3.474 \(\int (a+b \sinh ^2(e+f x))^{3/2} \tanh ^4(e+f x) \, dx\)

Optimal. Leaf size=305 \[ -\frac {\tanh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{3 f}+\frac {(a-2 b) \sinh ^2(e+f x) \tanh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{f}+\frac {8 (a-2 b) \tanh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}-\frac {(3 a-8 b) \sinh (e+f x) \cosh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}+\frac {(3 a-8 b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right )}{3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {8 (a-2 b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right )}{3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}} \]

[Out]

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

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Rubi [A]  time = 0.38, antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3196, 467, 577, 582, 531, 418, 492, 411} \[ -\frac {\tanh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{3 f}+\frac {(a-2 b) \sinh ^2(e+f x) \tanh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{f}+\frac {8 (a-2 b) \tanh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}-\frac {(3 a-8 b) \sinh (e+f x) \cosh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}+\frac {(3 a-8 b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right )}{3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {8 (a-2 b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right )}{3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sinh[e + f*x]^2)^(3/2)*Tanh[e + f*x]^4,x]

[Out]

-((3*a - 8*b)*Cosh[e + f*x]*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f) - (8*(a - 2*b)*EllipticE[ArcTan[S
inh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e +
f*x]^2))/a]) + ((3*a - 8*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2
])/(3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (8*(a - 2*b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e +
f*x])/(3*f) + ((a - 2*b)*Sinh[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/f - ((a + b*Sinh[e + f*x]
^2)^(3/2)*Tanh[e + f*x]^3)/(3*f)

Rule 411

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticE[ArcTan
[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticF[ArcT
an[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 467

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e^(n -
1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*n*(p + 1)), x] - Dist[e^n/(b*n*(p + 1)), Int[(e*x)^
(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m - n + 1) + d*(m + n*(q - 1) + 1)*x^n, x], x], x] /;
FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 0] && GtQ[m - n + 1, 0] &
& IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[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 531

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 577

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

Rule 582

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

Rule 3196

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_), 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)/(1 - ff^2*x^2)^((m + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2]
&&  !IntegerQ[p]

Rubi steps

\begin {align*} \int \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh ^4(e+f x) \, dx &=\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (a+b x^2\right )^{3/2}}{\left (1+x^2\right )^{5/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=-\frac {\left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh ^3(e+f x)}{3 f}+\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {a+b x^2} \left (3 a+6 b x^2\right )}{\left (1+x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 f}\\ &=\frac {(a-2 b) \sinh ^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{f}-\frac {\left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh ^3(e+f x)}{3 f}-\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (6 a (a-3 b)+3 (3 a-8 b) b x^2\right )}{\sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 f}\\ &=-\frac {(3 a-8 b) \cosh (e+f x) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}+\frac {(a-2 b) \sinh ^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{f}-\frac {\left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh ^3(e+f x)}{3 f}+\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {3 a (3 a-8 b) b+24 (a-2 b) b^2 x^2}{\sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{9 b f}\\ &=-\frac {(3 a-8 b) \cosh (e+f x) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}+\frac {(a-2 b) \sinh ^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{f}-\frac {\left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh ^3(e+f x)}{3 f}+\frac {\left (a (3 a-8 b) \sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 f}+\frac {\left (8 (a-2 b) b \sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 f}\\ &=-\frac {(3 a-8 b) \cosh (e+f x) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}+\frac {(3 a-8 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 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {8 (a-2 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 f}+\frac {(a-2 b) \sinh ^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{f}-\frac {\left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh ^3(e+f x)}{3 f}-\frac {\left (8 (a-2 b) \sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x^2}}{\left (1+x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 f}\\ &=-\frac {(3 a-8 b) \cosh (e+f x) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}-\frac {8 (a-2 b) 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 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(3 a-8 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 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {8 (a-2 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 f}+\frac {(a-2 b) \sinh ^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{f}-\frac {\left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh ^3(e+f x)}{3 f}\\ \end {align*}

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Mathematica [C]  time = 2.81, size = 224, normalized size = 0.73 \[ \frac {-\frac {\tanh (e+f x) \text {sech}^2(e+f x) \left (\left (64 a^2-160 a b+17 b^2\right ) \cosh (2 (e+f x))+32 a^2+2 b (6 a-17 b) \cosh (4 (e+f x))-108 a b-b^2 \cosh (6 (e+f x))+18 b^2\right )}{4 \sqrt {2}}+4 i a (5 a-8 b) \sqrt {\frac {2 a+b \cosh (2 (e+f x))-b}{a}} F\left (i (e+f x)\left |\frac {b}{a}\right .\right )-32 i a (a-2 b) \sqrt {\frac {2 a+b \cosh (2 (e+f x))-b}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )}{12 f \sqrt {2 a+b \cosh (2 (e+f x))-b}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sinh[e + f*x]^2)^(3/2)*Tanh[e + f*x]^4,x]

[Out]

((-32*I)*a*(a - 2*b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (4*I)*a*(5*a - 8*b)
*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] - ((32*a^2 - 108*a*b + 18*b^2 + (64*a^2 -
160*a*b + 17*b^2)*Cosh[2*(e + f*x)] + 2*(6*a - 17*b)*b*Cosh[4*(e + f*x)] - b^2*Cosh[6*(e + f*x)])*Sech[e + f*
x]^2*Tanh[e + f*x])/(4*Sqrt[2]))/(12*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])

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fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \tanh \left (f x + e\right )^{4}, x\right ) \]

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)^4,x, algorithm="fricas")

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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)^4,x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Evaluation time:
2.27Unable to divide, perhaps due to rounding error%%%{-786432,[8,13,12]%%%}+%%%{%%%{3932160,[1]%%%},[8,13,11]
%%%}+%%%{%%%{-7864320,[2]%%%},[8,13,10]%%%}+%%%{%%%{7864320,[3]%%%},[8,13,9]%%%}+%%%{%%%{-3932160,[4]%%%},[8,1
3,8]%%%}+%%%{%%%{786432,[5]%%%},[8,13,7]%%%}+%%%{%%{[6291456,0]:[1,0,%%%{-1,[1]%%%}]%%},[7,13,12]%%%}+%%%{%%{[
%%%{-31457280,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,13,11]%%%}+%%%{%%{[%%%{62914560,[2]%%%},0]:[1,0,%%%{-1,[1]
%%%}]%%},[7,13,10]%%%}+%%%{%%{[%%%{-62914560,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,13,9]%%%}+%%%{%%{[%%%{31457
280,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,13,8]%%%}+%%%{%%{[%%%{-6291456,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7
,13,7]%%%}+%%%{-12582912,[6,13,13]%%%}+%%%{%%%{53477376,[1]%%%},[6,13,12]%%%}+%%%{%%%{-78643200,[2]%%%},[6,13,
11]%%%}+%%%{%%%{31457280,[3]%%%},[6,13,10]%%%}+%%%{%%%{31457280,[4]%%%},[6,13,9]%%%}+%%%{%%%{-34603008,[5]%%%}
,[6,13,8]%%%}+%%%{%%%{9437184,[6]%%%},[6,13,7]%%%}+%%%{%%{[75497472,0]:[1,0,%%%{-1,[1]%%%}]%%},[5,13,13]%%%}+%
%%{%%{[%%%{-408944640,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,13,12]%%%}+%%%{%%{[%%%{912261120,[2]%%%},0]:[1,0,%
%%{-1,[1]%%%}]%%},[5,13,11]%%%}+%%%{%%{[%%%{-1069547520,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,13,10]%%%}+%%%{%
%{[%%%{692060160,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,13,9]%%%}+%%%{%%{[%%%{-232783872,[5]%%%},0]:[1,0,%%%{-1
,[1]%%%}]%%},[5,13,8]%%%}+%%%{%%{[%%%{31457280,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,13,7]%%%}+%%%{-75497472,[
4,13,14]%%%}+%%%{%%%{339738624,[1]%%%},[4,13,13]%%%}+%%%{%%%{-508035072,[2]%%%},[4,13,12]%%%}+%%%{%%%{86507520
,[3]%%%},[4,13,11]%%%}+%%%{%%%{581959680,[4]%%%},[4,13,10]%%%}+%%%{%%%{-695205888,[5]%%%},[4,13,9]%%%}+%%%{%%%
{328728576,[6]%%%},[4,13,8]%%%}+%%%{%%%{-58195968,[7]%%%},[4,13,7]%%%}+%%%{%%{[301989888,0]:[1,0,%%%{-1,[1]%%%
}]%%},[3,13,14]%%%}+%%%{%%{[%%%{-1862270976,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,13,13]%%%}+%%%{%%{[%%%{48758
78400,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,13,12]%%%}+%%%{%%{[%%%{-7014973440,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]
%%},[3,13,11]%%%}+%%%{%%{[%%%{5976883200,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,13,10]%%%}+%%%{%%{[%%%{-3007315
968,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,13,9]%%%}+%%%{%%{[%%%{824180736,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[
3,13,8]%%%}+%%%{%%{[%%%{-94371840,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,13,7]%%%}+%%%{-201326592,[2,13,15]%%%}
+%%%{%%%{1157627904,[1]%%%},[2,13,14]%%%}+%%%{%%%{-2654994432,[2]%%%},[2,13,13]%%%}+%%%{%%%{2872049664,[3]%%%}
,[2,13,12]%%%}+%%%{%%%{-959447040,[4]%%%},[2,13,11]%%%}+%%%{%%%{-1025507328,[5]%%%},[2,13,10]%%%}+%%%{%%%{1264
582656,[6]%%%},[2,13,9]%%%}+%%%{%%%{-537919488,[7]%%%},[2,13,8]%%%}+%%%{%%%{84934656,[8]%%%},[2,13,7]%%%}+%%%{
%%{[402653184,0]:[1,0,%%%{-1,[1]%%%}]%%},[1,13,15]%%%}+%%%{%%{[%%%{-2919235584,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]
%%},[1,13,14]%%%}+%%%{%%{[%%%{9235857408,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,13,13]%%%}+%%%{%%{[%%%{-1665348
4032,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,13,12]%%%}+%%%{%%{[%%%{18717081600,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%
%},[1,13,11]%%%}+%%%{%%{[%%%{-13425967104,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,13,10]%%%}+%%%{%%{[%%%{6002049
024,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,13,9]%%%}+%%%{%%{[%%%{-1528823808,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%}
,[1,13,8]%%%}+%%%{%%{[%%%{169869312,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,13,7]%%%}+%%%{-201326592,[0,13,16]%%
%}+%%%{%%%{1610612736,[1]%%%},[0,13,15]%%%}+%%%{%%%{-5712642048,[2]%%%},[0,13,14]%%%}+%%%{%%%{11790188544,[3]%
%%},[0,13,13]%%%}+%%%{%%%{-15603597312,[4]%%%},[0,13,12]%%%}+%%%{%%%{13731889152,[5]%%%},[0,13,11]%%%}+%%%{%%%
{-8035762176,[6]%%%},[0,13,10]%%%}+%%%{%%%{3015180288,[7]%%%},[0,13,9]%%%}+%%%{%%%{-658243584,[8]%%%},[0,13,8]
%%%}+%%%{%%%{63700992,[9]%%%},[0,13,7]%%%} / %%%{%%%{1,[2]%%%},[8,0,0]%%%}+%%%{%%{poly1[%%%{-8,[2]%%%},0]:[1,0
,%%%{-1,[1]%%%}]%%},[7,0,0]%%%}+%%%{%%%{16,[2]%%%},[6,0,1]%%%}+%%%{%%%{12,[3]%%%},[6,0,0]%%%}+%%%{%%{[%%%{-96,
[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,0,1]%%%}+%%%{%%{poly1[%%%{40,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,0,0]%
%%}+%%%{%%%{96,[2]%%%},[4,0,2]%%%}+%%%{%%%{48,[3]%%%},[4,0,1]%%%}+%%%{%%%{-74,[4]%%%},[4,0,0]%%%}+%%%{%%{poly1
[%%%{-384,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,2]%%%}+%%%{%%{[%%%{448,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3
,0,1]%%%}+%%%{%%{poly1[%%%{-120,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,0]%%%}+%%%{%%%{256,[2]%%%},[2,0,3]%%%}
+%%%{%%%{-192,[3]%%%},[2,0,2]%%%}+%%%{%%%{-144,[4]%%%},[2,0,1]%%%}+%%%{%%%{108,[5]%%%},[2,0,0]%%%}+%%%{%%{[%%%
{-512,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,0,3]%%%}+%%%{%%{poly1[%%%{1152,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},
[1,0,2]%%%}+%%%{%%{[%%%{-864,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,0,1]%%%}+%%%{%%{poly1[%%%{216,[5]%%%},0]:[1
,0,%%%{-1,[1]%%%}]%%},[1,0,0]%%%}+%%%{%%%{256,[2]%%%},[0,0,4]%%%}+%%%{%%%{-768,[3]%%%},[0,0,3]%%%}+%%%{%%%{864
,[4]%%%},[0,0,2]%%%}+%%%{%%%{-432,[5]%%%},[0,0,1]%%%}+%%%{%%%{81,[6]%%%},[0,0,0]%%%} Error: Bad Argument Value

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maple [A]  time = 0.31, size = 385, normalized size = 1.26 \[ \frac {\sqrt {-\frac {b}{a}}\, b^{2} \sinh \left (f x +e \right ) \left (\cosh ^{6}\left (f x +e \right )\right )+\left (-3 \sqrt {-\frac {b}{a}}\, a b +7 \sqrt {-\frac {b}{a}}\, b^{2}\right ) \left (\cosh ^{4}\left (f x +e \right )\right ) \sinh \left (f x +e \right )+\left (-4 \sqrt {-\frac {b}{a}}\, a^{2}+13 \sqrt {-\frac {b}{a}}\, a b -9 \sqrt {-\frac {b}{a}}\, b^{2}\right ) \left (\cosh ^{2}\left (f x +e \right )\right ) \sinh \left (f x +e \right )+\left (\sqrt {-\frac {b}{a}}\, a^{2}-2 \sqrt {-\frac {b}{a}}\, a b +\sqrt {-\frac {b}{a}}\, b^{2}\right ) \sinh \left (f x +e \right )+\sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \left (3 \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2}-16 \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b +16 \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}+8 \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b -16 \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}{3 \sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right )^{3} \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f} \]

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)^4,x)

[Out]

1/3*((-1/a*b)^(1/2)*b^2*sinh(f*x+e)*cosh(f*x+e)^6+(-3*(-1/a*b)^(1/2)*a*b+7*(-1/a*b)^(1/2)*b^2)*cosh(f*x+e)^4*s
inh(f*x+e)+(-4*(-1/a*b)^(1/2)*a^2+13*(-1/a*b)^(1/2)*a*b-9*(-1/a*b)^(1/2)*b^2)*cosh(f*x+e)^2*sinh(f*x+e)+((-1/a
*b)^(1/2)*a^2-2*(-1/a*b)^(1/2)*a*b+(-1/a*b)^(1/2)*b^2)*sinh(f*x+e)+(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x
+e)^2)^(1/2)*(3*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a^2-16*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),
(a/b)^(1/2))*a*b+16*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*b-16*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*b^2)*cosh(f*x+e)^2)/(-1/a*b)^(1/2)/co
sh(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(1/2)/f

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \tanh \left (f x + e\right )^{4}\,{d x} \]

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)^4,x, algorithm="maxima")

[Out]

integrate((b*sinh(f*x + e)^2 + a)^(3/2)*tanh(f*x + e)^4, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {tanh}\left (e+f\,x\right )}^4\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(e + f*x)^4*(a + b*sinh(e + f*x)^2)^(3/2),x)

[Out]

int(tanh(e + f*x)^4*(a + b*sinh(e + f*x)^2)^(3/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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)**4,x)

[Out]

Timed out

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