3.1.75 \(\int \text {csch}^4(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx\) [75]
Optimal. Leaf size=276 \[ \frac {(2 a-b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f}-\frac {\coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}+\frac {(2 a-b) E\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {b F\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {(2 a-b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a f} \]
[Out]
1/3*(2*a-b)*coth(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/a/f-1/3*coth(f*x+e)*csch(f*x+e)^2*(a+b*sinh(f*x+e)^2)^(1/2)/
f+1/3*(2*a-b)*(1/(1+sinh(f*x+e)^2))^(1/2)*(1+sinh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)/(1+sinh(f*x+e)^2)^(1/2
),(1-b/a)^(1/2))*sech(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/a/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)-1/3*b*(
1/(1+sinh(f*x+e)^2))^(1/2)*(1+sinh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)/(1+sinh(f*x+e)^2)^(1/2),(1-b/a)^(1/2)
)*sech(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/a/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)-1/3*(2*a-b)*(a+b*sinh(
f*x+e)^2)^(1/2)*tanh(f*x+e)/a/f
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Rubi [A]
time = 0.20, antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3267, 486, 597,
545, 429, 506, 422} \begin {gather*} -\frac {b \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} F\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{3 a f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(2 a-b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} E\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{3 a f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {(2 a-b) \tanh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f}+\frac {(2 a-b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f}-\frac {\coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Csch[e + f*x]^4*Sqrt[a + b*Sinh[e + f*x]^2],x]
[Out]
((2*a - b)*Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*f) - (Coth[e + f*x]*Csch[e + f*x]^2*Sqrt[a + b*Sinh
[e + f*x]^2])/(3*f) + ((2*a - b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f
*x]^2])/(3*a*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - (b*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/
a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - ((2*
a - b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*a*f)
Rule 422
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sq
rt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Rule 429
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*
Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Rule 486
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*e*(m + 1))), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*
x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*b*(m + 1) + n*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) + b*n*(p + q + 1))*x^n, x
], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[0, q, 1] && LtQ[m, -1] &&
IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Rule 506
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt
[c + d*x^2])), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
Rule 545
Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Dist[
e, Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x] + Dist[f, Int[x^n*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, n, p, q}, x]
Rule 597
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]
Rule 3267
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Sin[e + f*x], x]}, Dist[ff^(m + 1)*(Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])), Subst[Int[x^m*((a + b*ff^2*
x^2)^p/Sqrt[1 - ff^2*x^2]), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && !In
tegerQ[p]
Rubi steps
\begin {align*} \int \text {csch}^4(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx &=\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{x^4 \sqrt {1+x^2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=-\frac {\coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}+\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {-2 a+b-b x^2}{x^2 \sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 f}\\ &=\frac {(2 a-b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f}-\frac {\coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}-\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {a b+(2 a-b) b x^2}{\sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 a f}\\ &=\frac {(2 a-b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f}-\frac {\coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}-\frac {\left (b \sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 f}-\frac {\left ((2 a-b) b \sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 a f}\\ &=\frac {(2 a-b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f}-\frac {\coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}-\frac {b F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {(2 a-b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a f}+\frac {\left ((2 a-b) \sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{\left (1+x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 a f}\\ &=\frac {(2 a-b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f}-\frac {\coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}+\frac {(2 a-b) E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {b F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {(2 a-b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a f}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 2.01, size = 208, normalized size = 0.75 \begin {gather*} \frac {\frac {\left (4 \left (2 a^2-4 a b+b^2\right ) \cosh (2 (e+f x))-(2 a-b) (8 a-3 b-b \cosh (4 (e+f x)))\right ) \coth (e+f x) \text {csch}^2(e+f x)}{2 \sqrt {2}}+2 i a (2 a-b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )-4 i a (a-b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} F\left (i (e+f x)\left |\frac {b}{a}\right .\right )}{6 a f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Csch[e + f*x]^4*Sqrt[a + b*Sinh[e + f*x]^2],x]
[Out]
(((4*(2*a^2 - 4*a*b + b^2)*Cosh[2*(e + f*x)] - (2*a - b)*(8*a - 3*b - b*Cosh[4*(e + f*x)]))*Coth[e + f*x]*Csch
[e + f*x]^2)/(2*Sqrt[2]) + (2*I)*a*(2*a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/
a] - (4*I)*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a])/(6*a*f*Sqrt[2*a - b
+ b*Cosh[2*(e + f*x)]])
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Maple [A]
time = 5.33, size = 436, normalized size = 1.58
| | |
method |
result |
size |
| | |
default |
\(\frac {2 \sqrt {-\frac {b}{a}}\, a b \left (\sinh ^{6}\left (f x +e \right )\right )-\sqrt {-\frac {b}{a}}\, b^{2} \left (\sinh ^{6}\left (f x +e \right )\right )+b \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \left (\sinh ^{3}\left (f x +e \right )\right )-\sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2} \left (\sinh ^{3}\left (f x +e \right )\right )-2 \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b \left (\sinh ^{3}\left (f x +e \right )\right )+\sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2} \left (\sinh ^{3}\left (f x +e \right )\right )+2 \sqrt {-\frac {b}{a}}\, a^{2} \left (\sinh ^{4}\left (f x +e \right )\right )-\sqrt {-\frac {b}{a}}\, b^{2} \left (\sinh ^{4}\left (f x +e \right )\right )+\sqrt {-\frac {b}{a}}\, a^{2} \left (\sinh ^{2}\left (f x +e \right )\right )-2 \sqrt {-\frac {b}{a}}\, a b \left (\sinh ^{2}\left (f x +e \right )\right )-\sqrt {-\frac {b}{a}}\, a^{2}}{3 a \sinh \left (f x +e \right )^{3} \sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f}\) |
\(436\) |
risch |
\(\text {Expression too large to display}\) |
\(243160\) |
| | |
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)
[Out]
1/3*(2*(-1/a*b)^(1/2)*a*b*sinh(f*x+e)^6-(-1/a*b)^(1/2)*b^2*sinh(f*x+e)^6+b*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh
(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a*sinh(f*x+e)^3-((a+b*sinh(f*x+e)^2)/a)^(1/
2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*b^2*sinh(f*x+e)^3-2*((a+b*sinh(f*x+
e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a*b*sinh(f*x+e)^3+((a+b
*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*b^2*sinh(f*x+
e)^3+2*(-1/a*b)^(1/2)*a^2*sinh(f*x+e)^4-(-1/a*b)^(1/2)*b^2*sinh(f*x+e)^4+(-1/a*b)^(1/2)*a^2*sinh(f*x+e)^2-2*(-
1/a*b)^(1/2)*a*b*sinh(f*x+e)^2-(-1/a*b)^(1/2)*a^2)/a/sinh(f*x+e)^3/(-1/a*b)^(1/2)/cosh(f*x+e)/(a+b*sinh(f*x+e)
^2)^(1/2)/f
________________________________________________________________________________________
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)^4*(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)^4, x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2144 vs.
\(2 (280) = 560\).
time = 0.13, size = 2144, normalized size = 7.77 \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)^4*(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="fricas")
[Out]
-1/3*(((4*a^2 - 4*a*b + b^2)*cosh(f*x + e)^6 + 6*(4*a^2 - 4*a*b + b^2)*cosh(f*x + e)*sinh(f*x + e)^5 + (4*a^2
- 4*a*b + b^2)*sinh(f*x + e)^6 - 3*(4*a^2 - 4*a*b + b^2)*cosh(f*x + e)^4 + 3*(5*(4*a^2 - 4*a*b + b^2)*cosh(f*x
+ e)^2 - 4*a^2 + 4*a*b - b^2)*sinh(f*x + e)^4 + 4*(5*(4*a^2 - 4*a*b + b^2)*cosh(f*x + e)^3 - 3*(4*a^2 - 4*a*b
+ b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 3*(4*a^2 - 4*a*b + b^2)*cosh(f*x + e)^2 + 3*(5*(4*a^2 - 4*a*b + b^2)*
cosh(f*x + e)^4 - 6*(4*a^2 - 4*a*b + b^2)*cosh(f*x + e)^2 + 4*a^2 - 4*a*b + b^2)*sinh(f*x + e)^2 - 4*a^2 + 4*a
*b - b^2 + 6*((4*a^2 - 4*a*b + b^2)*cosh(f*x + e)^5 - 2*(4*a^2 - 4*a*b + b^2)*cosh(f*x + e)^3 + (4*a^2 - 4*a*b
+ b^2)*cosh(f*x + e))*sinh(f*x + e) - 2*((2*a*b - b^2)*cosh(f*x + e)^6 + 6*(2*a*b - b^2)*cosh(f*x + e)*sinh(f
*x + e)^5 + (2*a*b - b^2)*sinh(f*x + e)^6 - 3*(2*a*b - b^2)*cosh(f*x + e)^4 + 3*(5*(2*a*b - b^2)*cosh(f*x + e)
^2 - 2*a*b + b^2)*sinh(f*x + e)^4 + 4*(5*(2*a*b - b^2)*cosh(f*x + e)^3 - 3*(2*a*b - b^2)*cosh(f*x + e))*sinh(f
*x + e)^3 + 3*(2*a*b - b^2)*cosh(f*x + e)^2 + 3*(5*(2*a*b - b^2)*cosh(f*x + e)^4 - 6*(2*a*b - b^2)*cosh(f*x +
e)^2 + 2*a*b - b^2)*sinh(f*x + e)^2 - 2*a*b + b^2 + 6*((2*a*b - b^2)*cosh(f*x + e)^5 - 2*(2*a*b - b^2)*cosh(f*
x + e)^3 + (2*a*b - b^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt((a^2 - a*b)/b^2))*sqrt(b)*sqrt((2*b*sqrt((a^2 - a*
b)/b^2) - 2*a + b)/b)*elliptic_e(arcsin(sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*(cosh(f*x + e) + sinh(f*
x + e))), (8*a^2 - 8*a*b + b^2 + 4*(2*a*b - b^2)*sqrt((a^2 - a*b)/b^2))/b^2) - 2*((2*a^2 - a*b)*cosh(f*x + e)^
6 + 6*(2*a^2 - a*b)*cosh(f*x + e)*sinh(f*x + e)^5 + (2*a^2 - a*b)*sinh(f*x + e)^6 - 3*(2*a^2 - a*b)*cosh(f*x +
e)^4 + 3*(5*(2*a^2 - a*b)*cosh(f*x + e)^2 - 2*a^2 + a*b)*sinh(f*x + e)^4 + 4*(5*(2*a^2 - a*b)*cosh(f*x + e)^3
- 3*(2*a^2 - a*b)*cosh(f*x + e))*sinh(f*x + e)^3 + 3*(2*a^2 - a*b)*cosh(f*x + e)^2 + 3*(5*(2*a^2 - a*b)*cosh(
f*x + e)^4 - 6*(2*a^2 - a*b)*cosh(f*x + e)^2 + 2*a^2 - a*b)*sinh(f*x + e)^2 - 2*a^2 + a*b + 6*((2*a^2 - a*b)*c
osh(f*x + e)^5 - 2*(2*a^2 - a*b)*cosh(f*x + e)^3 + (2*a^2 - a*b)*cosh(f*x + e))*sinh(f*x + e) - 2*((a*b - b^2)
*cosh(f*x + e)^6 + 6*(a*b - b^2)*cosh(f*x + e)*sinh(f*x + e)^5 + (a*b - b^2)*sinh(f*x + e)^6 - 3*(a*b - b^2)*c
osh(f*x + e)^4 + 3*(5*(a*b - b^2)*cosh(f*x + e)^2 - a*b + b^2)*sinh(f*x + e)^4 + 4*(5*(a*b - b^2)*cosh(f*x + e
)^3 - 3*(a*b - b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 3*(a*b - b^2)*cosh(f*x + e)^2 + 3*(5*(a*b - b^2)*cosh(f*x
+ e)^4 - 6*(a*b - b^2)*cosh(f*x + e)^2 + a*b - b^2)*sinh(f*x + e)^2 - a*b + b^2 + 6*((a*b - b^2)*cosh(f*x + e
)^5 - 2*(a*b - b^2)*cosh(f*x + e)^3 + (a*b - b^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt((a^2 - a*b)/b^2))*sqrt(b)
*sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*elliptic_f(arcsin(sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)
*(cosh(f*x + e) + sinh(f*x + e))), (8*a^2 - 8*a*b + b^2 + 4*(2*a*b - b^2)*sqrt((a^2 - a*b)/b^2))/b^2) - sqrt(2
)*((2*a*b - b^2)*cosh(f*x + e)^5 + 5*(2*a*b - b^2)*cosh(f*x + e)*sinh(f*x + e)^4 + (2*a*b - b^2)*sinh(f*x + e)
^5 - 2*(3*a*b - b^2)*cosh(f*x + e)^3 + 2*(5*(2*a*b - b^2)*cosh(f*x + e)^2 - 3*a*b + b^2)*sinh(f*x + e)^3 - b^2
*cosh(f*x + e) + 2*(5*(2*a*b - b^2)*cosh(f*x + e)^3 - 3*(3*a*b - b^2)*cosh(f*x + e))*sinh(f*x + e)^2 + (5*(2*a
*b - b^2)*cosh(f*x + e)^4 - 6*(3*a*b - b^2)*cosh(f*x + e)^2 - b^2)*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*b*f*cosh(f
*x + e)^6 + 6*a*b*f*cosh(f*x + e)*sinh(f*x + e)^5 + a*b*f*sinh(f*x + e)^6 - 3*a*b*f*cosh(f*x + e)^4 + 3*a*b*f*
cosh(f*x + e)^2 + 3*(5*a*b*f*cosh(f*x + e)^2 - a*b*f)*sinh(f*x + e)^4 + 4*(5*a*b*f*cosh(f*x + e)^3 - 3*a*b*f*c
osh(f*x + e))*sinh(f*x + e)^3 - a*b*f + 3*(5*a*b*f*cosh(f*x + e)^4 - 6*a*b*f*cosh(f*x + e)^2 + a*b*f)*sinh(f*x
+ e)^2 + 6*(a*b*f*cosh(f*x + e)^5 - 2*a*b*f*cosh(f*x + e)^3 + a*b*f*cosh(f*x + e))*sinh(f*x + e))
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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)**4*(a+b*sinh(f*x+e)**2)**(1/2),x)
[Out]
Exception raised: SystemError >> excessive stack use: stack is 3004 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)^4*(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 = -0.00 \begin {gather*} \int \frac {\sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a}}{{\mathrm {sinh}\left (e+f\,x\right )}^4} \,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)^4,x)
[Out]
int((a + b*sinh(e + f*x)^2)^(1/2)/sinh(e + f*x)^4, x)
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