3.2.1 \(\int \frac {\text {csch}^3(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx\) [101]

Optimal. Leaf size=89 \[ \frac {(a+b) \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{2 a^{3/2} f}-\frac {\sqrt {a-b+b \cosh ^2(e+f x)} \coth (e+f x) \text {csch}(e+f x)}{2 a f} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3265, 390, 385, 212} \begin {gather*} \frac {(a+b) \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a+b \cosh ^2(e+f x)-b}}\right )}{2 a^{3/2} f}-\frac {\coth (e+f x) \text {csch}(e+f x) \sqrt {a+b \cosh ^2(e+f x)-b}}{2 a f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.23, size = 102, normalized size = 1.15 \begin {gather*} \frac {2 (a+b) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \cosh (e+f x)}{\sqrt {2 a-b+b \cosh (2 (e+f x))}}\right )-\sqrt {2} \sqrt {a} \sqrt {2 a-b+b \cosh (2 (e+f x))} \coth (e+f x) \text {csch}(e+f x)}{4 a^{3/2} f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + b)*ArcTanh[(Sqrt[2]*Sqrt[a]*Cosh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]] - Sqrt[2]*Sqrt[a]*Sqrt
[2*a - b + b*Cosh[2*(e + f*x)]]*Coth[e + f*x]*Csch[e + f*x])/(4*a^(3/2)*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(233\) vs. \(2(77)=154\).
time = 2.14, size = 234, normalized size = 2.63

method	result	size
default	\(-\frac {\sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}\, \left (-\ln \left (\frac {\left (a +b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {b \left (\cosh ^{4}\left (f x +e \right )\right )+\left (a -b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )}+a -b}{\sinh \left (f x +e \right )^{2}}\right ) \left (\sinh ^{2}\left (f x +e \right )\right ) a^{2}-\ln \left (\frac {\left (a +b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {b \left (\cosh ^{4}\left (f x +e \right )\right )+\left (a -b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )}+a -b}{\sinh \left (f x +e \right )^{2}}\right ) b \left (\sinh ^{2}\left (f x +e \right )\right ) a +2 a^{\frac {3}{2}} \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}\right )}{4 \sinh \left (f x +e \right )^{2} a^{\frac {5}{2}} \cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f}\)	\(234\)
risch	\(\text {Expression too large to display}\)	\(567767\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*(-ln(((a+b)*cosh(f*x+e)^2+2*a^(1/2)*(b*cosh(f*x+e)^4+(a-b)*cosh
(f*x+e)^2)^(1/2)+a-b)/sinh(f*x+e)^2)*sinh(f*x+e)^2*a^2-ln(((a+b)*cosh(f*x+e)^2+2*a^(1/2)*(b*cosh(f*x+e)^4+(a-b
)*cosh(f*x+e)^2)^(1/2)+a-b)/sinh(f*x+e)^2)*b*sinh(f*x+e)^2*a+2*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*a^(3/
2))/sinh(f*x+e)^2/a^(5/2)/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)^3/(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 591 vs. \(2 (77) = 154\).
time = 0.47, size = 1285, normalized size = 14.44 \begin {gather*} \left [\frac {{\left ({\left (a + b\right )} \cosh \left (f x + e\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + {\left (a + b\right )} \sinh \left (f x + e\right )^{4} - 2 \, {\left (a + b\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (f x + e\right )^{2} - a - b\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (f x + e\right )^{3} - {\left (a + b\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + a + b\right )} \sqrt {a} \log \left (-\frac {{\left (a + b\right )} \cosh \left (f x + e\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + {\left (a + b\right )} \sinh \left (f x + e\right )^{4} + 2 \, {\left (3 \, a - b\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (f x + e\right )^{2} + 3 \, a - b\right )} \sinh \left (f x + e\right )^{2} + 2 \, \sqrt {2} {\left (\cosh \left (f x + e\right )^{2} + 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2} + 1\right )} \sqrt {a} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}} + 4 \, {\left ({\left (a + b\right )} \cosh \left (f x + e\right )^{3} + {\left (3 \, a - b\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + a + b}{\cosh \left (f x + e\right )^{4} + 4 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + \sinh \left (f x + e\right )^{4} + 2 \, {\left (3 \, \cosh \left (f x + e\right )^{2} - 1\right )} \sinh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right )^{2} + 4 \, {\left (\cosh \left (f x + e\right )^{3} - \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + 1}\right ) - 2 \, \sqrt {2} {\left (a \cosh \left (f x + e\right )^{2} + 2 \, a \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + a \sinh \left (f x + e\right )^{2} + a\right )} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}}}{4 \, {\left (a^{2} f \cosh \left (f x + e\right )^{4} + 4 \, a^{2} f \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + a^{2} f \sinh \left (f x + e\right )^{4} - 2 \, a^{2} f \cosh \left (f x + e\right )^{2} + a^{2} f + 2 \, {\left (3 \, a^{2} f \cosh \left (f x + e\right )^{2} - a^{2} f\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left (a^{2} f \cosh \left (f x + e\right )^{3} - a^{2} f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )\right )}}, -\frac {{\left ({\left (a + b\right )} \cosh \left (f x + e\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + {\left (a + b\right )} \sinh \left (f x + e\right )^{4} - 2 \, {\left (a + b\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (f x + e\right )^{2} - a - b\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (f x + e\right )^{3} - {\left (a + b\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + a + b\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {2} {\left (\cosh \left (f x + e\right )^{2} + 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2} + 1\right )} \sqrt {-a} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}}}{b \cosh \left (f x + e\right )^{4} + 4 \, b \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + b \sinh \left (f x + e\right )^{4} + 2 \, {\left (2 \, a - b\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, b \cosh \left (f x + e\right )^{2} + 2 \, a - b\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left (b \cosh \left (f x + e\right )^{3} + {\left (2 \, a - b\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + b}\right ) + \sqrt {2} {\left (a \cosh \left (f x + e\right )^{2} + 2 \, a \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + a \sinh \left (f x + e\right )^{2} + a\right )} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}}}{2 \, {\left (a^{2} f \cosh \left (f x + e\right )^{4} + 4 \, a^{2} f \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + a^{2} f \sinh \left (f x + e\right )^{4} - 2 \, a^{2} f \cosh \left (f x + e\right )^{2} + a^{2} f + 2 \, {\left (3 \, a^{2} f \cosh \left (f x + e\right )^{2} - a^{2} f\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left (a^{2} f \cosh \left (f x + e\right )^{3} - a^{2} f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="fricas")

[Out]

[1/4*(((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*(a + b)
*cosh(f*x + e)^2 + 2*(3*(a + b)*cosh(f*x + e)^2 - a - b)*sinh(f*x + e)^2 + 4*((a + b)*cosh(f*x + e)^3 - (a + b
)*cosh(f*x + e))*sinh(f*x + e) + a + b)*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)*si
nh(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*cosh(f*x + e)^2 + 2*a*cosh(f*x + e)
*sinh(f*x + e) + a*sinh(f*x + e)^2 + 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)))/(a^2*f*cosh(f*x + e)^4 + 4*a^2*f*cosh(f*x + e)*sinh(f*x
+ e)^3 + a^2*f*sinh(f*x + e)^4 - 2*a^2*f*cosh(f*x + e)^2 + a^2*f + 2*(3*a^2*f*cosh(f*x + e)^2 - a^2*f)*sinh(f
*x + e)^2 + 4*(a^2*f*cosh(f*x + e)^3 - a^2*f*cosh(f*x + e))*sinh(f*x + e)), -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*(a + b)*cosh(f*x + e)^2 + 2*(3*(a + b)*co
sh(f*x + e)^2 - a - b)*sinh(f*x + e)^2 + 4*((a + b)*cosh(f*x + e)^3 - (a + b)*cosh(f*x + e))*sinh(f*x + e) + a
+ b)*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) + sin
h(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*cosh(f*x + e)^2 + 2*a*cosh(f*x + e)*sinh(f*x + e) + a*sinh(f*x + e)^2 + 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)))/(a^2*f*cosh(f*x + e)^4 + 4*a^2*f*cosh(f*x + e)*sinh(f*x + e)^3 + a^2*f*sinh(f*x + e)^4 - 2*a^2*
f*cosh(f*x + e)^2 + a^2*f + 2*(3*a^2*f*cosh(f*x + e)^2 - a^2*f)*sinh(f*x + e)^2 + 4*(a^2*f*cosh(f*x + e)^3 - a
^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}^{3}{\left (e + f x \right )}}{\sqrt {a + b \sinh ^{2}{\left (e + f x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)**3/(a+b*sinh(f*x+e)**2)**(1/2),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 669 vs. \(2 (77) = 154\).
time = 0.56, size = 669, normalized size = 7.52 \begin {gather*} -\frac {{\left (\frac {{\left (a + b\right )} \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} - \frac {2 \, {\left ({\left (\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}\right )}^{3} a + {\left (\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}\right )}^{3} b + 5 \, {\left (\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}\right )}^{2} a \sqrt {b} - 3 \, {\left (\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}\right )}^{2} b^{\frac {3}{2}} + 4 \, {\left (\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}\right )} a^{2} - 9 \, {\left (\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}\right )} a b + 3 \, {\left (\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}\right )} b^{2} - 4 \, a^{2} \sqrt {b} + 3 \, a b^{\frac {3}{2}} - b^{\frac {5}{2}}\right )} e^{\left (-4 \, e\right )}}{{\left ({\left (\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}\right )}^{2} - 2 \, {\left (\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}\right )} \sqrt {b} - 4 \, a + b\right )}^{2} 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)^3/(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="giac")

[Out]

-((a + b)*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))*e^(-4*e)/(sqrt(-a)*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*a + (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) - 3*(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 - 9*(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 + 3*(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) + 3*a*b^(3/2) - b^(5/2))*e^(-4*e)/(((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 + b)^
2*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 )}^3\,\sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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