3.1.69 \(\int \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx\) [69]

Optimal. Leaf size=88 \[ \frac {(a-b) \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{2 \sqrt {a} f}-\frac {\sqrt {a-b+b \cosh ^2(e+f x)} \coth (e+f x) \text {csch}(e+f x)}{2 f} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 88, 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, 386, 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 \sqrt {a} f}-\frac {\coth (e+f x) \text {csch}(e+f x) \sqrt {a+b \cosh ^2(e+f x)-b}}{2 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*Sqrt[a]*f) - (Sqrt[a - b + b*Cos
h[e + f*x]^2]*Coth[e + f*x]*Csch[e + f*x])/(2*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 386

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

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Mathematica [A]
time = 0.21, size = 104, normalized size = 1.18 \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 \sqrt {a} 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*Sqrt[a]*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(229\) vs. \(2(76)=152\).
time = 1.11, size = 230, normalized size = 2.61

method	result	size
default	\(-\frac {\sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}\, \left (-a \ln \left (\frac {\left (a +b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {b \left (\cosh ^{4}\left (f x +e \right )\right )+\left (a -b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )}+a -b}{\sinh \left (f x +e \right )^{2}}\right ) \left (\sinh ^{2}\left (f x +e \right )\right )+b \ln \left (\frac {\left (a +b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {b \left (\cosh ^{4}\left (f x +e \right )\right )+\left (a -b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )}+a -b}{\sinh \left (f x +e \right )^{2}}\right ) \left (\sinh ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}\right )}{4 \sqrt {a}\, \sinh \left (f x +e \right )^{2} \cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f}\)	\(230\)

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 587 vs. \(2 (76) = 152\).
time = 0.45, size = 1277, normalized size = 14.51 \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 f \cosh \left (f x + e\right )^{4} + 4 \, a f \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + a f \sinh \left (f x + e\right )^{4} - 2 \, a f \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, a f \cosh \left (f x + e\right )^{2} - a f\right )} \sinh \left (f x + e\right )^{2} + a f + 4 \, {\left (a f \cosh \left (f x + e\right )^{3} - a 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 f \cosh \left (f x + e\right )^{4} + 4 \, a f \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + a f \sinh \left (f x + e\right )^{4} - 2 \, a f \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, a f \cosh \left (f x + e\right )^{2} - a f\right )} \sinh \left (f x + e\right )^{2} + a f + 4 \, {\left (a f \cosh \left (f x + e\right )^{3} - a 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)*s
inh(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)*sqr
t((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*f*cosh(f*x + e)^4 + 4*a*f*cosh(f*x + e)*sinh(f*x +
e)^3 + a*f*sinh(f*x + e)^4 - 2*a*f*cosh(f*x + e)^2 + 2*(3*a*f*cosh(f*x + e)^2 - a*f)*sinh(f*x + e)^2 + a*f + 4
*(a*f*cosh(f*x + e)^3 - a*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)*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)*ar
ctan(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*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
*f*cosh(f*x + e)^4 + 4*a*f*cosh(f*x + e)*sinh(f*x + e)^3 + a*f*sinh(f*x + e)^4 - 2*a*f*cosh(f*x + e)^2 + 2*(3*
a*f*cosh(f*x + e)^2 - a*f)*sinh(f*x + e)^2 + a*f + 4*(a*f*cosh(f*x + e)^3 - a*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 \sqrt {a + b \sinh ^{2}{\left (e + f x \right )}} \operatorname {csch}^{3}{\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(sqrt(a + b*sinh(e + f*x)**2)*csch(e + f*x)**3, x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Error: Bad Argument Type

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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