3.2.21 \(\int \frac {\sinh ^4(e+f x)}{(a+b \sinh ^2(e+f x))^{5/2}} \, dx\) [121]

Optimal. Leaf size=244 \[ -\frac {a \cosh (e+f x) \sinh (e+f x)}{3 (a-b) b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {2 \sqrt {a} (a-2 b) \cosh (e+f x) E\left (\text {ArcTan}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{3 (a-b)^2 b^{3/2} f \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {(a-3 b) F\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a (a-b)^2 b f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}} \]

[Out]

-1/3*a*cosh(f*x+e)*sinh(f*x+e)/(a-b)/b/f/(a+b*sinh(f*x+e)^2)^(3/2)+2/3*(a-2*b)*cosh(f*x+e)*(1/(1+b*sinh(f*x+e)
^2/a))^(1/2)*(1+b*sinh(f*x+e)^2/a)^(1/2)*EllipticE(sinh(f*x+e)*b^(1/2)/a^(1/2)/(1+b*sinh(f*x+e)^2/a)^(1/2),(1-
a/b)^(1/2))*a^(1/2)/(a-b)^2/b^(3/2)/f/(a*cosh(f*x+e)^2/(a+b*sinh(f*x+e)^2))^(1/2)/(a+b*sinh(f*x+e)^2)^(1/2)-1/
3*(a-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/(a-b)^2/b/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)

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Rubi [A]
time = 0.17, antiderivative size = 244, 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 = {3267, 481, 539, 429, 422} \begin {gather*} \frac {2 \sqrt {a} (a-2 b) \cosh (e+f x) E\left (\text {ArcTan}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{3 b^{3/2} f (a-b)^2 \sqrt {a+b \sinh ^2(e+f x)} \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}}}-\frac {(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 b f (a-b)^2 \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {a \sinh (e+f x) \cosh (e+f x)}{3 b f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[e + f*x]^4/(a + b*Sinh[e + f*x]^2)^(5/2),x]

[Out]

-1/3*(a*Cosh[e + f*x]*Sinh[e + f*x])/((a - b)*b*f*(a + b*Sinh[e + f*x]^2)^(3/2)) + (2*Sqrt[a]*(a - 2*b)*Cosh[e
+ f*x]*EllipticE[ArcTan[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]], 1 - a/b])/(3*(a - b)^2*b^(3/2)*f*Sqrt[(a*Cosh[e + f
*x]^2)/(a + b*Sinh[e + f*x]^2)]*Sqrt[a + b*Sinh[e + f*x]^2]) - ((a - 3*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 -
b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*(a - b)^2*b*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]
^2))/a])

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 481

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-a)*e^(
2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Dist[e^
(2*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1)
+ (a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 539

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^(3/2)), x_Symbol] :> Dist[(b*e - a*
f)/(b*c - a*d), Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[Sqrt[a + b
*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] && PosQ[d/c]

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 \frac {\sinh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {1+x^2} \left (a+b x^2\right )^{5/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=-\frac {a \cosh (e+f x) \sinh (e+f x)}{3 (a-b) b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {a+(2 a-3 b) x^2}{\sqrt {1+x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 (a-b) b f}\\ &=-\frac {a \cosh (e+f x) \sinh (e+f x)}{3 (a-b) b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\left ((a-3 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 (a-b)^2 b f}+\frac {\left (2 a (a-2 b) \sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {\sqrt {1+x^2}}{\left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 (a-b)^2 b f}\\ &=-\frac {a \cosh (e+f x) \sinh (e+f x)}{3 (a-b) b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {2 \sqrt {a} (a-2 b) \cosh (e+f x) E\left (\tan ^{-1}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{3 (a-b)^2 b^{3/2} f \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {(a-3 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 (a-b)^2 b f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.13, size = 198, normalized size = 0.81 \begin {gather*} \frac {2 i a^2 (a-2 b) \left (\frac {2 a-b+b \cosh (2 (e+f x))}{a}\right )^{3/2} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )-i a \left (2 a^2-5 a b+3 b^2\right ) \left (\frac {2 a-b+b \cosh (2 (e+f x))}{a}\right )^{3/2} F\left (i (e+f x)\left |\frac {b}{a}\right .\right )-\sqrt {2} b \left (-a^2+4 a b-2 b^2-(a-2 b) b \cosh (2 (e+f x))\right ) \sinh (2 (e+f x))}{3 (a-b)^2 b^2 f (2 a-b+b \cosh (2 (e+f x)))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[e + f*x]^4/(a + b*Sinh[e + f*x]^2)^(5/2),x]

[Out]

((2*I)*a^2*(a - 2*b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticE[I*(e + f*x), b/a] - I*a*(2*a^2 - 5*a*
b + 3*b^2)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticF[I*(e + f*x), b/a] - Sqrt[2]*b*(-a^2 + 4*a*b - 2
*b^2 - (a - 2*b)*b*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)])/(3*(a - b)^2*b^2*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3
/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(658\) vs. \(2(314)=628\).
time = 1.16, size = 659, normalized size = 2.70

method	result	size
default	\(\frac {2 \sqrt {-\frac {b}{a}}\, a b \left (\sinh ^{5}\left (f x +e \right )\right )-4 \sqrt {-\frac {b}{a}}\, b^{2} \left (\sinh ^{5}\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 ) a b \left (\sinh ^{2}\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 ^{2}\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 ^{2}\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 ^{2}\left (f x +e \right )\right )+\sqrt {-\frac {b}{a}}\, a^{2} \left (\sinh ^{3}\left (f x +e \right )\right )-\sqrt {-\frac {b}{a}}\, a b \left (\sinh ^{3}\left (f x +e \right )\right )-4 \sqrt {-\frac {b}{a}}\, b^{2} \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 ) a^{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}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b -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^{2}+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 ) a b +\sinh \left (f x +e \right ) a^{2} \sqrt {-\frac {b}{a}}-3 \sinh \left (f x +e \right ) b a \sqrt {-\frac {b}{a}}}{3 \sqrt {-\frac {b}{a}}\, \left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}} \left (a -b \right )^{2} b \cosh \left (f x +e \right ) f}\)	\(659\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(f*x+e)^4/(a+b*sinh(f*x+e)^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

[Out]

integrate(sinh(f*x + e)^4/(b*sinh(f*x + e)^2 + a)^(5/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4985 vs. \(2 (252) = 504\).
time = 0.18, size = 4985, normalized size = 20.43 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(f*x+e)^4/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="fricas")

[Out]

-2/3*(((2*a^2*b^2 - 5*a*b^3 + 2*b^4)*cosh(f*x + e)^8 + 8*(2*a^2*b^2 - 5*a*b^3 + 2*b^4)*cosh(f*x + e)*sinh(f*x
+ e)^7 + (2*a^2*b^2 - 5*a*b^3 + 2*b^4)*sinh(f*x + e)^8 + 4*(4*a^3*b - 12*a^2*b^2 + 9*a*b^3 - 2*b^4)*cosh(f*x +
e)^6 + 4*(4*a^3*b - 12*a^2*b^2 + 9*a*b^3 - 2*b^4 + 7*(2*a^2*b^2 - 5*a*b^3 + 2*b^4)*cosh(f*x + e)^2)*sinh(f*x
+ e)^6 + 8*(7*(2*a^2*b^2 - 5*a*b^3 + 2*b^4)*cosh(f*x + e)^3 + 3*(4*a^3*b - 12*a^2*b^2 + 9*a*b^3 - 2*b^4)*cosh(
f*x + e))*sinh(f*x + e)^5 + 2*(16*a^4 - 56*a^3*b + 62*a^2*b^2 - 31*a*b^3 + 6*b^4)*cosh(f*x + e)^4 + 2*(35*(2*a
^2*b^2 - 5*a*b^3 + 2*b^4)*cosh(f*x + e)^4 + 16*a^4 - 56*a^3*b + 62*a^2*b^2 - 31*a*b^3 + 6*b^4 + 30*(4*a^3*b -
12*a^2*b^2 + 9*a*b^3 - 2*b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 2*a^2*b^2 - 5*a*b^3 + 2*b^4 + 8*(7*(2*a^2*b^2
- 5*a*b^3 + 2*b^4)*cosh(f*x + e)^5 + 10*(4*a^3*b - 12*a^2*b^2 + 9*a*b^3 - 2*b^4)*cosh(f*x + e)^3 + (16*a^4 -
56*a^3*b + 62*a^2*b^2 - 31*a*b^3 + 6*b^4)*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(4*a^3*b - 12*a^2*b^2 + 9*a*b^3 -
2*b^4)*cosh(f*x + e)^2 + 4*(7*(2*a^2*b^2 - 5*a*b^3 + 2*b^4)*cosh(f*x + e)^6 + 15*(4*a^3*b - 12*a^2*b^2 + 9*a*
b^3 - 2*b^4)*cosh(f*x + e)^4 + 4*a^3*b - 12*a^2*b^2 + 9*a*b^3 - 2*b^4 + 3*(16*a^4 - 56*a^3*b + 62*a^2*b^2 - 31
*a*b^3 + 6*b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 8*((2*a^2*b^2 - 5*a*b^3 + 2*b^4)*cosh(f*x + e)^7 + 3*(4*a^3
*b - 12*a^2*b^2 + 9*a*b^3 - 2*b^4)*cosh(f*x + e)^5 + (16*a^4 - 56*a^3*b + 62*a^2*b^2 - 31*a*b^3 + 6*b^4)*cosh(
f*x + e)^3 + (4*a^3*b - 12*a^2*b^2 + 9*a*b^3 - 2*b^4)*cosh(f*x + e))*sinh(f*x + e) - 2*((a*b^3 - 2*b^4)*cosh(f
*x + e)^8 + 8*(a*b^3 - 2*b^4)*cosh(f*x + e)*sinh(f*x + e)^7 + (a*b^3 - 2*b^4)*sinh(f*x + e)^8 + 4*(2*a^2*b^2 -
5*a*b^3 + 2*b^4)*cosh(f*x + e)^6 + 4*(2*a^2*b^2 - 5*a*b^3 + 2*b^4 + 7*(a*b^3 - 2*b^4)*cosh(f*x + e)^2)*sinh(f
*x + e)^6 + 8*(7*(a*b^3 - 2*b^4)*cosh(f*x + e)^3 + 3*(2*a^2*b^2 - 5*a*b^3 + 2*b^4)*cosh(f*x + e))*sinh(f*x + e
)^5 + 2*(8*a^3*b - 24*a^2*b^2 + 19*a*b^3 - 6*b^4)*cosh(f*x + e)^4 + 2*(35*(a*b^3 - 2*b^4)*cosh(f*x + e)^4 + 8*
a^3*b - 24*a^2*b^2 + 19*a*b^3 - 6*b^4 + 30*(2*a^2*b^2 - 5*a*b^3 + 2*b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + a*
b^3 - 2*b^4 + 8*(7*(a*b^3 - 2*b^4)*cosh(f*x + e)^5 + 10*(2*a^2*b^2 - 5*a*b^3 + 2*b^4)*cosh(f*x + e)^3 + (8*a^3
*b - 24*a^2*b^2 + 19*a*b^3 - 6*b^4)*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(2*a^2*b^2 - 5*a*b^3 + 2*b^4)*cosh(f*x
+ e)^2 + 4*(7*(a*b^3 - 2*b^4)*cosh(f*x + e)^6 + 15*(2*a^2*b^2 - 5*a*b^3 + 2*b^4)*cosh(f*x + e)^4 + 2*a^2*b^2 -
5*a*b^3 + 2*b^4 + 3*(8*a^3*b - 24*a^2*b^2 + 19*a*b^3 - 6*b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 8*((a*b^3 -
2*b^4)*cosh(f*x + e)^7 + 3*(2*a^2*b^2 - 5*a*b^3 + 2*b^4)*cosh(f*x + e)^5 + (8*a^3*b - 24*a^2*b^2 + 19*a*b^3 -
6*b^4)*cosh(f*x + e)^3 + (2*a^2*b^2 - 5*a*b^3 + 2*b^4)*cosh(f*x + e))*sinh(f*x + e))*sqrt((a^2 - a*b)/b^2))*sq
rt(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*b^2 - 7*a*b^3 + 3*b^4)*cosh(f*x + e)^8 + 8*(2*a^2*b^2 - 7*a*b^3 + 3*b^4)*cosh(f*x + e)*sinh(f*x + e)^7
+ (2*a^2*b^2 - 7*a*b^3 + 3*b^4)*sinh(f*x + e)^8 + 4*(4*a^3*b - 16*a^2*b^2 + 13*a*b^3 - 3*b^4)*cosh(f*x + e)^6
+ 4*(4*a^3*b - 16*a^2*b^2 + 13*a*b^3 - 3*b^4 + 7*(2*a^2*b^2 - 7*a*b^3 + 3*b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^
6 + 8*(7*(2*a^2*b^2 - 7*a*b^3 + 3*b^4)*cosh(f*x + e)^3 + 3*(4*a^3*b - 16*a^2*b^2 + 13*a*b^3 - 3*b^4)*cosh(f*x
+ e))*sinh(f*x + e)^5 + 2*(16*a^4 - 72*a^3*b + 86*a^2*b^2 - 45*a*b^3 + 9*b^4)*cosh(f*x + e)^4 + 2*(35*(2*a^2*b
^2 - 7*a*b^3 + 3*b^4)*cosh(f*x + e)^4 + 16*a^4 - 72*a^3*b + 86*a^2*b^2 - 45*a*b^3 + 9*b^4 + 30*(4*a^3*b - 16*a
^2*b^2 + 13*a*b^3 - 3*b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 2*a^2*b^2 - 7*a*b^3 + 3*b^4 + 8*(7*(2*a^2*b^2 -
7*a*b^3 + 3*b^4)*cosh(f*x + e)^5 + 10*(4*a^3*b - 16*a^2*b^2 + 13*a*b^3 - 3*b^4)*cosh(f*x + e)^3 + (16*a^4 - 72
*a^3*b + 86*a^2*b^2 - 45*a*b^3 + 9*b^4)*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(4*a^3*b - 16*a^2*b^2 + 13*a*b^3 -
3*b^4)*cosh(f*x + e)^2 + 4*(7*(2*a^2*b^2 - 7*a*b^3 + 3*b^4)*cosh(f*x + e)^6 + 15*(4*a^3*b - 16*a^2*b^2 + 13*a*
b^3 - 3*b^4)*cosh(f*x + e)^4 + 4*a^3*b - 16*a^2*b^2 + 13*a*b^3 - 3*b^4 + 3*(16*a^4 - 72*a^3*b + 86*a^2*b^2 - 4
5*a*b^3 + 9*b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 8*((2*a^2*b^2 - 7*a*b^3 + 3*b^4)*cosh(f*x + e)^7 + 3*(4*a^
3*b - 16*a^2*b^2 + 13*a*b^3 - 3*b^4)*cosh(f*x + e)^5 + (16*a^4 - 72*a^3*b + 86*a^2*b^2 - 45*a*b^3 + 9*b^4)*cos
h(f*x + e)^3 + (4*a^3*b - 16*a^2*b^2 + 13*a*b^3 - 3*b^4)*cosh(f*x + e))*sinh(f*x + e) - 2*((a*b^3 - b^4)*cosh(
f*x + e)^8 + 8*(a*b^3 - b^4)*cosh(f*x + e)*sinh(f*x + e)^7 + (a*b^3 - b^4)*sinh(f*x + e)^8 + 4*(2*a^2*b^2 - 3*
a*b^3 + b^4)*cosh(f*x + e)^6 + 4*(2*a^2*b^2 - 3*a*b^3 + b^4 + 7*(a*b^3 - b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^6
+ 8*(7*(a*b^3 - b^4)*cosh(f*x + e)^3 + 3*(2*a^2*b^2 - 3*a*b^3 + b^4)*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(8*a^
3*b - 16*a^2*b^2 + 11*a*b^3 - 3*b^4)*cosh(f*x + e)^4 + 2*(35*(a*b^3 - b^4)*cosh(f*x + e)^4 + 8*a^3*b - 16*a^2*
b^2 + 11*a*b^3 - 3*b^4 + 30*(2*a^2*b^2 - 3*a*b^...

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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(sinh(f*x+e)**4/(a+b*sinh(f*x+e)**2)**(5/2),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: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(f*x+e)^4/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Error: Bad Argume
nt Type

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {sinh}\left (e+f\,x\right )}^4}{{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(e + f*x)^4/(a + b*sinh(e + f*x)^2)^(5/2),x)

[Out]

int(sinh(e + f*x)^4/(a + b*sinh(e + f*x)^2)^(5/2), x)

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