3.76 \(\int \sinh ^3(e+f x) (a+b \sinh ^2(e+f x))^{3/2} \, dx\)

Optimal. Leaf size=177 \[ -\frac{(a-b)^2 (a+5 b) \tanh ^{-1}\left (\frac{\sqrt{b} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right )}{16 b^{3/2} f}+\frac{\cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{5/2}}{6 b f}-\frac{(a+5 b) \cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}}{24 b f}-\frac{(a-b) (a+5 b) \cosh (e+f x) \sqrt{a+b \cosh ^2(e+f x)-b}}{16 b f} \]

[Out]

-((a - b)^2*(a + 5*b)*ArcTanh[(Sqrt[b]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(16*b^(3/2)*f) - ((a -
b)*(a + 5*b)*Cosh[e + f*x]*Sqrt[a - b + b*Cosh[e + f*x]^2])/(16*b*f) - ((a + 5*b)*Cosh[e + f*x]*(a - b + b*Co
sh[e + f*x]^2)^(3/2))/(24*b*f) + (Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^(5/2))/(6*b*f)

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Rubi [A]  time = 0.181481, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3186, 388, 195, 217, 206} \[ -\frac{(a-b)^2 (a+5 b) \tanh ^{-1}\left (\frac{\sqrt{b} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right )}{16 b^{3/2} f}+\frac{\cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{5/2}}{6 b f}-\frac{(a+5 b) \cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}}{24 b f}-\frac{(a-b) (a+5 b) \cosh (e+f x) \sqrt{a+b \cosh ^2(e+f x)-b}}{16 b f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a - b)^2*(a + 5*b)*ArcTanh[(Sqrt[b]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(16*b^(3/2)*f) - ((a -
b)*(a + 5*b)*Cosh[e + f*x]*Sqrt[a - b + b*Cosh[e + f*x]^2])/(16*b*f) - ((a + 5*b)*Cosh[e + f*x]*(a - b + b*Co
sh[e + f*x]^2)^(3/2))/(24*b*f) + (Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^(5/2))/(6*b*f)

Rule 3186

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]

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \sinh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \left (a-b+b x^2\right )^{3/2} \, dx,x,\cosh (e+f x)\right )}{f}\\ &=\frac{\cosh (e+f x) \left (a-b+b \cosh ^2(e+f x)\right )^{5/2}}{6 b f}-\frac{(a+5 b) \operatorname{Subst}\left (\int \left (a-b+b x^2\right )^{3/2} \, dx,x,\cosh (e+f x)\right )}{6 b f}\\ &=-\frac{(a+5 b) \cosh (e+f x) \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}}{24 b f}+\frac{\cosh (e+f x) \left (a-b+b \cosh ^2(e+f x)\right )^{5/2}}{6 b f}-\frac{((a-b) (a+5 b)) \operatorname{Subst}\left (\int \sqrt{a-b+b x^2} \, dx,x,\cosh (e+f x)\right )}{8 b f}\\ &=-\frac{(a-b) (a+5 b) \cosh (e+f x) \sqrt{a-b+b \cosh ^2(e+f x)}}{16 b f}-\frac{(a+5 b) \cosh (e+f x) \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}}{24 b f}+\frac{\cosh (e+f x) \left (a-b+b \cosh ^2(e+f x)\right )^{5/2}}{6 b f}-\frac{\left ((a-b)^2 (a+5 b)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-b+b x^2}} \, dx,x,\cosh (e+f x)\right )}{16 b f}\\ &=-\frac{(a-b) (a+5 b) \cosh (e+f x) \sqrt{a-b+b \cosh ^2(e+f x)}}{16 b f}-\frac{(a+5 b) \cosh (e+f x) \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}}{24 b f}+\frac{\cosh (e+f x) \left (a-b+b \cosh ^2(e+f x)\right )^{5/2}}{6 b f}-\frac{\left ((a-b)^2 (a+5 b)\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\cosh (e+f x)}{\sqrt{a-b+b \cosh ^2(e+f x)}}\right )}{16 b f}\\ &=-\frac{(a-b)^2 (a+5 b) \tanh ^{-1}\left (\frac{\sqrt{b} \cosh (e+f x)}{\sqrt{a-b+b \cosh ^2(e+f x)}}\right )}{16 b^{3/2} f}-\frac{(a-b) (a+5 b) \cosh (e+f x) \sqrt{a-b+b \cosh ^2(e+f x)}}{16 b f}-\frac{(a+5 b) \cosh (e+f x) \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}}{24 b f}+\frac{\cosh (e+f x) \left (a-b+b \cosh ^2(e+f x)\right )^{5/2}}{6 b f}\\ \end{align*}

Mathematica [A]  time = 0.582482, size = 151, normalized size = 0.85 \[ \frac{\sqrt{2} \sqrt{b} \sqrt{2 a+b \cosh (2 (e+f x))-b} \left (\left (6 a^2-51 a b+37 b^2\right ) \cosh (e+f x)+b ((7 a-8 b) \cosh (3 (e+f x))+b \cosh (5 (e+f x)))\right )-12 (a-b)^2 (a+5 b) \log \left (\sqrt{2 a+b \cosh (2 (e+f x))-b}+\sqrt{2} \sqrt{b} \cosh (e+f x)\right )}{192 b^{3/2} f} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[2]*Sqrt[b]*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]*((6*a^2 - 51*a*b + 37*b^2)*Cosh[e + f*x] + b*((7*a - 8*b)
*Cosh[3*(e + f*x)] + b*Cosh[5*(e + f*x)])) - 12*(a - b)^2*(a + 5*b)*Log[Sqrt[2]*Sqrt[b]*Cosh[e + f*x] + Sqrt[2
*a - b + b*Cosh[2*(e + f*x)]]])/(192*b^(3/2)*f)

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Maple [B]  time = 0.091, size = 483, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*(16*b^(7/2)*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)*cosh(f*
x+e)^4+4*b^(5/2)*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)*(-13*b+7*a)*cosh(f*x+e)^2+66*b^(7/2)*(b*cosh(f*x+
e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)-72*a*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)*b^(5/2)+6*a^2*(b*cosh(f*x+e)^
4+(a-b)*cosh(f*x+e)^2)^(1/2)*b^(3/2)-3*ln(1/2*(2*b*cosh(f*x+e)^2+2*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)
*b^(1/2)+a-b)/b^(1/2))*a^3*b-9*ln(1/2*(2*b*cosh(f*x+e)^2+2*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)*b^(1/2)
+a-b)/b^(1/2))*a^2*b^2+27*b^3*a*ln(1/2*(2*b*cosh(f*x+e)^2+2*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)*b^(1/2
)+a-b)/b^(1/2))-15*b^4*ln(1/2*(2*b*cosh(f*x+e)^2+2*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)*b^(1/2)+a-b)/b^
(1/2)))/b^(5/2)/cosh(f*x+e)/(a+b*sinh(f*x+e)^2)^(1/2)/f

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} \sinh \left (f x + e\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 3.80683, size = 11580, normalized size = 65.42 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

[1/384*(6*((a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(f*x + e)^6 + 6*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(f*x +
e)^5*sinh(f*x + e) + 15*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(f*x + e)^4*sinh(f*x + e)^2 + 20*(a^3 + 3*a^2*b
- 9*a*b^2 + 5*b^3)*cosh(f*x + e)^3*sinh(f*x + e)^3 + 15*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(f*x + e)^2*sinh
(f*x + e)^4 + 6*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(f*x + e)*sinh(f*x + e)^5 + (a^3 + 3*a^2*b - 9*a*b^2 + 5
*b^3)*sinh(f*x + e)^6)*sqrt(b)*log((a^2*b*cosh(f*x + e)^8 + 8*a^2*b*cosh(f*x + e)*sinh(f*x + e)^7 + a^2*b*sinh
(f*x + e)^8 + 2*(a^3 + a^2*b)*cosh(f*x + e)^6 + 2*(14*a^2*b*cosh(f*x + e)^2 + a^3 + a^2*b)*sinh(f*x + e)^6 + 4
*(14*a^2*b*cosh(f*x + e)^3 + 3*(a^3 + a^2*b)*cosh(f*x + e))*sinh(f*x + e)^5 + (9*a^2*b - 4*a*b^2 + b^3)*cosh(f
*x + e)^4 + (70*a^2*b*cosh(f*x + e)^4 + 9*a^2*b - 4*a*b^2 + b^3 + 30*(a^3 + a^2*b)*cosh(f*x + e)^2)*sinh(f*x +
e)^4 + 4*(14*a^2*b*cosh(f*x + e)^5 + 10*(a^3 + a^2*b)*cosh(f*x + e)^3 + (9*a^2*b - 4*a*b^2 + b^3)*cosh(f*x +
e))*sinh(f*x + e)^3 + b^3 + 2*(3*a*b^2 - b^3)*cosh(f*x + e)^2 + 2*(14*a^2*b*cosh(f*x + e)^6 + 15*(a^3 + a^2*b)
*cosh(f*x + e)^4 + 3*a*b^2 - b^3 + 3*(9*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 - sqrt(2)*(a^2
*cosh(f*x + e)^6 + 6*a^2*cosh(f*x + e)*sinh(f*x + e)^5 + a^2*sinh(f*x + e)^6 + 3*a^2*cosh(f*x + e)^4 + 3*(5*a^
2*cosh(f*x + e)^2 + a^2)*sinh(f*x + e)^4 + 4*(5*a^2*cosh(f*x + e)^3 + 3*a^2*cosh(f*x + e))*sinh(f*x + e)^3 + (
4*a*b - b^2)*cosh(f*x + e)^2 + (15*a^2*cosh(f*x + e)^4 + 18*a^2*cosh(f*x + e)^2 + 4*a*b - b^2)*sinh(f*x + e)^2
+ b^2 + 2*(3*a^2*cosh(f*x + e)^5 + 6*a^2*cosh(f*x + e)^3 + (4*a*b - b^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt(b
)*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) + si
nh(f*x + e)^2)) + 4*(2*a^2*b*cosh(f*x + e)^7 + 3*(a^3 + a^2*b)*cosh(f*x + e)^5 + (9*a^2*b - 4*a*b^2 + b^3)*cos
h(f*x + e)^3 + (3*a*b^2 - b^3)*cosh(f*x + e))*sinh(f*x + e))/(cosh(f*x + e)^6 + 6*cosh(f*x + e)^5*sinh(f*x + e
) + 15*cosh(f*x + e)^4*sinh(f*x + e)^2 + 20*cosh(f*x + e)^3*sinh(f*x + e)^3 + 15*cosh(f*x + e)^2*sinh(f*x + e)
^4 + 6*cosh(f*x + e)*sinh(f*x + e)^5 + sinh(f*x + e)^6)) + 6*((a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(f*x + e)^
6 + 6*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(f*x + e)^5*sinh(f*x + e) + 15*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*c
osh(f*x + e)^4*sinh(f*x + e)^2 + 20*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(f*x + e)^3*sinh(f*x + e)^3 + 15*(a^
3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(f*x + e)^2*sinh(f*x + e)^4 + 6*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(f*x
+ e)*sinh(f*x + e)^5 + (a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*sinh(f*x + e)^6)*sqrt(b)*log(-(b*cosh(f*x + e)^4 + 4*
b*cosh(f*x + e)*sinh(f*x + e)^3 + b*sinh(f*x + e)^4 + 2*(a - b)*cosh(f*x + e)^2 + 2*(3*b*cosh(f*x + e)^2 + a -
b)*sinh(f*x + e)^2 - sqrt(2)*(cosh(f*x + e)^2 + 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 - 1)*sqrt(b)*
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*(b*cosh(f*x + e)^3 + (a - b)*cosh(f*x + e))*sinh(f*x + e) + b)/(cosh(f*x + e)^2 + 2*cosh(f*x
+ e)*sinh(f*x + e) + sinh(f*x + e)^2)) + sqrt(2)*(b^3*cosh(f*x + e)^10 + 10*b^3*cosh(f*x + e)*sinh(f*x + e)^9
+ b^3*sinh(f*x + e)^10 + (7*a*b^2 - 8*b^3)*cosh(f*x + e)^8 + (45*b^3*cosh(f*x + e)^2 + 7*a*b^2 - 8*b^3)*sinh(
f*x + e)^8 + 8*(15*b^3*cosh(f*x + e)^3 + (7*a*b^2 - 8*b^3)*cosh(f*x + e))*sinh(f*x + e)^7 + (6*a^2*b - 51*a*b^
2 + 37*b^3)*cosh(f*x + e)^6 + (210*b^3*cosh(f*x + e)^4 + 6*a^2*b - 51*a*b^2 + 37*b^3 + 28*(7*a*b^2 - 8*b^3)*co
sh(f*x + e)^2)*sinh(f*x + e)^6 + 2*(126*b^3*cosh(f*x + e)^5 + 28*(7*a*b^2 - 8*b^3)*cosh(f*x + e)^3 + 3*(6*a^2*
b - 51*a*b^2 + 37*b^3)*cosh(f*x + e))*sinh(f*x + e)^5 + (6*a^2*b - 51*a*b^2 + 37*b^3)*cosh(f*x + e)^4 + (210*b
^3*cosh(f*x + e)^6 + 70*(7*a*b^2 - 8*b^3)*cosh(f*x + e)^4 + 6*a^2*b - 51*a*b^2 + 37*b^3 + 15*(6*a^2*b - 51*a*b
^2 + 37*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 4*(30*b^3*cosh(f*x + e)^7 + 14*(7*a*b^2 - 8*b^3)*cosh(f*x + e)
^5 + 5*(6*a^2*b - 51*a*b^2 + 37*b^3)*cosh(f*x + e)^3 + (6*a^2*b - 51*a*b^2 + 37*b^3)*cosh(f*x + e))*sinh(f*x +
e)^3 + b^3 + (7*a*b^2 - 8*b^3)*cosh(f*x + e)^2 + (45*b^3*cosh(f*x + e)^8 + 28*(7*a*b^2 - 8*b^3)*cosh(f*x + e)
^6 + 15*(6*a^2*b - 51*a*b^2 + 37*b^3)*cosh(f*x + e)^4 + 7*a*b^2 - 8*b^3 + 6*(6*a^2*b - 51*a*b^2 + 37*b^3)*cosh
(f*x + e)^2)*sinh(f*x + e)^2 + 2*(5*b^3*cosh(f*x + e)^9 + 4*(7*a*b^2 - 8*b^3)*cosh(f*x + e)^7 + 3*(6*a^2*b - 5
1*a*b^2 + 37*b^3)*cosh(f*x + e)^5 + 2*(6*a^2*b - 51*a*b^2 + 37*b^3)*cosh(f*x + e)^3 + (7*a*b^2 - 8*b^3)*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)))/(b^2*f*cosh(f*x + e)^6 + 6*b^2*f*cosh(f*x + e)^5*sinh(f*x + e) + 15*b^2
*f*cosh(f*x + e)^4*sinh(f*x + e)^2 + 20*b^2*f*cosh(f*x + e)^3*sinh(f*x + e)^3 + 15*b^2*f*cosh(f*x + e)^2*sinh(
f*x + e)^4 + 6*b^2*f*cosh(f*x + e)*sinh(f*x + e)^5 + b^2*f*sinh(f*x + e)^6), 1/384*(12*((a^3 + 3*a^2*b - 9*a*b
^2 + 5*b^3)*cosh(f*x + e)^6 + 6*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(f*x + e)^5*sinh(f*x + e) + 15*(a^3 + 3*
a^2*b - 9*a*b^2 + 5*b^3)*cosh(f*x + e)^4*sinh(f*x + e)^2 + 20*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(f*x + e)^
3*sinh(f*x + e)^3 + 15*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(f*x + e)^2*sinh(f*x + e)^4 + 6*(a^3 + 3*a^2*b -
9*a*b^2 + 5*b^3)*cosh(f*x + e)*sinh(f*x + e)^5 + (a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*sinh(f*x + e)^6)*sqrt(-b)*a
rctan(sqrt(2)*(a*cosh(f*x + e)^2 + 2*a*cosh(f*x + e)*sinh(f*x + e) + a*sinh(f*x + e)^2 + b)*sqrt(-b)*sqrt((b*c
osh(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*cosh(f*x + e)^4 + 4*a*b*cosh(f*x + e)*sinh(f*x + e)^3 + a*b*sinh(f*x + e)^4 + (3*a*b - b^2)*cosh(f*x
+ e)^2 + (6*a*b*cosh(f*x + e)^2 + 3*a*b - b^2)*sinh(f*x + e)^2 + b^2 + 2*(2*a*b*cosh(f*x + e)^3 + (3*a*b - b^
2)*cosh(f*x + e))*sinh(f*x + e))) + 12*((a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(f*x + e)^6 + 6*(a^3 + 3*a^2*b -
9*a*b^2 + 5*b^3)*cosh(f*x + e)^5*sinh(f*x + e) + 15*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(f*x + e)^4*sinh(f*
x + e)^2 + 20*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(f*x + e)^3*sinh(f*x + e)^3 + 15*(a^3 + 3*a^2*b - 9*a*b^2
+ 5*b^3)*cosh(f*x + e)^2*sinh(f*x + e)^4 + 6*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(f*x + e)*sinh(f*x + e)^5 +
(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*sinh(f*x + e)^6)*sqrt(-b)*arctan(sqrt(2)*(cosh(f*x + e)^2 + 2*cosh(f*x + e)
*sinh(f*x + e) + sinh(f*x + e)^2 - 1)*sqrt(-b)*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)*(b^3*cosh(f*x + e)^10 + 10*b^3*c
osh(f*x + e)*sinh(f*x + e)^9 + b^3*sinh(f*x + e)^10 + (7*a*b^2 - 8*b^3)*cosh(f*x + e)^8 + (45*b^3*cosh(f*x + e
)^2 + 7*a*b^2 - 8*b^3)*sinh(f*x + e)^8 + 8*(15*b^3*cosh(f*x + e)^3 + (7*a*b^2 - 8*b^3)*cosh(f*x + e))*sinh(f*x
+ e)^7 + (6*a^2*b - 51*a*b^2 + 37*b^3)*cosh(f*x + e)^6 + (210*b^3*cosh(f*x + e)^4 + 6*a^2*b - 51*a*b^2 + 37*b
^3 + 28*(7*a*b^2 - 8*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^6 + 2*(126*b^3*cosh(f*x + e)^5 + 28*(7*a*b^2 - 8*b^3)
*cosh(f*x + e)^3 + 3*(6*a^2*b - 51*a*b^2 + 37*b^3)*cosh(f*x + e))*sinh(f*x + e)^5 + (6*a^2*b - 51*a*b^2 + 37*b
^3)*cosh(f*x + e)^4 + (210*b^3*cosh(f*x + e)^6 + 70*(7*a*b^2 - 8*b^3)*cosh(f*x + e)^4 + 6*a^2*b - 51*a*b^2 + 3
7*b^3 + 15*(6*a^2*b - 51*a*b^2 + 37*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 4*(30*b^3*cosh(f*x + e)^7 + 14*(7*
a*b^2 - 8*b^3)*cosh(f*x + e)^5 + 5*(6*a^2*b - 51*a*b^2 + 37*b^3)*cosh(f*x + e)^3 + (6*a^2*b - 51*a*b^2 + 37*b^
3)*cosh(f*x + e))*sinh(f*x + e)^3 + b^3 + (7*a*b^2 - 8*b^3)*cosh(f*x + e)^2 + (45*b^3*cosh(f*x + e)^8 + 28*(7*
a*b^2 - 8*b^3)*cosh(f*x + e)^6 + 15*(6*a^2*b - 51*a*b^2 + 37*b^3)*cosh(f*x + e)^4 + 7*a*b^2 - 8*b^3 + 6*(6*a^2
*b - 51*a*b^2 + 37*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*(5*b^3*cosh(f*x + e)^9 + 4*(7*a*b^2 - 8*b^3)*cosh
(f*x + e)^7 + 3*(6*a^2*b - 51*a*b^2 + 37*b^3)*cosh(f*x + e)^5 + 2*(6*a^2*b - 51*a*b^2 + 37*b^3)*cosh(f*x + e)^
3 + (7*a*b^2 - 8*b^3)*cosh(f*x + e))*sinh(f*x + e))*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(co
sh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)))/(b^2*f*cosh(f*x + e)^6 + 6*b^2*f*cosh(f*x +
e)^5*sinh(f*x + e) + 15*b^2*f*cosh(f*x + e)^4*sinh(f*x + e)^2 + 20*b^2*f*cosh(f*x + e)^3*sinh(f*x + e)^3 + 15
*b^2*f*cosh(f*x + e)^2*sinh(f*x + e)^4 + 6*b^2*f*cosh(f*x + e)*sinh(f*x + e)^5 + b^2*f*sinh(f*x + e)^6)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(f*x+e)**3*(a+b*sinh(f*x+e)**2)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} \sinh \left (f x + e\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

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