3.1.70 \(\int \text {csch}^5(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx\) [70]
Optimal. Leaf size=144 \[ -\frac {(a-b) (3 a+b) \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{8 a^{3/2} f}+\frac {(3 a+b) \sqrt {a-b+b \cosh ^2(e+f x)} \coth (e+f x) \text {csch}(e+f x)}{8 a f}-\frac {\left (a-b+b \cosh ^2(e+f x)\right )^{3/2} \coth (e+f x) \text {csch}^3(e+f x)}{4 a f} \]
[Out]
-1/8*(a-b)*(3*a+b)*arctanh(cosh(f*x+e)*a^(1/2)/(a-b+b*cosh(f*x+e)^2)^(1/2))/a^(3/2)/f-1/4*(a-b+b*cosh(f*x+e)^2
)^(3/2)*coth(f*x+e)*csch(f*x+e)^3/a/f+1/8*(3*a+b)*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 = 0.10, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3265, 390, 386,
385, 212} \begin {gather*} -\frac {(a-b) (3 a+b) \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a+b \cosh ^2(e+f x)-b}}\right )}{8 a^{3/2} f}-\frac {\coth (e+f x) \text {csch}^3(e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}}{4 a f}+\frac {(3 a+b) \coth (e+f x) \text {csch}(e+f x) \sqrt {a+b \cosh ^2(e+f x)-b}}{8 a f} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Csch[e + f*x]^5*Sqrt[a + b*Sinh[e + f*x]^2],x]
[Out]
-1/8*((a - b)*(3*a + b)*ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(a^(3/2)*f) + ((3*a
+ b)*Sqrt[a - b + b*Cosh[e + f*x]^2]*Coth[e + f*x]*Csch[e + f*x])/(8*a*f) - ((a - b + b*Cosh[e + f*x]^2)^(3/2)
*Coth[e + f*x]*Csch[e + f*x]^3)/(4*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 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 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 \text {csch}^5(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 )^3} \, dx,x,\cosh (e+f x)\right )}{f}\\ &=-\frac {\left (a-b+b \cosh ^2(e+f x)\right )^{3/2} \coth (e+f x) \text {csch}^3(e+f x)}{4 a f}-\frac {(3 a+b) \text {Subst}\left (\int \frac {\sqrt {a-b+b x^2}}{\left (1-x^2\right )^2} \, dx,x,\cosh (e+f x)\right )}{4 a f}\\ &=\frac {(3 a+b) \sqrt {a-b+b \cosh ^2(e+f x)} \coth (e+f x) \text {csch}(e+f x)}{8 a f}-\frac {\left (a-b+b \cosh ^2(e+f x)\right )^{3/2} \coth (e+f x) \text {csch}^3(e+f x)}{4 a f}-\frac {((a-b) (3 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 )}{8 a f}\\ &=\frac {(3 a+b) \sqrt {a-b+b \cosh ^2(e+f x)} \coth (e+f x) \text {csch}(e+f x)}{8 a f}-\frac {\left (a-b+b \cosh ^2(e+f x)\right )^{3/2} \coth (e+f x) \text {csch}^3(e+f x)}{4 a f}-\frac {((a-b) (3 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 )}{8 a f}\\ &=-\frac {(a-b) (3 a+b) \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{8 a^{3/2} f}+\frac {(3 a+b) \sqrt {a-b+b \cosh ^2(e+f x)} \coth (e+f x) \text {csch}(e+f x)}{8 a f}-\frac {\left (a-b+b \cosh ^2(e+f x)\right )^{3/2} \coth (e+f x) \text {csch}^3(e+f x)}{4 a f}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 129, normalized size = 0.90 \begin {gather*} \frac {\left (-6 a^2+4 a b+2 b^2\right ) \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) \left (-3 a+b+2 a \text {csch}^2(e+f x)\right )}{16 a^{3/2} f} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Csch[e + f*x]^5*Sqrt[a + b*Sinh[e + f*x]^2],x]
[Out]
((-6*a^2 + 4*a*b + 2*b^2)*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]*(-3*a + b + 2*a*Csch[e + f*x]^2))/(
16*a^(3/2)*f)
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(380\) vs.
\(2(128)=256\).
time = 103.82, size = 381, normalized size = 2.65
| | |
| method |
result |
size |
| | |
| default |
\(\frac {\sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}\, \left (6 \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}\, \left (\sinh ^{2}\left (f x +e \right )\right ) a^{\frac {5}{2}}-3 a^{3} \ln \left (\frac {\left (a +b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {b \left (\cosh ^{4}\left (f x +e \right )\right )+\left (a -b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )}+a -b}{\sinh \left (f x +e \right )^{2}}\right ) \left (\sinh ^{4}\left (f x +e \right )\right )+2 b \ln \left (\frac {\left (a +b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {b \left (\cosh ^{4}\left (f x +e \right )\right )+\left (a -b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )}+a -b}{\sinh \left (f x +e \right )^{2}}\right ) \left (\sinh ^{4}\left (f x +e \right )\right ) a^{2}+\ln \left (\frac {\left (a +b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {b \left (\cosh ^{4}\left (f x +e \right )\right )+\left (a -b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )}+a -b}{\sinh \left (f x +e \right )^{2}}\right ) b^{2} \left (\sinh ^{4}\left (f x +e \right )\right ) a -2 b \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}\, \left (\sinh ^{2}\left (f x +e \right )\right ) a^{\frac {3}{2}}-4 \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}\, a^{\frac {5}{2}}\right )}{16 \sinh \left (f x +e \right )^{4} a^{\frac {5}{2}} \cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f}\) |
\(381\) |
| risch |
\(\text {Expression too large to display}\) |
\(190014651\) |
| | |
|
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|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(f*x+e)^5*(a+b*sinh(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)
[Out]
1/16*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*(6*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*sinh(f*x+e)^2*a^(5
/2)-3*a^3*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)*si
nh(f*x+e)^4+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)^4*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^2*sinh(f*x+e)^4*a-2*b*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*sinh(f*x+e)^2*a^(3/2)-4*((a+b*sinh
(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*a^(5/2))/sinh(f*x+e)^4/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)^5*(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)^5, x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1646 vs.
\(2 (128) = 256\).
time = 0.61, size = 3395, normalized size = 23.58 \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)^5*(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="fricas")
[Out]
[-1/16*(((3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^8 + 8*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)*sinh(f*x + e)^7 + (3*a^
2 - 2*a*b - b^2)*sinh(f*x + e)^8 - 4*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^6 + 4*(7*(3*a^2 - 2*a*b - b^2)*cosh(f
*x + e)^2 - 3*a^2 + 2*a*b + b^2)*sinh(f*x + e)^6 + 8*(7*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^3 - 3*(3*a^2 - 2*a
*b - b^2)*cosh(f*x + e))*sinh(f*x + e)^5 + 6*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^4 + 2*(35*(3*a^2 - 2*a*b - b^
2)*cosh(f*x + e)^4 - 30*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^2 + 9*a^2 - 6*a*b - 3*b^2)*sinh(f*x + e)^4 + 8*(7*
(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^5 - 10*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^3 + 3*(3*a^2 - 2*a*b - b^2)*cos
h(f*x + e))*sinh(f*x + e)^3 - 4*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^2 + 4*(7*(3*a^2 - 2*a*b - b^2)*cosh(f*x +
e)^6 - 15*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^4 + 9*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^2 - 3*a^2 + 2*a*b + b^
2)*sinh(f*x + e)^2 + 3*a^2 - 2*a*b - b^2 + 8*((3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^7 - 3*(3*a^2 - 2*a*b - b^2)*
cosh(f*x + e)^5 + 3*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^3 - (3*a^2 - 2*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)*((3*a^2 - a*b)*cosh(f*x + e)^6 + 6*(3*a^2 - a*b)*cosh(f*x + e)*sinh(f*x + e)
^5 + (3*a^2 - a*b)*sinh(f*x + e)^6 - (11*a^2 - a*b)*cosh(f*x + e)^4 + (15*(3*a^2 - a*b)*cosh(f*x + e)^2 - 11*a
^2 + a*b)*sinh(f*x + e)^4 + 4*(5*(3*a^2 - a*b)*cosh(f*x + e)^3 - (11*a^2 - a*b)*cosh(f*x + e))*sinh(f*x + e)^3
- (11*a^2 - a*b)*cosh(f*x + e)^2 + (15*(3*a^2 - a*b)*cosh(f*x + e)^4 - 6*(11*a^2 - a*b)*cosh(f*x + e)^2 - 11*
a^2 + a*b)*sinh(f*x + e)^2 + 3*a^2 - a*b + 2*(3*(3*a^2 - a*b)*cosh(f*x + e)^5 - 2*(11*a^2 - a*b)*cosh(f*x + e)
^3 - (11*a^2 - a*b)*cosh(f*x + e))*sinh(f*x + e))*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)^8 + 8*a^2*f*cosh(f*x + e
)*sinh(f*x + e)^7 + a^2*f*sinh(f*x + e)^8 - 4*a^2*f*cosh(f*x + e)^6 + 6*a^2*f*cosh(f*x + e)^4 + 4*(7*a^2*f*cos
h(f*x + e)^2 - a^2*f)*sinh(f*x + e)^6 + 8*(7*a^2*f*cosh(f*x + e)^3 - 3*a^2*f*cosh(f*x + e))*sinh(f*x + e)^5 -
4*a^2*f*cosh(f*x + e)^2 + 2*(35*a^2*f*cosh(f*x + e)^4 - 30*a^2*f*cosh(f*x + e)^2 + 3*a^2*f)*sinh(f*x + e)^4 +
8*(7*a^2*f*cosh(f*x + e)^5 - 10*a^2*f*cosh(f*x + e)^3 + 3*a^2*f*cosh(f*x + e))*sinh(f*x + e)^3 + a^2*f + 4*(7*
a^2*f*cosh(f*x + e)^6 - 15*a^2*f*cosh(f*x + e)^4 + 9*a^2*f*cosh(f*x + e)^2 - a^2*f)*sinh(f*x + e)^2 + 8*(a^2*f
*cosh(f*x + e)^7 - 3*a^2*f*cosh(f*x + e)^5 + 3*a^2*f*cosh(f*x + e)^3 - a^2*f*cosh(f*x + e))*sinh(f*x + e)), 1/
8*(((3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^8 + 8*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)*sinh(f*x + e)^7 + (3*a^2 - 2
*a*b - b^2)*sinh(f*x + e)^8 - 4*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^6 + 4*(7*(3*a^2 - 2*a*b - b^2)*cosh(f*x +
e)^2 - 3*a^2 + 2*a*b + b^2)*sinh(f*x + e)^6 + 8*(7*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^3 - 3*(3*a^2 - 2*a*b -
b^2)*cosh(f*x + e))*sinh(f*x + e)^5 + 6*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^4 + 2*(35*(3*a^2 - 2*a*b - b^2)*co
sh(f*x + e)^4 - 30*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^2 + 9*a^2 - 6*a*b - 3*b^2)*sinh(f*x + e)^4 + 8*(7*(3*a^
2 - 2*a*b - b^2)*cosh(f*x + e)^5 - 10*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^3 + 3*(3*a^2 - 2*a*b - b^2)*cosh(f*x
+ e))*sinh(f*x + e)^3 - 4*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^2 + 4*(7*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^6
- 15*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^4 + 9*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^2 - 3*a^2 + 2*a*b + b^2)*si
nh(f*x + e)^2 + 3*a^2 - 2*a*b - b^2 + 8*((3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^7 - 3*(3*a^2 - 2*a*b - b^2)*cosh(
f*x + e)^5 + 3*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^3 - (3*a^2 - 2*a*b - b^2)*cosh(f*x + e))*sinh(f*x + e))*sqr
t(-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)*((3*a^2 - a*b)*cosh(f*x + e)^6 + 6*(3*a^2 - a*b)*cosh(f*x + e)*sinh(f*x + e)^5 + (3*a^2
- a*b)*sinh(f*x + e)^6 - (11*a^2 - a*b)*cosh(f*x + e)^4 + (15*(3*a^2 - a*b)*cosh(f*x + e)^2 - 11*a^2 + a*b)*s
inh(f*x + e)^4 + 4*(5*(3*a^2 - a*b)*cosh(f*x + ...
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(f*x+e)**5*(a+b*sinh(f*x+e)**2)**(1/2),x)
[Out]
Exception raised: SystemError >> excessive stack use: stack is 5006 deep
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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)^5*(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 )}^5} \,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)^5,x)
[Out]
int((a + b*sinh(e + f*x)^2)^(1/2)/sinh(e + f*x)^5, x)
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