3.5.68 \(\int (a+b \sinh ^2(e+f x))^{3/2} \tanh ^5(e+f x) \, dx\) [468]

Optimal. Leaf size=232 \[ -\frac {\left (8 a^2-40 a b+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{8 \sqrt {a-b} f}+\frac {\left (8 a^2-40 a b+35 b^2\right ) \sqrt {a+b \sinh ^2(e+f x)}}{8 (a-b) f}+\frac {\left (8 a^2-40 a b+35 b^2\right ) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{24 (a-b)^2 f}+\frac {(8 a-9 b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{8 (a-b)^2 f}-\frac {\text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{4 (a-b) f} \]

[Out]

1/24*(8*a^2-40*a*b+35*b^2)*(a+b*sinh(f*x+e)^2)^(3/2)/(a-b)^2/f+1/8*(8*a-9*b)*sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)
^(5/2)/(a-b)^2/f-1/4*sech(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(5/2)/(a-b)/f-1/8*(8*a^2-40*a*b+35*b^2)*arctanh((a+b*si
nh(f*x+e)^2)^(1/2)/(a-b)^(1/2))/f/(a-b)^(1/2)+1/8*(8*a^2-40*a*b+35*b^2)*(a+b*sinh(f*x+e)^2)^(1/2)/(a-b)/f

________________________________________________________________________________________

Rubi [A]
time = 0.19, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3273, 91, 79, 52, 65, 214} \begin {gather*} \frac {\left (8 a^2-40 a b+35 b^2\right ) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{24 f (a-b)^2}+\frac {\left (8 a^2-40 a b+35 b^2\right ) \sqrt {a+b \sinh ^2(e+f x)}}{8 f (a-b)}-\frac {\left (8 a^2-40 a b+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{8 f \sqrt {a-b}}-\frac {\text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{4 f (a-b)}+\frac {(8 a-9 b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{8 f (a-b)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Sinh[e + f*x]^2)^(3/2)*Tanh[e + f*x]^5,x]

[Out]

-1/8*((8*a^2 - 40*a*b + 35*b^2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]])/(Sqrt[a - b]*f) + ((8*a^2 -
40*a*b + 35*b^2)*Sqrt[a + b*Sinh[e + f*x]^2])/(8*(a - b)*f) + ((8*a^2 - 40*a*b + 35*b^2)*(a + b*Sinh[e + f*x]^
2)^(3/2))/(24*(a - b)^2*f) + ((8*a - 9*b)*Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(5/2))/(8*(a - b)^2*f) - (Se
ch[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(5/2))/(4*(a - b)*f)

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 214

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

Rule 3273

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

Rubi steps

\begin {align*} \int \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh ^5(e+f x) \, dx &=\frac {\text {Subst}\left (\int \frac {x^2 (a+b x)^{3/2}}{(1+x)^3} \, dx,x,\sinh ^2(e+f x)\right )}{2 f}\\ &=-\frac {\text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{4 (a-b) f}+\frac {\text {Subst}\left (\int \frac {\left (\frac {1}{2} (-4 a+5 b)+2 (a-b) x\right ) (a+b x)^{3/2}}{(1+x)^2} \, dx,x,\sinh ^2(e+f x)\right )}{4 (a-b) f}\\ &=\frac {(8 a-9 b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{8 (a-b)^2 f}-\frac {\text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{4 (a-b) f}+\frac {\left (8 a^2-40 a b+35 b^2\right ) \text {Subst}\left (\int \frac {(a+b x)^{3/2}}{1+x} \, dx,x,\sinh ^2(e+f x)\right )}{16 (a-b)^2 f}\\ &=\frac {\left (8 a^2-40 a b+35 b^2\right ) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{24 (a-b)^2 f}+\frac {(8 a-9 b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{8 (a-b)^2 f}-\frac {\text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{4 (a-b) f}+\frac {\left (8 a^2-40 a b+35 b^2\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{1+x} \, dx,x,\sinh ^2(e+f x)\right )}{16 (a-b) f}\\ &=\frac {\left (8 a^2-40 a b+35 b^2\right ) \sqrt {a+b \sinh ^2(e+f x)}}{8 (a-b) f}+\frac {\left (8 a^2-40 a b+35 b^2\right ) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{24 (a-b)^2 f}+\frac {(8 a-9 b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{8 (a-b)^2 f}-\frac {\text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{4 (a-b) f}+\frac {\left (8 a^2-40 a b+35 b^2\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x}} \, dx,x,\sinh ^2(e+f x)\right )}{16 f}\\ &=\frac {\left (8 a^2-40 a b+35 b^2\right ) \sqrt {a+b \sinh ^2(e+f x)}}{8 (a-b) f}+\frac {\left (8 a^2-40 a b+35 b^2\right ) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{24 (a-b)^2 f}+\frac {(8 a-9 b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{8 (a-b)^2 f}-\frac {\text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{4 (a-b) f}+\frac {\left (8 a^2-40 a b+35 b^2\right ) \text {Subst}\left (\int \frac {1}{1-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^2(e+f x)}\right )}{8 b f}\\ &=-\frac {\left (8 a^2-40 a b+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{8 \sqrt {a-b} f}+\frac {\left (8 a^2-40 a b+35 b^2\right ) \sqrt {a+b \sinh ^2(e+f x)}}{8 (a-b) f}+\frac {\left (8 a^2-40 a b+35 b^2\right ) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{24 (a-b)^2 f}+\frac {(8 a-9 b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{8 (a-b)^2 f}-\frac {\text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{4 (a-b) f}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.53, size = 169, normalized size = 0.73 \begin {gather*} -\frac {-3 (8 a-9 b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}+6 (a-b) \text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}-\left (8 a^2-40 a b+35 b^2\right ) \left (-3 (a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )+\sqrt {a+b \sinh ^2(e+f x)} \left (4 a-3 b+b \sinh ^2(e+f x)\right )\right )}{24 (a-b)^2 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sinh[e + f*x]^2)^(3/2)*Tanh[e + f*x]^5,x]

[Out]

-1/24*(-3*(8*a - 9*b)*Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(5/2) + 6*(a - b)*Sech[e + f*x]^4*(a + b*Sinh[e
+ f*x]^2)^(5/2) - (8*a^2 - 40*a*b + 35*b^2)*(-3*(a - b)^(3/2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]]
+ Sqrt[a + b*Sinh[e + f*x]^2]*(4*a - 3*b + b*Sinh[e + f*x]^2)))/((a - b)^2*f)

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.53, size = 71, normalized size = 0.31

method	result	size
default	\(\frac {\mathit {`\,int/indef0`\,}\left (\frac {\left (\sinh ^{5}\left (f x +e \right )\right ) \left (b^{2} \left (\sinh ^{4}\left (f x +e \right )\right )+2 a b \left (\sinh ^{2}\left (f x +e \right )\right )+a^{2}\right )}{\cosh \left (f x +e \right )^{6} \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f}\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sinh(f*x+e)^2)^(3/2)*tanh(f*x+e)^5,x,method=_RETURNVERBOSE)

[Out]

`int/indef0`(sinh(f*x+e)^5*(b^2*sinh(f*x+e)^4+2*a*b*sinh(f*x+e)^2+a^2)/cosh(f*x+e)^6/(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((a+b*sinh(f*x+e)^2)^(3/2)*tanh(f*x+e)^5,x, algorithm="maxima")

[Out]

integrate((b*sinh(f*x + e)^2 + a)^(3/2)*tanh(f*x + e)^5, x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3092 vs. \(2 (208) = 416\).
time = 0.94, size = 6380, normalized size = 27.50 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sinh(f*x+e)^2)^(3/2)*tanh(f*x+e)^5,x, algorithm="fricas")

[Out]

[1/48*(3*((8*a^2 - 40*a*b + 35*b^2)*cosh(f*x + e)^11 + 11*(8*a^2 - 40*a*b + 35*b^2)*cosh(f*x + e)*sinh(f*x + e
)^10 + (8*a^2 - 40*a*b + 35*b^2)*sinh(f*x + e)^11 + 4*(8*a^2 - 40*a*b + 35*b^2)*cosh(f*x + e)^9 + (55*(8*a^2 -
40*a*b + 35*b^2)*cosh(f*x + e)^2 + 32*a^2 - 160*a*b + 140*b^2)*sinh(f*x + e)^9 + 3*(55*(8*a^2 - 40*a*b + 35*b
^2)*cosh(f*x + e)^3 + 12*(8*a^2 - 40*a*b + 35*b^2)*cosh(f*x + e))*sinh(f*x + e)^8 + 6*(8*a^2 - 40*a*b + 35*b^2
)*cosh(f*x + e)^7 + 6*(55*(8*a^2 - 40*a*b + 35*b^2)*cosh(f*x + e)^4 + 24*(8*a^2 - 40*a*b + 35*b^2)*cosh(f*x +
e)^2 + 8*a^2 - 40*a*b + 35*b^2)*sinh(f*x + e)^7 + 42*(11*(8*a^2 - 40*a*b + 35*b^2)*cosh(f*x + e)^5 + 8*(8*a^2
- 40*a*b + 35*b^2)*cosh(f*x + e)^3 + (8*a^2 - 40*a*b + 35*b^2)*cosh(f*x + e))*sinh(f*x + e)^6 + 4*(8*a^2 - 40*
a*b + 35*b^2)*cosh(f*x + e)^5 + 2*(231*(8*a^2 - 40*a*b + 35*b^2)*cosh(f*x + e)^6 + 252*(8*a^2 - 40*a*b + 35*b^
2)*cosh(f*x + e)^4 + 63*(8*a^2 - 40*a*b + 35*b^2)*cosh(f*x + e)^2 + 16*a^2 - 80*a*b + 70*b^2)*sinh(f*x + e)^5
+ 2*(165*(8*a^2 - 40*a*b + 35*b^2)*cosh(f*x + e)^7 + 252*(8*a^2 - 40*a*b + 35*b^2)*cosh(f*x + e)^5 + 105*(8*a^
2 - 40*a*b + 35*b^2)*cosh(f*x + e)^3 + 10*(8*a^2 - 40*a*b + 35*b^2)*cosh(f*x + e))*sinh(f*x + e)^4 + (8*a^2 -
40*a*b + 35*b^2)*cosh(f*x + e)^3 + (165*(8*a^2 - 40*a*b + 35*b^2)*cosh(f*x + e)^8 + 336*(8*a^2 - 40*a*b + 35*b
^2)*cosh(f*x + e)^6 + 210*(8*a^2 - 40*a*b + 35*b^2)*cosh(f*x + e)^4 + 40*(8*a^2 - 40*a*b + 35*b^2)*cosh(f*x +
e)^2 + 8*a^2 - 40*a*b + 35*b^2)*sinh(f*x + e)^3 + (55*(8*a^2 - 40*a*b + 35*b^2)*cosh(f*x + e)^9 + 144*(8*a^2 -
40*a*b + 35*b^2)*cosh(f*x + e)^7 + 126*(8*a^2 - 40*a*b + 35*b^2)*cosh(f*x + e)^5 + 40*(8*a^2 - 40*a*b + 35*b^
2)*cosh(f*x + e)^3 + 3*(8*a^2 - 40*a*b + 35*b^2)*cosh(f*x + e))*sinh(f*x + e)^2 + (11*(8*a^2 - 40*a*b + 35*b^2
)*cosh(f*x + e)^10 + 36*(8*a^2 - 40*a*b + 35*b^2)*cosh(f*x + e)^8 + 42*(8*a^2 - 40*a*b + 35*b^2)*cosh(f*x + e)
^6 + 20*(8*a^2 - 40*a*b + 35*b^2)*cosh(f*x + e)^4 + 3*(8*a^2 - 40*a*b + 35*b^2)*cosh(f*x + e)^2)*sinh(f*x + e)
)*sqrt(a - 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*(4*a - 3*b)*c
osh(f*x + e)^2 + 2*(3*b*cosh(f*x + e)^2 + 4*a - 3*b)*sinh(f*x + e)^2 - 4*sqrt(2)*sqrt(a - 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))*(co
sh(f*x + e) + sinh(f*x + e)) + 4*(b*cosh(f*x + e)^3 + (4*a - 3*b)*cosh(f*x + e))*sinh(f*x + e) + b)/(cosh(f*x
+ e)^4 + 4*cosh(f*x + e)*sinh(f*x + e)^3 + sinh(f*x + e)^4 + 2*(3*cosh(f*x + e)^2 + 1)*sinh(f*x + e)^2 + 2*cos
h(f*x + e)^2 + 4*(cosh(f*x + e)^3 + cosh(f*x + e))*sinh(f*x + e) + 1)) + 2*sqrt(2)*((a*b - b^2)*cosh(f*x + e)^
12 + 12*(a*b - b^2)*cosh(f*x + e)*sinh(f*x + e)^11 + (a*b - b^2)*sinh(f*x + e)^12 + 2*(8*a^2 - 25*a*b + 17*b^2
)*cosh(f*x + e)^10 + 2*(33*(a*b - b^2)*cosh(f*x + e)^2 + 8*a^2 - 25*a*b + 17*b^2)*sinh(f*x + e)^10 + 20*(11*(a
*b - b^2)*cosh(f*x + e)^3 + (8*a^2 - 25*a*b + 17*b^2)*cosh(f*x + e))*sinh(f*x + e)^9 + (112*a^2 - 335*a*b + 22
3*b^2)*cosh(f*x + e)^8 + (495*(a*b - b^2)*cosh(f*x + e)^4 + 90*(8*a^2 - 25*a*b + 17*b^2)*cosh(f*x + e)^2 + 112
*a^2 - 335*a*b + 223*b^2)*sinh(f*x + e)^8 + 8*(99*(a*b - b^2)*cosh(f*x + e)^5 + 30*(8*a^2 - 25*a*b + 17*b^2)*c
osh(f*x + e)^3 + (112*a^2 - 335*a*b + 223*b^2)*cosh(f*x + e))*sinh(f*x + e)^7 + 8*(18*a^2 - 59*a*b + 41*b^2)*c
osh(f*x + e)^6 + 4*(231*(a*b - b^2)*cosh(f*x + e)^6 + 105*(8*a^2 - 25*a*b + 17*b^2)*cosh(f*x + e)^4 + 7*(112*a
^2 - 335*a*b + 223*b^2)*cosh(f*x + e)^2 + 36*a^2 - 118*a*b + 82*b^2)*sinh(f*x + e)^6 + 8*(99*(a*b - b^2)*cosh(
f*x + e)^7 + 63*(8*a^2 - 25*a*b + 17*b^2)*cosh(f*x + e)^5 + 7*(112*a^2 - 335*a*b + 223*b^2)*cosh(f*x + e)^3 +
6*(18*a^2 - 59*a*b + 41*b^2)*cosh(f*x + e))*sinh(f*x + e)^5 + (112*a^2 - 335*a*b + 223*b^2)*cosh(f*x + e)^4 +
(495*(a*b - b^2)*cosh(f*x + e)^8 + 420*(8*a^2 - 25*a*b + 17*b^2)*cosh(f*x + e)^6 + 70*(112*a^2 - 335*a*b + 223
*b^2)*cosh(f*x + e)^4 + 120*(18*a^2 - 59*a*b + 41*b^2)*cosh(f*x + e)^2 + 112*a^2 - 335*a*b + 223*b^2)*sinh(f*x
+ e)^4 + 4*(55*(a*b - b^2)*cosh(f*x + e)^9 + 60*(8*a^2 - 25*a*b + 17*b^2)*cosh(f*x + e)^7 + 14*(112*a^2 - 335
*a*b + 223*b^2)*cosh(f*x + e)^5 + 40*(18*a^2 - 59*a*b + 41*b^2)*cosh(f*x + e)^3 + (112*a^2 - 335*a*b + 223*b^2
)*cosh(f*x + e))*sinh(f*x + e)^3 + 2*(8*a^2 - 25*a*b + 17*b^2)*cosh(f*x + e)^2 + 2*(33*(a*b - b^2)*cosh(f*x +
e)^10 + 45*(8*a^2 - 25*a*b + 17*b^2)*cosh(f*x + e)^8 + 14*(112*a^2 - 335*a*b + 223*b^2)*cosh(f*x + e)^6 + 60*(
18*a^2 - 59*a*b + 41*b^2)*cosh(f*x + e)^4 + 3*(112*a^2 - 335*a*b + 223*b^2)*cosh(f*x + e)^2 + 8*a^2 - 25*a*b +
17*b^2)*sinh(f*x + e)^2 + a*b - b^2 + 4*(3*(a*b - b^2)*cosh(f*x + e)^11 + 5*(8*a^2 - 25*a*b + 17*b^2)*cosh(f*
x + e)^9 + 2*(112*a^2 - 335*a*b + 223*b^2)*cosh(f*x + e)^7 + 12*(18*a^2 - 59*a*b + 41*b^2)*cosh(f*x + e)^5 + (
112*a^2 - 335*a*b + 223*b^2)*cosh(f*x + e)^3 + (8*a^2 - 25*a*b + 17*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)))/((a - b)*f*cosh(f*x + e)^11 + 11*(a - b...

________________________________________________________________________________________

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((a+b*sinh(f*x+e)**2)**(3/2)*tanh(f*x+e)**5,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4370 deep

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sinh(f*x+e)^2)^(3/2)*tanh(f*x+e)^5,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Evaluation time: 12.72Error: Bad Argument Type

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {tanh}\left (e+f\,x\right )}^5\,{\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(tanh(e + f*x)^5*(a + b*sinh(e + f*x)^2)^(3/2),x)

[Out]

int(tanh(e + f*x)^5*(a + b*sinh(e + f*x)^2)^(3/2), x)

________________________________________________________________________________________