTanh[Sin[c + d*x]])/(2*d) + (2*a*b*Sec[c + d*x])/d + (b^2*Sec[c + d*x]
*Tan[c + d*x])/(2*d)

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Maple [A]  time = 0.021, size = 98, normalized size = 1.5 \begin{align*}{\frac{{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{\sin \left ( dx+c \right ){b}^{2}}{2\,d}}-{\frac{{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+2\,{\frac{ab}{d\cos \left ( dx+c \right ) }}+{\frac{{a}^{2}\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(sec(d*x+c)*(a+b*tan(d*x+c))^2,x)

[Out]

1/2/d*b^2*sin(d*x+c)^3/cos(d*x+c)^2+1/2/d*sin(d*x+c)*b^2-1/2/d*b^2*ln(sec(d*x+c)+tan(d*x+c))+2/d*a*b/cos(d*x+c
)+1/d*a^2*ln(sec(d*x+c)+tan(d*x+c))

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Maxima [A]  time = 1.36742, size = 111, normalized size = 1.71 \begin{align*} -\frac{b^{2}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + \log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 4 \, a^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) - \frac{8 \, a b}{\cos \left (d x + c\right )}}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(b^2*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) + log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) - 4*a^2*log(se
c(d*x + c) + tan(d*x + c)) - 8*a*b/cos(d*x + c))/d

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Fricas [A]  time = 1.79038, size = 234, normalized size = 3.6 \begin{align*} \frac{{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 8 \, a b \cos \left (d x + c\right ) + 2 \, b^{2} \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(sec(d*x+c)*(a+b*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

1/4*((2*a^2 - b^2)*cos(d*x + c)^2*log(sin(d*x + c) + 1) - (2*a^2 - b^2)*cos(d*x + c)^2*log(-sin(d*x + c) + 1)
+ 8*a*b*cos(d*x + c) + 2*b^2*sin(d*x + c))/(d*cos(d*x + c)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.48975, size = 165, normalized size = 2.54 \begin{align*} \frac{{\left (2 \, a^{2} - b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) -{\left (2 \, a^{2} - b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 4 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 4 \, a b\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((2*a^2 - b^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - (2*a^2 - b^2)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*(
b^2*tan(1/2*d*x + 1/2*c)^3 - 4*a*b*tan(1/2*d*x + 1/2*c)^2 + b^2*tan(1/2*d*x + 1/2*c) + 4*a*b)/(tan(1/2*d*x + 1
/2*c)^2 - 1)^2)/d