3.4.84 \(\int \frac {\text {sech}(e+f x)}{(a+b \sinh ^2(e+f x))^{3/2}} \, dx\) [384]

Optimal. Leaf size=85 \[ \frac {\text {ArcTan}\left (\frac {\sqrt {a-b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{(a-b)^{3/2} f}-\frac {b \sinh (e+f x)}{a (a-b) f \sqrt {a+b \sinh ^2(e+f x)}} \]

[Out]

arctan(sinh(f*x+e)*(a-b)^(1/2)/(a+b*sinh(f*x+e)^2)^(1/2))/(a-b)^(3/2)/f-b*sinh(f*x+e)/a/(a-b)/f/(a+b*sinh(f*x+
e)^2)^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3269, 390, 385, 209} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {a-b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{f (a-b)^{3/2}}-\frac {b \sinh (e+f x)}{a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sech[e + f*x]/(a + b*Sinh[e + f*x]^2)^(3/2),x]

[Out]

ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]]/((a - b)^(3/2)*f) - (b*Sinh[e + f*x])/(a*(a -
b)*f*Sqrt[a + b*Sinh[e + f*x]^2])

Rule 209

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

Rule 385

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 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a
*d)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && Eq
Q[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rule 3269

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 \frac {\text {sech}(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=-\frac {b \sinh (e+f x)}{a (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{(a-b) f}\\ &=-\frac {b \sinh (e+f x)}{a (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\text {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 )}{(a-b) f}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {a-b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{(a-b)^{3/2} f}-\frac {b \sinh (e+f x)}{a (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 7.51, size = 315, normalized size = 3.71 \begin {gather*} \frac {\text {sech}^7(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x) \left (4 (a-b)^2 \, _2F_1\left (2,2;\frac {7}{2};\frac {(a-b) \tanh ^2(e+f x)}{a}\right ) \sinh ^4(e+f x) \left (a+b \sinh ^2(e+f x)\right ) \sqrt {\frac {\text {sech}^2(e+f x) \left (a^2-b^2 \sinh ^2(e+f x)+a b \left (-1+\sinh ^2(e+f x)\right )\right ) \tanh ^2(e+f x)}{a^2}}+15 a \cosh ^2(e+f x) \left (3 a+2 b \sinh ^2(e+f x)\right ) \left (-\text {ArcSin}\left (\sqrt {\frac {(a-b) \tanh ^2(e+f x)}{a}}\right ) \left (a+b \sinh ^2(e+f x)\right )+a \cosh ^2(e+f x) \sqrt {\frac {\text {sech}^2(e+f x) \left (a^2-b^2 \sinh ^2(e+f x)+a b \left (-1+\sinh ^2(e+f x)\right )\right ) \tanh ^2(e+f x)}{a^2}}\right )\right )}{15 a^5 f \left (\frac {(a-b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right ) \tanh ^2(e+f x)}{a^2}\right )^{3/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Sech[e + f*x]/(a + b*Sinh[e + f*x]^2)^(3/2),x]

[Out]

(Sech[e + f*x]^7*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x]*(4*(a - b)^2*Hypergeometric2F1[2, 2, 7/2, ((a - b)*
Tanh[e + f*x]^2)/a]*Sinh[e + f*x]^4*(a + b*Sinh[e + f*x]^2)*Sqrt[(Sech[e + f*x]^2*(a^2 - b^2*Sinh[e + f*x]^2 +
a*b*(-1 + Sinh[e + f*x]^2))*Tanh[e + f*x]^2)/a^2] + 15*a*Cosh[e + f*x]^2*(3*a + 2*b*Sinh[e + f*x]^2)*(-(ArcSi
n[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*(a + b*Sinh[e + f*x]^2)) + a*Cosh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a^2 -
b^2*Sinh[e + f*x]^2 + a*b*(-1 + Sinh[e + f*x]^2))*Tanh[e + f*x]^2)/a^2])))/(15*a^5*f*(((a - b)*Sech[e + f*x]^
2*(a + b*Sinh[e + f*x]^2)*Tanh[e + f*x]^2)/a^2)^(3/2))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.96, size = 101, normalized size = 1.19

method	result	size
default	\(\frac {\mathit {`\,int/indef0`\,}\left (\frac {-b \left (\sinh ^{2}\left (f x +e \right )\right )-a}{\left (-b^{2} \left (\sinh ^{6}\left (f x +e \right )\right )+\left (-2 a b -b^{2}\right ) \left (\sinh ^{4}\left (f x +e \right )\right )+\left (-a^{2}-2 a b \right ) \left (\sinh ^{2}\left (f x +e \right )\right )-a^{2}\right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f}\)	\(101\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(f*x+e)/(a+b*sinh(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

`int/indef0`((-b*sinh(f*x+e)^2-a)/(-b^2*sinh(f*x+e)^6+(-2*a*b-b^2)*sinh(f*x+e)^4+(-a^2-2*a*b)*sinh(f*x+e)^2-a^
2)/(a+b*sinh(f*x+e)^2)^(1/2),sinh(f*x+e))/f

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(f*x+e)/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate(sech(f*x + e)/(b*sinh(f*x + e)^2 + a)^(3/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 800 vs. \(2 (77) = 154\).
time = 0.46, size = 1717, normalized size = 20.20 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(f*x+e)/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

[1/2*((a*b*cosh(f*x + e)^4 + 4*a*b*cosh(f*x + e)*sinh(f*x + e)^3 + a*b*sinh(f*x + e)^4 + 2*(2*a^2 - a*b)*cosh(
f*x + e)^2 + 2*(3*a*b*cosh(f*x + e)^2 + 2*a^2 - a*b)*sinh(f*x + e)^2 + a*b + 4*(a*b*cosh(f*x + e)^3 + (2*a^2 -
a*b)*cosh(f*x + e))*sinh(f*x + e))*sqrt(-a + b)*log(((a - 2*b)*cosh(f*x + e)^4 + 4*(a - 2*b)*cosh(f*x + e)*si
nh(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)*cosh(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 - b^2)*co
sh(f*x + e)^2 + 2*(a*b - b^2)*cosh(f*x + e)*sinh(f*x + e) + (a*b - b^2)*sinh(f*x + e)^2 - a*b + b^2)*sqrt((b*c
osh(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^3*b - 2*a^2*b^2 + a*b^3)*f*cosh(f*x + e)^4 + 4*(a^3*b - 2*a^2*b^2 + a*b^3)*f*cosh(f*x + e)*sinh(f*x
+ e)^3 + (a^3*b - 2*a^2*b^2 + a*b^3)*f*sinh(f*x + e)^4 + 2*(2*a^4 - 5*a^3*b + 4*a^2*b^2 - a*b^3)*f*cosh(f*x +
e)^2 + 2*(3*(a^3*b - 2*a^2*b^2 + a*b^3)*f*cosh(f*x + e)^2 + (2*a^4 - 5*a^3*b + 4*a^2*b^2 - a*b^3)*f)*sinh(f*x
+ e)^2 + (a^3*b - 2*a^2*b^2 + a*b^3)*f + 4*((a^3*b - 2*a^2*b^2 + a*b^3)*f*cosh(f*x + e)^3 + (2*a^4 - 5*a^3*b +
4*a^2*b^2 - a*b^3)*f*cosh(f*x + e))*sinh(f*x + e)), ((a*b*cosh(f*x + e)^4 + 4*a*b*cosh(f*x + e)*sinh(f*x + e)
^3 + a*b*sinh(f*x + e)^4 + 2*(2*a^2 - a*b)*cosh(f*x + e)^2 + 2*(3*a*b*cosh(f*x + e)^2 + 2*a^2 - a*b)*sinh(f*x
+ e)^2 + a*b + 4*(a*b*cosh(f*x + e)^3 + (2*a^2 - a*b)*cosh(f*x + e))*sinh(f*x + e))*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)) - sq
rt(2)*((a*b - b^2)*cosh(f*x + e)^2 + 2*(a*b - b^2)*cosh(f*x + e)*sinh(f*x + e) + (a*b - b^2)*sinh(f*x + e)^2 -
a*b + b^2)*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^3*b - 2*a^2*b^2 + a*b^3)*f*cosh(f*x + e)^4 + 4*(a^3*b - 2*a^2*b^2 + a*b^3)*f*co
sh(f*x + e)*sinh(f*x + e)^3 + (a^3*b - 2*a^2*b^2 + a*b^3)*f*sinh(f*x + e)^4 + 2*(2*a^4 - 5*a^3*b + 4*a^2*b^2 -
a*b^3)*f*cosh(f*x + e)^2 + 2*(3*(a^3*b - 2*a^2*b^2 + a*b^3)*f*cosh(f*x + e)^2 + (2*a^4 - 5*a^3*b + 4*a^2*b^2
- a*b^3)*f)*sinh(f*x + e)^2 + (a^3*b - 2*a^2*b^2 + a*b^3)*f + 4*((a^3*b - 2*a^2*b^2 + a*b^3)*f*cosh(f*x + e)^3
+ (2*a^4 - 5*a^3*b + 4*a^2*b^2 - a*b^3)*f*cosh(f*x + e))*sinh(f*x + e))]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {sech}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(f*x+e)/(a+b*sinh(f*x+e)**2)**(3/2),x)

[Out]

Integral(sech(e + f*x)/(a + b*sinh(e + f*x)**2)**(3/2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (77) = 154\).
time = 0.54, size = 294, normalized size = 3.46 \begin {gather*} -\frac {{\left (\frac {\frac {{\left (a^{2} b e^{\left (4 \, e\right )} - 2 \, a b^{2} e^{\left (4 \, e\right )} + b^{3} e^{\left (4 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a^{4} e^{\left (6 \, e\right )} - 3 \, a^{3} b e^{\left (6 \, e\right )} + 3 \, a^{2} b^{2} e^{\left (6 \, e\right )} - a b^{3} e^{\left (6 \, e\right )}} - \frac {a^{2} b e^{\left (2 \, e\right )} - 2 \, a b^{2} e^{\left (2 \, e\right )} + b^{3} e^{\left (2 \, e\right )}}{a^{4} e^{\left (6 \, e\right )} - 3 \, a^{3} b e^{\left (6 \, e\right )} + 3 \, a^{2} b^{2} e^{\left (6 \, e\right )} - a b^{3} e^{\left (6 \, e\right )}}}{\sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}} - \frac {2 \, \arctan \left (-\frac {\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b} + \sqrt {b}}{2 \, \sqrt {a - b}}\right )}{{\left (a e^{\left (4 \, e\right )} - b e^{\left (4 \, e\right )}\right )} \sqrt {a - b}}\right )} e^{\left (4 \, e\right )}}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(f*x+e)/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

-(((a^2*b*e^(4*e) - 2*a*b^2*e^(4*e) + b^3*e^(4*e))*e^(2*f*x)/(a^4*e^(6*e) - 3*a^3*b*e^(6*e) + 3*a^2*b^2*e^(6*e
) - a*b^3*e^(6*e)) - (a^2*b*e^(2*e) - 2*a*b^2*e^(2*e) + b^3*e^(2*e))/(a^4*e^(6*e) - 3*a^3*b*e^(6*e) + 3*a^2*b^
2*e^(6*e) - a*b^3*e^(6*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*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) + sqr
t(b))/sqrt(a - b))/((a*e^(4*e) - b*e^(4*e))*sqrt(a - b)))*e^(4*e)/f

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\mathrm {cosh}\left (e+f\,x\right )\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(e + f*x)*(a + b*sinh(e + f*x)^2)^(3/2)),x)

[Out]

int(1/(cosh(e + f*x)*(a + b*sinh(e + f*x)^2)^(3/2)), x)

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