3.2.9 \(\int \frac {\text {csch}(e+f x)}{(a+b \sinh ^2(e+f x))^{3/2}} \, dx\) [109]

Optimal. Leaf size=84 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{a^{3/2} f}-\frac {b \cosh (e+f x)}{a (a-b) f \sqrt {a-b+b \cosh ^2(e+f x)}} \]

[Out]

-arctanh(cosh(f*x+e)*a^(1/2)/(a-b+b*cosh(f*x+e)^2)^(1/2))/a^(3/2)/f-b*cosh(f*x+e)/a/(a-b)/f/(a-b+b*cosh(f*x+e)
^2)^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 84, 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 = {3265, 390, 385, 212} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a+b \cosh ^2(e+f x)-b}}\right )}{a^{3/2} f}-\frac {b \cosh (e+f x)}{a f (a-b) \sqrt {a+b \cosh ^2(e+f x)-b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[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 3265

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

Rubi steps

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

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Mathematica [A]
time = 0.27, size = 98, normalized size = 1.17 \begin {gather*} \frac {-\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \cosh (e+f x)}{\sqrt {2 a-b+b \cosh (2 (e+f x))}}\right )-\frac {\sqrt {2} \sqrt {a} b \cosh (e+f x)}{(a-b) \sqrt {2 a-b+b \cosh (2 (e+f x))}}}{a^{3/2} f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(153\) vs. \(2(76)=152\).
time = 1.23, size = 154, normalized size = 1.83

method	result	size
default	\(\frac {\sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}\, \left (-\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a \left (a -b \right ) \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}}-\frac {\ln \left (\frac {2 a +\left (a +b \right ) \left (\sinh ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}}{\sinh \left (f x +e \right )^{2}}\right )}{2 a^{\frac {3}{2}}}\right )}{\cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f}\)	\(154\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*(-1/a*b*cosh(f*x+e)^2/(a-b)/((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2
)-1/2/a^(3/2)*ln((2*a+(a+b)*sinh(f*x+e)^2+2*a^(1/2)*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2))/sinh(f*x+e)^2))
/cosh(f*x+e)/(a+b*sinh(f*x+e)^2)^(1/2)/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(csch(f*x+e)/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate(csch(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. 769 vs. \(2 (76) = 152\).
time = 0.48, size = 1641, normalized size = 19.54 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(((a*b - b^2)*cosh(f*x + e)^4 + 4*(a*b - b^2)*cosh(f*x + e)*sinh(f*x + e)^3 + (a*b - b^2)*sinh(f*x + e)^4
+ 2*(2*a^2 - 3*a*b + b^2)*cosh(f*x + e)^2 + 2*(3*(a*b - b^2)*cosh(f*x + e)^2 + 2*a^2 - 3*a*b + b^2)*sinh(f*x
+ e)^2 + a*b - b^2 + 4*((a*b - b^2)*cosh(f*x + e)^3 + (2*a^2 - 3*a*b + b^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt
(a)*log(-((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*(3*a
- b)*cosh(f*x + e)^2 + 2*(3*(a + b)*cosh(f*x + e)^2 + 3*a - 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)*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 + b)*cosh(f*x + e)^3 + (3*
a - b)*cosh(f*x + e))*sinh(f*x + e) + a + 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^3*b - a^2*b^2)*f*cosh(f*x + e)^4 + 4*(a^3*b - a^2*b^2)*f*cosh(f*x + e)*sinh(f*x + e)^3
+ (a^3*b - a^2*b^2)*f*sinh(f*x + e)^4 + 2*(2*a^4 - 3*a^3*b + a^2*b^2)*f*cosh(f*x + e)^2 + 2*(3*(a^3*b - a^2*b^
2)*f*cosh(f*x + e)^2 + (2*a^4 - 3*a^3*b + a^2*b^2)*f)*sinh(f*x + e)^2 + (a^3*b - a^2*b^2)*f + 4*((a^3*b - a^2*
b^2)*f*cosh(f*x + e)^3 + (2*a^4 - 3*a^3*b + a^2*b^2)*f*cosh(f*x + e))*sinh(f*x + e)), (((a*b - b^2)*cosh(f*x +
e)^4 + 4*(a*b - b^2)*cosh(f*x + e)*sinh(f*x + e)^3 + (a*b - b^2)*sinh(f*x + e)^4 + 2*(2*a^2 - 3*a*b + b^2)*co
sh(f*x + e)^2 + 2*(3*(a*b - b^2)*cosh(f*x + e)^2 + 2*a^2 - 3*a*b + b^2)*sinh(f*x + e)^2 + a*b - b^2 + 4*((a*b
- b^2)*cosh(f*x + e)^3 + (2*a^2 - 3*a*b + b^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt(-a)*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)*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*co
sh(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*sin
h(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 - a^2*b^
2)*f*cosh(f*x + e)^4 + 4*(a^3*b - a^2*b^2)*f*cosh(f*x + e)*sinh(f*x + e)^3 + (a^3*b - a^2*b^2)*f*sinh(f*x + e)
^4 + 2*(2*a^4 - 3*a^3*b + a^2*b^2)*f*cosh(f*x + e)^2 + 2*(3*(a^3*b - a^2*b^2)*f*cosh(f*x + e)^2 + (2*a^4 - 3*a
^3*b + a^2*b^2)*f)*sinh(f*x + e)^2 + (a^3*b - a^2*b^2)*f + 4*((a^3*b - a^2*b^2)*f*cosh(f*x + e)^3 + (2*a^4 - 3
*a^3*b + a^2*b^2)*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 {csch}{\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(csch(f*x+e)/(a+b*sinh(f*x+e)**2)**(3/2),x)

[Out]

Integral(csch(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. 198 vs. \(2 (76) = 152\).
time = 0.57, size = 198, normalized size = 2.36 \begin {gather*} -\frac {{\left (\frac {\frac {a^{2} b e^{\left (2 \, f x + 4 \, e\right )}}{a^{4} e^{\left (6 \, e\right )} - a^{3} b e^{\left (6 \, e\right )}} + \frac {a^{2} b e^{\left (2 \, e\right )}}{a^{4} e^{\left (6 \, e\right )} - a^{3} b 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}}\right ) e^{\left (-4 \, e\right )}}{\sqrt {-a} a}\right )} e^{\left (4 \, e\right )}}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((a^2*b*e^(2*f*x + 4*e)/(a^4*e^(6*e) - a^3*b*e^(6*e)) + a^2*b*e^(2*e)/(a^4*e^(6*e) - a^3*b*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) - sqr
t(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))*e^(-4*e)/(sqrt(-a)*a
))*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 {sinh}\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/(sinh(e + f*x)*(a + b*sinh(e + f*x)^2)^(3/2)),x)

[Out]

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

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