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

Optimal. Leaf size=130 \[ -\frac{(a-b) (a+3 b) \tanh ^{-1}\left (\frac{\sqrt{b} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right )}{8 b^{3/2} f}+\frac{\cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}}{4 b f}-\frac{(a+3 b) \cosh (e+f x) \sqrt{a+b \cosh ^2(e+f x)-b}}{8 b f} \]

[Out]

-((a - b)*(a + 3*b)*ArcTanh[(Sqrt[b]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(8*b^(3/2)*f) - ((a + 3*
b)*Cosh[e + f*x]*Sqrt[a - b + b*Cosh[e + f*x]^2])/(8*b*f) + (Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^(3/2))/
(4*b*f)

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Rubi [A]  time = 0.148348, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ -\frac{(a-b) (a+3 b) \tanh ^{-1}\left (\frac{\sqrt{b} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right )}{8 b^{3/2} f}+\frac{\cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}}{4 b f}-\frac{(a+3 b) \cosh (e+f x) \sqrt{a+b \cosh ^2(e+f x)-b}}{8 b f} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[e + f*x]^3*Sqrt[a + b*Sinh[e + f*x]^2],x]

[Out]

-((a - b)*(a + 3*b)*ArcTanh[(Sqrt[b]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(8*b^(3/2)*f) - ((a + 3*
b)*Cosh[e + f*x]*Sqrt[a - b + b*Cosh[e + f*x]^2])/(8*b*f) + (Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^(3/2))/
(4*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) \sqrt{a+b \sinh ^2(e+f x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \sqrt{a-b+b x^2} \, dx,x,\cosh (e+f x)\right )}{f}\\ &=\frac{\cosh (e+f x) \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}}{4 b f}-\frac{(a+3 b) \operatorname{Subst}\left (\int \sqrt{a-b+b x^2} \, dx,x,\cosh (e+f x)\right )}{4 b f}\\ &=-\frac{(a+3 b) \cosh (e+f x) \sqrt{a-b+b \cosh ^2(e+f x)}}{8 b f}+\frac{\cosh (e+f x) \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}}{4 b f}-\frac{((a-b) (a+3 b)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-b+b x^2}} \, dx,x,\cosh (e+f x)\right )}{8 b f}\\ &=-\frac{(a+3 b) \cosh (e+f x) \sqrt{a-b+b \cosh ^2(e+f x)}}{8 b f}+\frac{\cosh (e+f x) \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}}{4 b f}-\frac{((a-b) (a+3 b)) \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 )}{8 b f}\\ &=-\frac{(a-b) (a+3 b) \tanh ^{-1}\left (\frac{\sqrt{b} \cosh (e+f x)}{\sqrt{a-b+b \cosh ^2(e+f x)}}\right )}{8 b^{3/2} f}-\frac{(a+3 b) \cosh (e+f x) \sqrt{a-b+b \cosh ^2(e+f x)}}{8 b f}+\frac{\cosh (e+f x) \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}}{4 b f}\\ \end{align*}

Mathematica [A]  time = 0.651416, size = 114, normalized size = 0.88 \[ \frac{\frac{(b-a) (a+3 b) \log \left (\sqrt{2 a+b \cosh (2 (e+f x))-b}+\sqrt{2} \sqrt{b} \cosh (e+f x)\right )}{b^{3/2}}+\frac{\cosh (e+f x) \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b} (a+b \cosh (2 (e+f x))-4 b)}{2 b}}{8 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[e + f*x]^3*Sqrt[a + b*Sinh[e + f*x]^2],x]

[Out]

((Cosh[e + f*x]*(a - 4*b + b*Cosh[2*(e + f*x)])*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]])/(2*b) + ((-a + b)*(a
+ 3*b)*Log[Sqrt[2]*Sqrt[b]*Cosh[e + f*x] + Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]])/b^(3/2))/(8*f)

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Maple [B]  time = 0.114, size = 339, normalized size = 2.6 \begin{align*}{\frac{1}{16\,f\cosh \left ( fx+e \right ) }\sqrt{ \left ( a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2} \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}} \left ( 4\,\sqrt{b \left ( \cosh \left ( fx+e \right ) \right ) ^{4}+ \left ( a-b \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{b}^{5/2} \left ( \cosh \left ( fx+e \right ) \right ) ^{2}-10\,\sqrt{b \left ( \cosh \left ( fx+e \right ) \right ) ^{4}+ \left ( a-b \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{b}^{5/2}+2\,a\sqrt{b \left ( \cosh \left ( fx+e \right ) \right ) ^{4}+ \left ( a-b \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{b}^{3/2}-\ln \left ({\frac{1}{2} \left ( 2\,b \left ( \cosh \left ( fx+e \right ) \right ) ^{2}+2\,\sqrt{b \left ( \cosh \left ( fx+e \right ) \right ) ^{4}+ \left ( a-b \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}\sqrt{b}+a-b \right ){\frac{1}{\sqrt{b}}}} \right ){a}^{2}b-2\,a\ln \left ( 1/2\,{\frac{2\,b \left ( \cosh \left ( fx+e \right ) \right ) ^{2}+2\,\sqrt{b \left ( \cosh \left ( fx+e \right ) \right ) ^{4}+ \left ( a-b \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}\sqrt{b}+a-b}{\sqrt{b}}} \right ){b}^{2}+3\,{b}^{3}\ln \left ( 1/2\,{\frac{2\,b \left ( \cosh \left ( fx+e \right ) \right ) ^{2}+2\,\sqrt{b \left ( \cosh \left ( fx+e \right ) \right ) ^{4}+ \left ( a-b \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}\sqrt{b}+a-b}{\sqrt{b}}} \right ) \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2}}}}} \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)^(1/2),x)

[Out]

1/16*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*(4*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)*b^(5/2)*cosh(f*x
+e)^2-10*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)*b^(5/2)+2*a*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)*b
^(3/2)-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*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))*b^2+3*b^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)))/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 \sqrt{b \sinh \left (f x + e\right )^{2} + a} \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)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 3.89635, size = 7794, normalized size = 59.95 \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)^(1/2),x, algorithm="fricas")

[Out]

[-1/64*(2*((a^2 + 2*a*b - 3*b^2)*cosh(f*x + e)^4 + 4*(a^2 + 2*a*b - 3*b^2)*cosh(f*x + e)^3*sinh(f*x + e) + 6*(
a^2 + 2*a*b - 3*b^2)*cosh(f*x + e)^2*sinh(f*x + e)^2 + 4*(a^2 + 2*a*b - 3*b^2)*cosh(f*x + e)*sinh(f*x + e)^3 +
(a^2 + 2*a*b - 3*b^2)*sinh(f*x + e)^4)*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)*si
nh(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))*si
nh(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)*s
inh(f*x + e) + sinh(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)*cosh(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)) + 2*((a^2 + 2*a*b - 3*b^2)*cosh(f*x
+ e)^4 + 4*(a^2 + 2*a*b - 3*b^2)*cosh(f*x + e)^3*sinh(f*x + e) + 6*(a^2 + 2*a*b - 3*b^2)*cosh(f*x + e)^2*sinh(
f*x + e)^2 + 4*(a^2 + 2*a*b - 3*b^2)*cosh(f*x + e)*sinh(f*x + e)^3 + (a^2 + 2*a*b - 3*b^2)*sinh(f*x + e)^4)*sq
rt(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^2*cosh(f*x + e)^6 + 6*b
^2*cosh(f*x + e)*sinh(f*x + e)^5 + b^2*sinh(f*x + e)^6 + (2*a*b - 7*b^2)*cosh(f*x + e)^4 + (15*b^2*cosh(f*x +
e)^2 + 2*a*b - 7*b^2)*sinh(f*x + e)^4 + 4*(5*b^2*cosh(f*x + e)^3 + (2*a*b - 7*b^2)*cosh(f*x + e))*sinh(f*x + e
)^3 + (2*a*b - 7*b^2)*cosh(f*x + e)^2 + (15*b^2*cosh(f*x + e)^4 + 6*(2*a*b - 7*b^2)*cosh(f*x + e)^2 + 2*a*b -
7*b^2)*sinh(f*x + e)^2 + b^2 + 2*(3*b^2*cosh(f*x + e)^5 + 2*(2*a*b - 7*b^2)*cosh(f*x + e)^3 + (2*a*b - 7*b^2)*
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)^4 + 4*b^2*f*cosh(f*x + e)^3*sinh(f*x + e) +
6*b^2*f*cosh(f*x + e)^2*sinh(f*x + e)^2 + 4*b^2*f*cosh(f*x + e)*sinh(f*x + e)^3 + b^2*f*sinh(f*x + e)^4), 1/64
*(4*((a^2 + 2*a*b - 3*b^2)*cosh(f*x + e)^4 + 4*(a^2 + 2*a*b - 3*b^2)*cosh(f*x + e)^3*sinh(f*x + e) + 6*(a^2 +
2*a*b - 3*b^2)*cosh(f*x + e)^2*sinh(f*x + e)^2 + 4*(a^2 + 2*a*b - 3*b^2)*cosh(f*x + e)*sinh(f*x + e)^3 + (a^2
+ 2*a*b - 3*b^2)*sinh(f*x + e)^4)*sqrt(-b)*arctan(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*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))/(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))) + 4*((a^2 + 2*a*b - 3*b^2)*cosh(
f*x + e)^4 + 4*(a^2 + 2*a*b - 3*b^2)*cosh(f*x + e)^3*sinh(f*x + e) + 6*(a^2 + 2*a*b - 3*b^2)*cosh(f*x + e)^2*s
inh(f*x + e)^2 + 4*(a^2 + 2*a*b - 3*b^2)*cosh(f*x + e)*sinh(f*x + e)^3 + (a^2 + 2*a*b - 3*b^2)*sinh(f*x + e)^4
)*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)*sqr
t((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))*si
nh(f*x + e) + b)) + sqrt(2)*(b^2*cosh(f*x + e)^6 + 6*b^2*cosh(f*x + e)*sinh(f*x + e)^5 + b^2*sinh(f*x + e)^6 +
(2*a*b - 7*b^2)*cosh(f*x + e)^4 + (15*b^2*cosh(f*x + e)^2 + 2*a*b - 7*b^2)*sinh(f*x + e)^4 + 4*(5*b^2*cosh(f*
x + e)^3 + (2*a*b - 7*b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + (2*a*b - 7*b^2)*cosh(f*x + e)^2 + (15*b^2*cosh(f*x
+ e)^4 + 6*(2*a*b - 7*b^2)*cosh(f*x + e)^2 + 2*a*b - 7*b^2)*sinh(f*x + e)^2 + b^2 + 2*(3*b^2*cosh(f*x + e)^5
+ 2*(2*a*b - 7*b^2)*cosh(f*x + e)^3 + (2*a*b - 7*b^2)*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)^4 + 4*b^2*f*cosh(f*x + e)^3*sinh(f*x + e) + 6*b^2*f*cosh(f*x + e)^2*sinh(f*x + e)^2 + 4*b^2*f*cosh(f
*x + e)*sinh(f*x + e)^3 + b^2*f*sinh(f*x + e)^4)]

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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)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sinh \left (f x + e\right )^{2} + a} \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)^(1/2),x, algorithm="giac")

[Out]

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