3.1.86 \(\int \text {csch}^4(e+f x) (a+b \sinh ^2(e+f x))^{3/2} \, dx\) [86]

Optimal. Leaf size=267 \[ \frac {2 (a-2 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}-\frac {a \coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}+\frac {2 (a-2 b) E\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {(a-3 b) b F\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {2 (a-2 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 f} \]

[Out]

2/3*(a-2*b)*coth(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/f-1/3*a*coth(f*x+e)*csch(f*x+e)^2*(a+b*sinh(f*x+e)^2)^(1/2)/
f+2/3*(a-2*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)/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)-1/3*(a-3*
b)*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)-2/3*(a-2*b)*(a+b
*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)/f

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Rubi [A]
time = 0.21, antiderivative size = 267, 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, 485, 597, 545, 429, 506, 422} \begin {gather*} -\frac {b (a-3 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-2 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 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {2 (a-2 b) \tanh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}+\frac {2 (a-2 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}-\frac {a \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*(a + b*Sinh[e + f*x]^2)^(3/2),x]

[Out]

(2*(a - 2*b)*Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f) - (a*Coth[e + f*x]*Csch[e + f*x]^2*Sqrt[a + b*Si
nh[e + f*x]^2])/(3*f) + (2*(a - 2*b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e
+ f*x]^2])/(3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - ((a - 3*b)*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 - 2*b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*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 485

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[c*(e*x)^(
m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(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 - 2)*Simp[c*(c*b - a*d)*(m + 1) + c*n*(b*c*(p + 1) + a*d*(q - 1)) + d*((c*b - a*
d)*(m + 1) + c*b*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0]
&& GtQ[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) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx &=\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {\left (a+b x^2\right )^{3/2}}{x^4 \sqrt {1+x^2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=-\frac {a \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 (a-2 b)-(a-3 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-2 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}-\frac {a \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 (a-3 b) b+2 a (a-2 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-2 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}-\frac {a \coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}-\frac {\left ((a-3 b) 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-2 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 f}\\ &=\frac {2 (a-2 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}-\frac {a \coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}-\frac {(a-3 b) 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-2 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 f}+\frac {\left (2 (a-2 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 f}\\ &=\frac {2 (a-2 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}-\frac {a \coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}+\frac {2 (a-2 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 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {(a-3 b) 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-2 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 f}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.66, size = 213, normalized size = 0.80 \begin {gather*} \frac {\frac {\left (-8 a^2+13 a b-6 b^2+2 \left (2 a^2-7 a b+4 b^2\right ) \cosh (2 (e+f x))+(a-2 b) b \cosh (4 (e+f x))\right ) \coth (e+f x) \text {csch}^2(e+f x)}{\sqrt {2}}+4 i a (a-2 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 )-2 i \left (2 a^2-5 a b+3 b^2\right ) \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 f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(3/2),x]

[Out]

(((-8*a^2 + 13*a*b - 6*b^2 + 2*(2*a^2 - 7*a*b + 4*b^2)*Cosh[2*(e + f*x)] + (a - 2*b)*b*Cosh[4*(e + f*x)])*Coth
[e + f*x]*Csch[e + f*x]^2)/Sqrt[2] + (4*I)*a*(a - 2*b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e
+ f*x), b/a] - (2*I)*(2*a^2 - 5*a*b + 3*b^2)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/
a])/(6*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])

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Maple [A]
time = 1.34, size = 454, normalized size = 1.70

method	result	size
default	\(\frac {2 \sqrt {-\frac {b}{a}}\, a b \left (\sinh ^{6}\left (f x +e \right )\right )-4 \sqrt {-\frac {b}{a}}\, b^{2} \left (\sinh ^{6}\left (f x +e \right )\right )+b \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \left (\sinh ^{3}\left (f x +e \right )\right )-\sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2} \left (\sinh ^{3}\left (f x +e \right )\right )-2 \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b \left (\sinh ^{3}\left (f x +e \right )\right )+4 \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2} \left (\sinh ^{3}\left (f x +e \right )\right )+2 \sqrt {-\frac {b}{a}}\, a^{2} \left (\sinh ^{4}\left (f x +e \right )\right )-3 \sqrt {-\frac {b}{a}}\, a b \left (\sinh ^{4}\left (f x +e \right )\right )-4 \sqrt {-\frac {b}{a}}\, b^{2} \left (\sinh ^{4}\left (f x +e \right )\right )+\sqrt {-\frac {b}{a}}\, a^{2} \left (\sinh ^{2}\left (f x +e \right )\right )-5 \sqrt {-\frac {b}{a}}\, a b \left (\sinh ^{2}\left (f x +e \right )\right )-\sqrt {-\frac {b}{a}}\, a^{2}}{3 \sinh \left (f x +e \right )^{3} \sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f}\)	\(454\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/3*(2*(-1/a*b)^(1/2)*a*b*sinh(f*x+e)^6-4*(-1/a*b)^(1/2)*b^2*sinh(f*x+e)^6+b*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(co
sh(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+4*(
(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)^4-3*(-1/a*b)^(1/2)*a*b*sinh(f*x+e)^4-4*(-1/a*b)^(1/2)*b^2*sinh(f*x+e
)^4+(-1/a*b)^(1/2)*a^2*sinh(f*x+e)^2-5*(-1/a*b)^(1/2)*a*b*sinh(f*x+e)^2-(-1/a*b)^(1/2)*a^2)/sinh(f*x+e)^3/(-1/
a*b)^(1/2)/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)^4*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*sinh(f*x + e)^2 + a)^(3/2)*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. 2222 vs. \(2 (271) = 542\).
time = 0.14, size = 2222, normalized size = 8.32 \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)^(3/2),x, algorithm="fricas")

[Out]

-2/3*(((2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)^6 + 6*(2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)*sinh(f*x + e)^5 + (2*
a^2 - 5*a*b + 2*b^2)*sinh(f*x + e)^6 - 3*(2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)^4 + 3*(5*(2*a^2 - 5*a*b + 2*b^2
)*cosh(f*x + e)^2 - 2*a^2 + 5*a*b - 2*b^2)*sinh(f*x + e)^4 + 4*(5*(2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)^3 - 3*
(2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 3*(2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)^2 + 3*(5*(2*a
^2 - 5*a*b + 2*b^2)*cosh(f*x + e)^4 - 6*(2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)^2 + 2*a^2 - 5*a*b + 2*b^2)*sinh(
f*x + e)^2 - 2*a^2 + 5*a*b - 2*b^2 + 6*((2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)^5 - 2*(2*a^2 - 5*a*b + 2*b^2)*co
sh(f*x + e)^3 + (2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e))*sinh(f*x + e) - 2*((a*b - 2*b^2)*cosh(f*x + e)^6 + 6*(a
*b - 2*b^2)*cosh(f*x + e)*sinh(f*x + e)^5 + (a*b - 2*b^2)*sinh(f*x + e)^6 - 3*(a*b - 2*b^2)*cosh(f*x + e)^4 +
3*(5*(a*b - 2*b^2)*cosh(f*x + e)^2 - a*b + 2*b^2)*sinh(f*x + e)^4 + 4*(5*(a*b - 2*b^2)*cosh(f*x + e)^3 - 3*(a*
b - 2*b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 3*(a*b - 2*b^2)*cosh(f*x + e)^2 + 3*(5*(a*b - 2*b^2)*cosh(f*x + e)
^4 - 6*(a*b - 2*b^2)*cosh(f*x + e)^2 + a*b - 2*b^2)*sinh(f*x + e)^2 - a*b + 2*b^2 + 6*((a*b - 2*b^2)*cosh(f*x
+ e)^5 - 2*(a*b - 2*b^2)*cosh(f*x + e)^3 + (a*b - 2*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*a^2 - 7*a*b + 3*b^2)*cosh(f*x + e)^6 + 6*(2*a^2 - 7*a*b + 3*b^2)*cosh(f*x + e)*sinh(f*x + e)^5 + (2*a^2 -
7*a*b + 3*b^2)*sinh(f*x + e)^6 - 3*(2*a^2 - 7*a*b + 3*b^2)*cosh(f*x + e)^4 + 3*(5*(2*a^2 - 7*a*b + 3*b^2)*cos
h(f*x + e)^2 - 2*a^2 + 7*a*b - 3*b^2)*sinh(f*x + e)^4 + 4*(5*(2*a^2 - 7*a*b + 3*b^2)*cosh(f*x + e)^3 - 3*(2*a^
2 - 7*a*b + 3*b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 3*(2*a^2 - 7*a*b + 3*b^2)*cosh(f*x + e)^2 + 3*(5*(2*a^2 -
7*a*b + 3*b^2)*cosh(f*x + e)^4 - 6*(2*a^2 - 7*a*b + 3*b^2)*cosh(f*x + e)^2 + 2*a^2 - 7*a*b + 3*b^2)*sinh(f*x +
e)^2 - 2*a^2 + 7*a*b - 3*b^2 + 6*((2*a^2 - 7*a*b + 3*b^2)*cosh(f*x + e)^5 - 2*(2*a^2 - 7*a*b + 3*b^2)*cosh(f*
x + e)^3 + (2*a^2 - 7*a*b + 3*b^2)*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)*cosh(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)*((a*b - 2*b^2)*cosh(f*x + e)
^5 + 5*(a*b - 2*b^2)*cosh(f*x + e)*sinh(f*x + e)^4 + (a*b - 2*b^2)*sinh(f*x + e)^5 - (3*a*b - 4*b^2)*cosh(f*x
+ e)^3 + (10*(a*b - 2*b^2)*cosh(f*x + e)^2 - 3*a*b + 4*b^2)*sinh(f*x + e)^3 - 2*b^2*cosh(f*x + e) + (10*(a*b -
2*b^2)*cosh(f*x + e)^3 - 3*(3*a*b - 4*b^2)*cosh(f*x + e))*sinh(f*x + e)^2 + (5*(a*b - 2*b^2)*cosh(f*x + e)^4
- 3*(3*a*b - 4*b^2)*cosh(f*x + e)^2 - 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)))/(b*f*cosh(f*x + e)^6 + 6*b*f*cosh(f
*x + e)*sinh(f*x + e)^5 + b*f*sinh(f*x + e)^6 - 3*b*f*cosh(f*x + e)^4 + 3*(5*b*f*cosh(f*x + e)^2 - b*f)*sinh(f
*x + e)^4 + 3*b*f*cosh(f*x + e)^2 + 4*(5*b*f*cosh(f*x + e)^3 - 3*b*f*cosh(f*x + e))*sinh(f*x + e)^3 + 3*(5*b*f
*cosh(f*x + e)^4 - 6*b*f*cosh(f*x + e)^2 + b*f)*sinh(f*x + e)^2 - b*f + 6*(b*f*cosh(f*x + e)^5 - 2*b*f*cosh(f*
x + e)^3 + b*f*cosh(f*x + e))*sinh(f*x + e))

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**(3/2),x)

[Out]

Timed out

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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)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Evaluation time: 0.54Unable to divide, perhaps due to rounding error%%%{1024,[8,10,8]%%%}+%%%{%%%{-4096,[1]
%%%},[8,10,

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}}{{\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)^(3/2)/sinh(e + f*x)^4,x)

[Out]

int((a + b*sinh(e + f*x)^2)^(3/2)/sinh(e + f*x)^4, x)

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