3.288 \(\int \text{sech}^2(c+d x) (a+b \sinh ^2(c+d x)) \, dx\)

Optimal. Leaf size=19 \[ \frac{(a-b) \tanh (c+d x)}{d}+b x \]

[Out]

b*x + ((a - b)*Tanh[c + d*x])/d

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Rubi [A]  time = 0.0340843, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3191, 388, 206} \[ \frac{(a-b) \tanh (c+d x)}{d}+b x \]

Antiderivative was successfully verified.

[In]

Int[Sech[c + d*x]^2*(a + b*Sinh[c + d*x]^2),x]

[Out]

b*x + ((a - b)*Tanh[c + d*x])/d

Rule 3191

Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, T
an[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \text{sech}^2(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a-(a-b) x^2}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{(a-b) \tanh (c+d x)}{d}+\frac{b \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=b x+\frac{(a-b) \tanh (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.0138384, size = 36, normalized size = 1.89 \[ \frac{a \tanh (c+d x)}{d}+\frac{b \tanh ^{-1}(\tanh (c+d x))}{d}-\frac{b \tanh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[c + d*x]^2*(a + b*Sinh[c + d*x]^2),x]

[Out]

(b*ArcTanh[Tanh[c + d*x]])/d + (a*Tanh[c + d*x])/d - (b*Tanh[c + d*x])/d

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Maple [A]  time = 0.056, size = 29, normalized size = 1.5 \begin{align*}{\frac{\tanh \left ( dx+c \right ) a+b \left ( dx+c-\tanh \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^2*(a+b*sinh(d*x+c)^2),x)

[Out]

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

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Maxima [B]  time = 1.04978, size = 63, normalized size = 3.32 \begin{align*} b{\left (x + \frac{c}{d} - \frac{2}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} + \frac{2 \, a}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.51005, size = 101, normalized size = 5.32 \begin{align*} \frac{{\left (b d x - a + b\right )} \cosh \left (d x + c\right ) +{\left (a - b\right )} \sinh \left (d x + c\right )}{d \cosh \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*d*x - a + b)*cosh(d*x + c) + (a - b)*sinh(d*x + c))/(d*cosh(d*x + c))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)**2*(a+b*sinh(d*x+c)**2),x)

[Out]

Integral((a + b*sinh(c + d*x)**2)*sech(c + d*x)**2, x)

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Giac [A]  time = 1.15137, size = 46, normalized size = 2.42 \begin{align*} \frac{{\left (d x + c\right )} b}{d} - \frac{2 \,{\left (a - b\right )}}{d{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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