3.355 \(\int \text{sech}^3(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx\)
Optimal. Leaf size=86 \[ \frac{a \tan ^{-1}\left (\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right )}{2 f \sqrt{a-b}}+\frac{\tanh (e+f x) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{2 f} \]
[Out]
(a*ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(2*Sqrt[a - b]*f) + (Sech[e + f*x]*Sqrt[a
+ b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(2*f)
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Rubi [A] time = 0.0964859, antiderivative size = 86, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used =
{3190, 378, 377, 203} \[ \frac{a \tan ^{-1}\left (\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right )}{2 f \sqrt{a-b}}+\frac{\tanh (e+f x) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{2 f} \]
Antiderivative was successfully verified.
[In]
Int[Sech[e + f*x]^3*Sqrt[a + b*Sinh[e + f*x]^2],x]
[Out]
(a*ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(2*Sqrt[a - b]*f) + (Sech[e + f*x]*Sqrt[a
+ b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(2*f)
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]
Rule 378
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^q)/(a*n*(p + 1)), x] - Dist[(c*q)/(a*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && GtQ[q, 0] && NeQ[p, -1]
Rule 377
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
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])
Rubi steps
\begin{align*} \int \text{sech}^3(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b x^2}}{\left (1+x^2\right )^2} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac{\text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \tanh (e+f x)}{2 f}+\frac{a \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{2 f}\\ &=\frac{\text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \tanh (e+f x)}{2 f}+\frac{a \operatorname{Subst}\left (\int \frac{1}{1-(-a+b) x^2} \, dx,x,\frac{\sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right )}{2 f}\\ &=\frac{a \tan ^{-1}\left (\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right )}{2 \sqrt{a-b} f}+\frac{\text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \tanh (e+f x)}{2 f}\\ \end{align*}
Mathematica [B] time = 1.51378, size = 175, normalized size = 2.03 \[ \frac{\sinh (e+f x) \left (\sqrt{2} a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} \tanh ^{-1}\left (\frac{\sqrt{-\frac{(a-b) \sinh ^2(e+f x)}{a}}}{\sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}}\right )+\text{sech}^2(e+f x) \sqrt{-\frac{(a-b) \sinh ^2(e+f x)}{a}} (2 a+b \cosh (2 (e+f x))-b)\right )}{4 f \sqrt{-\frac{(a-b) \sinh ^2(e+f x)}{a}} \sqrt{a+b \sinh ^2(e+f x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[Sech[e + f*x]^3*Sqrt[a + b*Sinh[e + f*x]^2],x]
[Out]
(Sinh[e + f*x]*(Sqrt[2]*a*ArcTanh[Sqrt[-(((a - b)*Sinh[e + f*x]^2)/a)]/Sqrt[1 + (b*Sinh[e + f*x]^2)/a]]*Sqrt[(
2*a - b + b*Cosh[2*(e + f*x)])/a] + (2*a - b + b*Cosh[2*(e + f*x)])*Sech[e + f*x]^2*Sqrt[-(((a - b)*Sinh[e + f
*x]^2)/a)]))/(4*f*Sqrt[-(((a - b)*Sinh[e + f*x]^2)/a)]*Sqrt[a + b*Sinh[e + f*x]^2])
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Maple [C] time = 0.085, size = 35, normalized size = 0.4 \begin{align*}{\frac{1}{f}\mbox{{\tt ` int/indef0`}} \left ({\frac{1}{ \left ( \cosh \left ( fx+e \right ) \right ) ^{4}}\sqrt{a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2}}},\sinh \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sech(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(1/2),x)
[Out]
`int/indef0`(1/cosh(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(1/2),sinh(f*x+e))/f
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sinh \left (f x + e\right )^{2} + a} \operatorname{sech}\left (f x + e\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(b*sinh(f*x + e)^2 + a)*sech(f*x + e)^3, x)
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Fricas [B] time = 1.97148, size = 3534, normalized size = 41.09 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="fricas")
[Out]
[-1/4*((a*cosh(f*x + e)^4 + 4*a*cosh(f*x + e)*sinh(f*x + e)^3 + a*sinh(f*x + e)^4 + 2*a*cosh(f*x + e)^2 + 2*(3
*a*cosh(f*x + e)^2 + a)*sinh(f*x + e)^2 + 4*(a*cosh(f*x + e)^3 + a*cosh(f*x + e))*sinh(f*x + e) + a)*sqrt(-a +
b)*log(((a - 2*b)*cosh(f*x + e)^4 + 4*(a - 2*b)*cosh(f*x + e)*sinh(f*x + e)^3 + (a - 2*b)*sinh(f*x + e)^4 - 2
*(3*a - 2*b)*cosh(f*x + e)^2 + 2*(3*(a - 2*b)*cosh(f*x + e)^2 - 3*a + 2*b)*sinh(f*x + e)^2 - 2*sqrt(2)*(cosh(f
*x + e)^2 + 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 - 1)*sqrt(-a + b)*sqrt((b*cosh(f*x + e)^2 + b*sinh
(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)) + 4*((a - 2*b)*cos
h(f*x + e)^3 - (3*a - 2*b)*cosh(f*x + e))*sinh(f*x + e) + a - 2*b)/(cosh(f*x + e)^4 + 4*cosh(f*x + e)*sinh(f*x
+ e)^3 + sinh(f*x + e)^4 + 2*(3*cosh(f*x + e)^2 + 1)*sinh(f*x + e)^2 + 2*cosh(f*x + e)^2 + 4*(cosh(f*x + e)^3
+ cosh(f*x + e))*sinh(f*x + e) + 1)) - 2*sqrt(2)*((a - b)*cosh(f*x + e)^2 + 2*(a - b)*cosh(f*x + e)*sinh(f*x
+ e) + (a - b)*sinh(f*x + e)^2 - a + b)*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^
2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)))/((a - b)*f*cosh(f*x + e)^4 + 4*(a - b)*f*cosh(f*x + e)*
sinh(f*x + e)^3 + (a - b)*f*sinh(f*x + e)^4 + 2*(a - b)*f*cosh(f*x + e)^2 + 2*(3*(a - b)*f*cosh(f*x + e)^2 + (
a - b)*f)*sinh(f*x + e)^2 + (a - b)*f + 4*((a - b)*f*cosh(f*x + e)^3 + (a - b)*f*cosh(f*x + e))*sinh(f*x + e))
, 1/2*((a*cosh(f*x + e)^4 + 4*a*cosh(f*x + e)*sinh(f*x + e)^3 + a*sinh(f*x + e)^4 + 2*a*cosh(f*x + e)^2 + 2*(3
*a*cosh(f*x + e)^2 + a)*sinh(f*x + e)^2 + 4*(a*cosh(f*x + e)^3 + a*cosh(f*x + e))*sinh(f*x + e) + a)*sqrt(a -
b)*arctan(sqrt(2)*(cosh(f*x + e)^2 + 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 - 1)*sqrt(a - b)*sqrt((b*
cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e
)^2))/(b*cosh(f*x + e)^4 + 4*b*cosh(f*x + e)*sinh(f*x + e)^3 + b*sinh(f*x + e)^4 + 2*(2*a - b)*cosh(f*x + e)^2
+ 2*(3*b*cosh(f*x + e)^2 + 2*a - b)*sinh(f*x + e)^2 + 4*(b*cosh(f*x + e)^3 + (2*a - b)*cosh(f*x + e))*sinh(f*
x + e) + b)) + sqrt(2)*((a - b)*cosh(f*x + e)^2 + 2*(a - b)*cosh(f*x + e)*sinh(f*x + e) + (a - b)*sinh(f*x + e
)^2 - a + b)*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*
x + e) + sinh(f*x + e)^2)))/((a - b)*f*cosh(f*x + e)^4 + 4*(a - b)*f*cosh(f*x + e)*sinh(f*x + e)^3 + (a - b)*f
*sinh(f*x + e)^4 + 2*(a - b)*f*cosh(f*x + e)^2 + 2*(3*(a - b)*f*cosh(f*x + e)^2 + (a - b)*f)*sinh(f*x + e)^2 +
(a - b)*f + 4*((a - b)*f*cosh(f*x + e)^3 + (a - b)*f*cosh(f*x + e))*sinh(f*x + e))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \sinh ^{2}{\left (e + f x \right )}} \operatorname{sech}^{3}{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(f*x+e)**3*(a+b*sinh(f*x+e)**2)**(1/2),x)
[Out]
Integral(sqrt(a + b*sinh(e + f*x)**2)*sech(e + f*x)**3, x)
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Giac [B] time = 1.29754, size = 938, normalized size = 10.91 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="giac")
[Out]
a*arctan(-1/2*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) +
b) + sqrt(b))/sqrt(a - b))/(sqrt(a - b)*f) - 2*((sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f
*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^3*a - 2*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x
+ 2*e) - 2*b*e^(2*f*x + 2*e) + b))^3*b - 5*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x +
2*e) - 2*b*e^(2*f*x + 2*e) + b))^2*a*sqrt(b) + 2*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2
*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^2*b^(3/2) - 4*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*
e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))*a^2 - (sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2
*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))*a*b + 2*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f
*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))*b^2 - 4*a^2*sqrt(b) + 5*a*b^(3/2) - 2*b^(5/2))/(((sqrt(b)*e^(2*f*x + 2*e
) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^2 + 2*(sqrt(b)*e^(2*f*x + 2*e) -
sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))*sqrt(b) + 4*a - 3*b)^2*f)