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

Optimal. Leaf size=28 \[ \frac {(a-b) \tan ^{-1}(\sinh (c+d x))}{d}+\frac {b \sinh (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 28, normalized size of antiderivative = 1.00, 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 = {3190, 388, 203} \[ \frac {(a-b) \tan ^{-1}(\sinh (c+d x))}{d}+\frac {b \sinh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a - b)*ArcTan[Sinh[c + d*x]])/d + (b*Sinh[c + d*x])/d

Rule 203

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

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 3190

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 37, normalized size = 1.32 \[ \frac {a \tan ^{-1}(\sinh (c+d x))}{d}+\frac {b \sinh (c+d x)}{d}-\frac {b \tan ^{-1}(\sinh (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*ArcTan[Sinh[c + d*x]])/d - (b*ArcTan[Sinh[c + d*x]])/d + (b*Sinh[c + d*x])/d

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fricas [B]  time = 0.58, size = 101, normalized size = 3.61 \[ \frac {b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a - b\right )} \cosh \left (d x + c\right ) + {\left (a - b\right )} \sinh \left (d x + c\right )\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) - b}{2 \, {\left (d \cosh \left (d x + c\right ) + d \sinh \left (d x + c\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + 4*((a - b)*cosh(d*x + c) + (a -
b)*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) - b)/(d*cosh(d*x + c) + d*sinh(d*x + c))

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giac [A]  time = 0.14, size = 40, normalized size = 1.43 \[ \frac {4 \, {\left (a - b\right )} \arctan \left (e^{\left (d x + c\right )}\right ) + b e^{\left (d x + c\right )} - b e^{\left (-d x - c\right )}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(4*(a - b)*arctan(e^(d*x + c)) + b*e^(d*x + c) - b*e^(-d*x - c))/d

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maple [A]  time = 0.06, size = 39, normalized size = 1.39 \[ \frac {b \sinh \left (d x +c \right )}{d}+\frac {2 a \arctan \left ({\mathrm e}^{d x +c}\right )}{d}-\frac {2 b \arctan \left ({\mathrm e}^{d x +c}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*sinh(d*x+c)/d+2/d*a*arctan(exp(d*x+c))-2/d*b*arctan(exp(d*x+c))

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maxima [A]  time = 0.45, size = 56, normalized size = 2.00 \[ \frac {1}{2} \, b {\left (\frac {4 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac {e^{\left (d x + c\right )}}{d} - \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {a \arctan \left (\sinh \left (d x + c\right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*(4*arctan(e^(-d*x - c))/d + e^(d*x + c)/d - e^(-d*x - c)/d) + a*arctan(sinh(d*x + c))/d

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mupad [B]  time = 1.78, size = 88, normalized size = 3.14 \[ \frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a\,\sqrt {d^2}-b\,\sqrt {d^2}\right )}{d\,\sqrt {a^2-2\,a\,b+b^2}}\right )\,\sqrt {a^2-2\,a\,b+b^2}}{\sqrt {d^2}}-\frac {b\,{\mathrm {e}}^{-c-d\,x}}{2\,d}+\frac {b\,{\mathrm {e}}^{c+d\,x}}{2\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sinh ^{2}{\left (c + d x \right )}\right ) \operatorname {sech}{\left (c + d x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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