3.4.62 \(\int \cosh ^3(e+f x) (a+b \sinh ^2(e+f x))^{3/2} \, dx\) [362]

Optimal. Leaf size=157 \[ -\frac {a^2 (a-6 b) \tanh ^{-1}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{16 b^{3/2} f}-\frac {a (a-6 b) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{16 b f}-\frac {(a-6 b) \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{24 b f}+\frac {\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{6 b f} \]

[Out]

-1/16*a^2*(a-6*b)*arctanh(sinh(f*x+e)*b^(1/2)/(a+b*sinh(f*x+e)^2)^(1/2))/b^(3/2)/f-1/24*(a-6*b)*sinh(f*x+e)*(a
+b*sinh(f*x+e)^2)^(3/2)/b/f+1/6*sinh(f*x+e)*(a+b*sinh(f*x+e)^2)^(5/2)/b/f-1/16*a*(a-6*b)*sinh(f*x+e)*(a+b*sinh
(f*x+e)^2)^(1/2)/b/f

________________________________________________________________________________________

Rubi [A]
time = 0.09, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3269, 396, 201, 223, 212} \begin {gather*} -\frac {a^2 (a-6 b) \tanh ^{-1}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{16 b^{3/2} f}+\frac {\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{6 b f}-\frac {(a-6 b) \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{24 b f}-\frac {a (a-6 b) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{16 b f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*(a^2*(a - 6*b)*ArcTanh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(b^(3/2)*f) - (a*(a - 6*b)*
Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(16*b*f) - ((a - 6*b)*Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^(3/2))/
(24*b*f) + (Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^(5/2))/(6*b*f)

Rule 201

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 212

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

Rule 223

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 396

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 3269

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e +
f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.88, size = 149, normalized size = 0.95 \begin {gather*} \frac {\sqrt {a+b \sinh ^2(e+f x)} \left (-3 a^{3/2} (a-6 b) \sinh ^{-1}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )+\sqrt {b} \sinh (e+f x) \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}} \left (3 a (a+10 b)+2 b (7 a+6 b) \sinh ^2(e+f x)+8 b^2 \sinh ^4(e+f x)\right )\right )}{48 b^{3/2} f \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*Sinh[e + f*x]^2]*(-3*a^(3/2)*(a - 6*b)*ArcSinh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]] + Sqrt[b]*Sinh[e +
f*x]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a]*(3*a*(a + 10*b) + 2*b*(7*a + 6*b)*Sinh[e + f*x]^2 + 8*b^2*Sinh[e + f*x]^
4)))/(48*b^(3/2)*f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.29, size = 77, normalized size = 0.49

method	result	size
default	\(\frac {\mathit {`\,int/indef0`\,}\left (\frac {b^{2} \left (\sinh ^{6}\left (f x +e \right )\right )+\left (2 a b +b^{2}\right ) \left (\sinh ^{4}\left (f x +e \right )\right )+\left (a^{2}+2 a b \right ) \left (\sinh ^{2}\left (f x +e \right )\right )+a^{2}}{\sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

`int/indef0`((b^2*sinh(f*x+e)^6+(2*a*b+b^2)*sinh(f*x+e)^4+(a^2+2*a*b)*sinh(f*x+e)^2+a^2)/(a+b*sinh(f*x+e)^2)^(
1/2),sinh(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(cosh(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)*cosh(f*x + e)^3, x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1846 vs. \(2 (137) = 274\).
time = 0.50, size = 4603, normalized size = 29.32 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/384*(6*((a^3 - 6*a^2*b)*cosh(f*x + e)^6 + 6*(a^3 - 6*a^2*b)*cosh(f*x + e)^5*sinh(f*x + e) + 15*(a^3 - 6*a^
2*b)*cosh(f*x + e)^4*sinh(f*x + e)^2 + 20*(a^3 - 6*a^2*b)*cosh(f*x + e)^3*sinh(f*x + e)^3 + 15*(a^3 - 6*a^2*b)
*cosh(f*x + e)^2*sinh(f*x + e)^4 + 6*(a^3 - 6*a^2*b)*cosh(f*x + e)*sinh(f*x + e)^5 + (a^3 - 6*a^2*b)*sinh(f*x
+ e)^6)*sqrt(b)*log(-((a^2*b - 2*a*b^2 + b^3)*cosh(f*x + e)^8 + 8*(a^2*b - 2*a*b^2 + b^3)*cosh(f*x + e)*sinh(f
*x + e)^7 + (a^2*b - 2*a*b^2 + b^3)*sinh(f*x + e)^8 + 2*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e)^6 + 2*
(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3 + 14*(a^2*b - 2*a*b^2 + b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^6 + 4*(14*(a^2*b
- 2*a*b^2 + b^3)*cosh(f*x + e)^3 + 3*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e))*sinh(f*x + e)^5 + (9*a^2
*b - 14*a*b^2 + 6*b^3)*cosh(f*x + e)^4 + (70*(a^2*b - 2*a*b^2 + b^3)*cosh(f*x + e)^4 + 9*a^2*b - 14*a*b^2 + 6*
b^3 + 30*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 4*(14*(a^2*b - 2*a*b^2 + b^3)*co
sh(f*x + e)^5 + 10*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e)^3 + (9*a^2*b - 14*a*b^2 + 6*b^3)*cosh(f*x +
e))*sinh(f*x + e)^3 + b^3 + 2*(3*a*b^2 - 2*b^3)*cosh(f*x + e)^2 + 2*(14*(a^2*b - 2*a*b^2 + b^3)*cosh(f*x + e)
^6 + 15*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e)^4 + 3*a*b^2 - 2*b^3 + 3*(9*a^2*b - 14*a*b^2 + 6*b^3)*c
osh(f*x + e)^2)*sinh(f*x + e)^2 + sqrt(2)*((a^2 - 2*a*b + b^2)*cosh(f*x + e)^6 + 6*(a^2 - 2*a*b + b^2)*cosh(f*
x + e)*sinh(f*x + e)^5 + (a^2 - 2*a*b + b^2)*sinh(f*x + e)^6 - 3*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^4 + 3*(5*(a
^2 - 2*a*b + b^2)*cosh(f*x + e)^2 - a^2 + 2*a*b - b^2)*sinh(f*x + e)^4 + 4*(5*(a^2 - 2*a*b + b^2)*cosh(f*x + e
)^3 - 3*(a^2 - 2*a*b + b^2)*cosh(f*x + e))*sinh(f*x + e)^3 - (4*a*b - 3*b^2)*cosh(f*x + e)^2 + (15*(a^2 - 2*a*
b + b^2)*cosh(f*x + e)^4 - 18*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^2 - 4*a*b + 3*b^2)*sinh(f*x + e)^2 - b^2 + 2*(
3*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^5 - 6*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^3 - (4*a*b - 3*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) + sinh(f*x + e)^2)) + 4*(2*(a^2*b - 2*a*b^2 + b^3)*cosh(f*x + e)^7 + 3*(a^3 - 4*a^2*b + 5*a*
b^2 - 2*b^3)*cosh(f*x + e)^5 + (9*a^2*b - 14*a*b^2 + 6*b^3)*cosh(f*x + e)^3 + (3*a*b^2 - 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*c
osh(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 - 6*a^2*b)*cosh(f*x + e)^6 + 6*(a^3 - 6*a^2*b)*cosh(f*x + e)^5*sinh(f*x + e) + 15*(a^3 -
6*a^2*b)*cosh(f*x + e)^4*sinh(f*x + e)^2 + 20*(a^3 - 6*a^2*b)*cosh(f*x + e)^3*sinh(f*x + e)^3 + 15*(a^3 - 6*a
^2*b)*cosh(f*x + e)^2*sinh(f*x + e)^4 + 6*(a^3 - 6*a^2*b)*cosh(f*x + e)*sinh(f*x + e)^5 + (a^3 - 6*a^2*b)*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*cosh
(f*x + e)^2 + 2*(3*b*cosh(f*x + e)^2 + a)*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*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 + b^3)*cosh(f*x + e)^8 + (45*b^3*cosh(f*x +
e)^2 + 7*a*b^2 + b^3)*sinh(f*x + e)^8 + 8*(15*b^3*cosh(f*x + e)^3 + (7*a*b^2 + b^3)*cosh(f*x + e))*sinh(f*x +
e)^7 + (6*a^2*b + 39*a*b^2 - 8*b^3)*cosh(f*x + e)^6 + (210*b^3*cosh(f*x + e)^4 + 6*a^2*b + 39*a*b^2 - 8*b^3 +
28*(7*a*b^2 + 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 + b^3)*cosh(f*
x + e)^3 + 3*(6*a^2*b + 39*a*b^2 - 8*b^3)*cosh(f*x + e))*sinh(f*x + e)^5 - (6*a^2*b + 39*a*b^2 - 8*b^3)*cosh(f
*x + e)^4 + (210*b^3*cosh(f*x + e)^6 + 70*(7*a*b^2 + b^3)*cosh(f*x + e)^4 - 6*a^2*b - 39*a*b^2 + 8*b^3 + 15*(6
*a^2*b + 39*a*b^2 - 8*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 + b^3)*c
osh(f*x + e)^5 + 5*(6*a^2*b + 39*a*b^2 - 8*b^3)*cosh(f*x + e)^3 - (6*a^2*b + 39*a*b^2 - 8*b^3)*cosh(f*x + e))*
sinh(f*x + e)^3 - b^3 - (7*a*b^2 + b^3)*cosh(f*x + e)^2 + (45*b^3*cosh(f*x + e)^8 + 28*(7*a*b^2 + b^3)*cosh(f*
x + e)^6 + 15*(6*a^2*b + 39*a*b^2 - 8*b^3)*cosh(f*x + e)^4 - 7*a*b^2 - b^3 - 6*(6*a^2*b + 39*a*b^2 - 8*b^3)*co
sh(f*x + e)^2)*sinh(f*x + e)^2 + 2*(5*b^3*cosh(f*x + e)^9 + 4*(7*a*b^2 + b^3)*cosh(f*x + e)^7 + 3*(6*a^2*b + 3
9*a*b^2 - 8*b^3)*cosh(f*x + e)^5 - 2*(6*a^2*b + 39*a*b^2 - 8*b^3)*cosh(f*x + e)^3 - (7*a*b^2 + 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*c
osh(f*x + e)^4*sinh(f*x + e)^2 + 20*b^2*f*cosh(...

________________________________________________________________________________________

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4847 deep

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1546 vs. \(2 (137) = 274\).
time = 1.03, size = 1546, normalized size = 9.85 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(((b*e^(2*f*x + 10*e) + (7*a*b^2*e^(14*e) + b^3*e^(14*e))*e^(-6*e)/b^2)*e^(2*f*x) + (6*a^2*b*e^(12*e) +
39*a*b^2*e^(12*e) - 8*b^3*e^(12*e))*e^(-6*e)/b^2)*sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x
+ 2*e) + b) - 24*(a^3*e^(6*e) - 6*a^2*b*e^(6*e))*arctan(-(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4
*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))/sqrt(-b))/(sqrt(-b)*b) + 12*(a^3*sqrt(b)*e^(6*e) - 6*a^2*b^(3/2
)*e^(6*e))*log(abs(-(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2
*e) + b))*b - 2*a*sqrt(b) + b^(3/2)))/b^2 + 2*(12*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2
*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^5*a^3*e^(6*e) + 72*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) +
4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^5*a^2*b*e^(6*e) - 48*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f
*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^5*a*b^2*e^(6*e) + 9*(sqrt(b)*e^(2*f*x + 2*e) - sqr
t(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^5*b^3*e^(6*e) + 48*(sqrt(b)*e^(2*f*x + 2
*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^4*a^2*b^(3/2)*e^(6*e) + 24*(sqr
t(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^4*a*b^(5/2)*e^
(6*e) - 9*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^
4*b^(7/2)*e^(6*e) + 32*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x
+ 2*e) + b))^3*a^3*b*e^(6*e) - 192*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2
*b*e^(2*f*x + 2*e) + b))^3*a^2*b^2*e^(6*e) + 132*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*
f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^3*a*b^3*e^(6*e) - 22*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e)
+ 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^3*b^4*e^(6*e) - 108*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f
*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^2*a*b^(7/2)*e^(6*e) + 30*(sqrt(b)*e^(2*f*x + 2*e)
- sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^2*b^(9/2)*e^(6*e) - 12*(sqrt(b)*e^(
2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))*a^3*b^2*e^(6*e) + 72*(
sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))*a^2*b^3*e^(
6*e) - 36*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))*
a*b^4*e^(6*e) - 3*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e
) + b))*b^5*e^(6*e) + 36*a*b^(9/2)*e^(6*e) - 5*b^(11/2)*e^(6*e))/(((sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x
+ 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^2 - b)^3*b))*e^(-6*e)/f

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {cosh}\left (e+f\,x\right )}^3\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________