3.2.20 \(\int \frac {\sinh ^6(e+f x)}{(a+b \sinh ^2(e+f x))^{5/2}} \, dx\) [120]

Optimal. Leaf size=344 \[ -\frac {a \cosh (e+f x) \sinh ^3(e+f x)}{3 (a-b) b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {2 a (2 a-3 b) \cosh (e+f x) \sinh (e+f x)}{3 (a-b)^2 b^2 f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {\left (8 a^2-13 a b+3 b^2\right ) E\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 (a-b)^2 b^3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {2 (2 a-3 b) F\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 (a-b)^2 b^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {\left (8 a^2-13 a b+3 b^2\right ) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 (a-b)^2 b^3 f} \]

[Out]

-1/3*a*cosh(f*x+e)*sinh(f*x+e)^3/(a-b)/b/f/(a+b*sinh(f*x+e)^2)^(3/2)-2/3*a*(2*a-3*b)*cosh(f*x+e)*sinh(f*x+e)/(
a-b)^2/b^2/f/(a+b*sinh(f*x+e)^2)^(1/2)-1/3*(8*a^2-13*a*b+3*b^2)*(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-b)
^2/b^3/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+2/3*(2*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
-b)^2/b^2/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+1/3*(8*a^2-13*a*b+3*b^2)*(a+b*sinh(f*x+e)^2)^(1/2)*tan
h(f*x+e)/(a-b)^2/b^3/f

________________________________________________________________________________________

Rubi [A]
time = 0.25, antiderivative size = 344, 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, 481, 592, 545, 429, 506, 422} \begin {gather*} -\frac {\left (8 a^2-13 a b+3 b^2\right ) \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 b^3 f (a-b)^2 \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {\left (8 a^2-13 a b+3 b^2\right ) \tanh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 b^3 f (a-b)^2}+\frac {2 (2 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 b^2 f (a-b)^2 \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {2 a (2 a-3 b) \sinh (e+f x) \cosh (e+f x)}{3 b^2 f (a-b)^2 \sqrt {a+b \sinh ^2(e+f x)}}-\frac {a \sinh ^3(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]^6/(a + b*Sinh[e + f*x]^2)^(5/2),x]

[Out]

-1/3*(a*Cosh[e + f*x]*Sinh[e + f*x]^3)/((a - b)*b*f*(a + b*Sinh[e + f*x]^2)^(3/2)) - (2*a*(2*a - 3*b)*Cosh[e +
f*x]*Sinh[e + f*x])/(3*(a - b)^2*b^2*f*Sqrt[a + b*Sinh[e + f*x]^2]) - ((8*a^2 - 13*a*b + 3*b^2)*EllipticE[Arc
Tan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*(a - b)^2*b^3*f*Sqrt[(Sech[e + f*x]
^2*(a + b*Sinh[e + f*x]^2))/a]) + (2*(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 - b)^2*b^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((8*a^2 - 13*a
*b + 3*b^2)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*(a - b)^2*b^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 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 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 592

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> Simp[g^(n - 1)*(b*e - a*f)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c -
a*d)*(p + 1))), x] - Dist[g^n/(b*n*(b*c - a*d)*(p + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*S
imp[c*(b*e - a*f)*(m - n + 1) + (d*(b*e - a*f)*(m + n*q + 1) - b*n*(c*f - d*e)*(p + 1))*x^n, x], x], x] /; Fre
eQ[{a, b, c, d, e, f, g, q}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, 0]

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 ^6(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^6}{\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 ^3(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 {x^2 \left (3 a+(4 a-3 b) x^2\right )}{\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 ^3(e+f x)}{3 (a-b) b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {2 a (2 a-3 b) \cosh (e+f x) \sinh (e+f x)}{3 (a-b)^2 b^2 f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {-2 a (2 a-3 b)+\left (-8 a^2+13 a b-3 b^2\right ) x^2}{\sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 (a-b)^2 b^2 f}\\ &=-\frac {a \cosh (e+f x) \sinh ^3(e+f x)}{3 (a-b) b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {2 a (2 a-3 b) \cosh (e+f x) \sinh (e+f x)}{3 (a-b)^2 b^2 f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\left (2 a (2 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^2 f}-\frac {\left (\left (-8 a^2+13 a b-3 b^2\right ) \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-b)^2 b^2 f}\\ &=-\frac {a \cosh (e+f x) \sinh ^3(e+f x)}{3 (a-b) b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {2 a (2 a-3 b) \cosh (e+f x) \sinh (e+f x)}{3 (a-b)^2 b^2 f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {2 (2 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-b)^2 b^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {\left (8 a^2-13 a b+3 b^2\right ) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 (a-b)^2 b^3 f}+\frac {\left (\left (-8 a^2+13 a b-3 b^2\right ) \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-b)^2 b^3 f}\\ &=-\frac {a \cosh (e+f x) \sinh ^3(e+f x)}{3 (a-b) b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {2 a (2 a-3 b) \cosh (e+f x) \sinh (e+f x)}{3 (a-b)^2 b^2 f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {\left (8 a^2-13 a b+3 b^2\right ) 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-b)^2 b^3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {2 (2 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-b)^2 b^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {\left (8 a^2-13 a b+3 b^2\right ) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 (a-b)^2 b^3 f}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 1.40, size = 207, normalized size = 0.60 \begin {gather*} \frac {a \left (-2 i a \left (8 a^2-13 a b+3 b^2\right ) \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 )+2 i a \left (8 a^2-17 a b+9 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 (-8 a^2+17 a b-7 b^2+b (-5 a+7 b) \cosh (2 (e+f x))\right ) \sinh (2 (e+f x))\right )}{6 (a-b)^2 b^3 f (2 a-b+b \cosh (2 (e+f x)))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[e + f*x]^6/(a + b*Sinh[e + f*x]^2)^(5/2),x]

[Out]

(a*((-2*I)*a*(8*a^2 - 13*a*b + 3*b^2)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticE[I*(e + f*x), b/a] +
(2*I)*a*(8*a^2 - 17*a*b + 9*b^2)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticF[I*(e + f*x), b/a] + Sqrt[
2]*b*(-8*a^2 + 17*a*b - 7*b^2 + b*(-5*a + 7*b)*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)]))/(6*(a - b)^2*b^3*f*(2*a
- b + b*Cosh[2*(e + f*x)])^(3/2))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(867\) vs. \(2(400)=800\).
time = 1.44, size = 868, normalized size = 2.52

method	result	size
default	\(\text {Expression too large to display}\)	\(868\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(f*x+e)^6/(a+b*sinh(f*x+e)^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*((5*(-1/a*b)^(1/2)*a^2*b-7*(-1/a*b)^(1/2)*a*b^2)*cosh(f*x+e)^4*sinh(f*x+e)+(4*(-1/a*b)^(1/2)*a^3-11*(-1/a
*b)^(1/2)*a^2*b+7*(-1/a*b)^(1/2)*a*b^2)*cosh(f*x+e)^2*sinh(f*x+e)+(cosh(f*x+e)^2)^(1/2)*(b/a*cosh(f*x+e)^2+(a-
b)/a)^(1/2)*b*(4*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a^2-7*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),
(a/b)^(1/2))*a*b+3*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*b^2-8*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2
),(a/b)^(1/2))*a^2+13*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a*b-3*EllipticE(sinh(f*x+e)*(-1/a*b)^(
1/2),(a/b)^(1/2))*b^2)*cosh(f*x+e)^2+4*(b/a*cosh(f*x+e)^2+(a-b)/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^3-11*(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(si
nh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a^2*b+10*(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*Ellipti
cF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a*b^2-3*(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*Ell
ipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*b^3-8*(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*E
llipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a^3+21*(b/a*cosh(f*x+e)^2+(a-b)/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*b-16*(b/a*cosh(f*x+e)^2+(a-b)/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^2+3*(b/a*cosh(f*x+e)^2+(a-b)/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^3)/(-1/a*b)^(1/2)/(a+b*sinh(f*x+e)^2)^(3/2)/(a-
b)^2/b^2/cosh(f*x+e)/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(sinh(f*x+e)^6/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(sinh(f*x + e)^6/(b*sinh(f*x + e)^2 + a)^(5/2), x)

________________________________________________________________________________________

Fricas [F]
time = 0.13, size = 71, normalized size = 0.21 \begin {gather*} {\rm integral}\left (\frac {\sqrt {b \sinh \left (f x + e\right )^{2} + a} \sinh \left (f x + e\right )^{6}}{b^{3} \sinh \left (f x + e\right )^{6} + 3 \, a b^{2} \sinh \left (f x + e\right )^{4} + 3 \, a^{2} b \sinh \left (f x + e\right )^{2} + a^{3}}, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(f*x+e)^6/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*sinh(f*x + e)^2 + a)*sinh(f*x + e)^6/(b^3*sinh(f*x + e)^6 + 3*a*b^2*sinh(f*x + e)^4 + 3*a^2*b*
sinh(f*x + e)^2 + a^3), x)

________________________________________________________________________________________

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)**6/(a+b*sinh(f*x+e)**2)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

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)^6/(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:Evaluation time:
0.61index.cc index_m i_lex_is_greater Error: Bad Argument Value

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {sinh}\left (e+f\,x\right )}^6}{{\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)^6/(a + b*sinh(e + f*x)^2)^(5/2),x)

[Out]

int(sinh(e + f*x)^6/(a + b*sinh(e + f*x)^2)^(5/2), x)

________________________________________________________________________________________