3.463 \(\int \sqrt {a+b \sinh ^2(e+f x)} \tanh ^4(e+f x) \, dx\)

Optimal. Leaf size=292 \[ -\frac {\tanh ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}+\frac {(7 a-8 b) \tanh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f (a-b)}-\frac {(3 a-4 b) \tanh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f (a-b)}+\frac {(3 a-4 b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right )}{3 f (a-b) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {(7 a-8 b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right )}{3 f (a-b) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}} \]

[Out]

-1/3*(7*a-8*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)/(a-b)/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+1/
3*(3*a-4*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)/(a-b)/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+1/3*(
7*a-8*b)*(a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)/(a-b)/f-1/3*(3*a-4*b)*(a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)/(a-
b)/f-1/3*(a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)^3/f

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Rubi [A]  time = 0.31, antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3196, 467, 578, 531, 418, 492, 411} \[ -\frac {\tanh ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}+\frac {(7 a-8 b) \tanh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f (a-b)}-\frac {(3 a-4 b) \tanh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f (a-b)}+\frac {(3 a-4 b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right )}{3 f (a-b) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {(7 a-8 b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right )}{3 f (a-b) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x]^4,x]

[Out]

-((7*a - 8*b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*(a - b)*
f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((3*a - 4*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*
Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*(a - b)*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (
(7*a - 8*b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*(a - b)*f) - ((3*a - 4*b)*Sqrt[a + b*Sinh[e + f*x]^2
]*Tanh[e + f*x])/(3*(a - b)*f) - (Sqrt[a + b*Sinh[e + f*x]^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 578

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

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 \sqrt {a+b \sinh ^2(e+f x)} \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 \sqrt {a+b x^2}}{\left (1+x^2\right )^{5/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=-\frac {\sqrt {a+b \sinh ^2(e+f x)} \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 (3 a+4 b x^2\right )}{\left (1+x^2\right )^{3/2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 f}\\ &=-\frac {(3 a-4 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 (a-b) f}-\frac {\sqrt {a+b \sinh ^2(e+f x)} \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 {a (3 a-4 b)+(7 a-8 b) b x^2}{\sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 (-a+b) f}\\ &=-\frac {(3 a-4 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 (a-b) f}-\frac {\sqrt {a+b \sinh ^2(e+f x)} \tanh ^3(e+f x)}{3 f}-\frac {\left (a (3 a-4 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 (-a+b) f}-\frac {\left ((7 a-8 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 (-a+b) f}\\ &=\frac {(3 a-4 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 (a-b) f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(7 a-8 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 (a-b) f}-\frac {(3 a-4 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 (a-b) f}-\frac {\sqrt {a+b \sinh ^2(e+f x)} \tanh ^3(e+f x)}{3 f}+\frac {\left ((7 a-8 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 (-a+b) f}\\ &=-\frac {(7 a-8 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 (a-b) f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(3 a-4 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 (a-b) f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(7 a-8 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 (a-b) f}-\frac {(3 a-4 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 (a-b) f}-\frac {\sqrt {a+b \sinh ^2(e+f x)} \tanh ^3(e+f x)}{3 f}\\ \end {align*}

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Mathematica [C]  time = 2.03, size = 214, normalized size = 0.73 \[ \frac {-\frac {\tanh (e+f x) \text {sech}^2(e+f x) \left (4 \left (4 a^2-6 a b+b^2\right ) \cosh (2 (e+f x))+8 a^2+b (4 a-5 b) \cosh (4 (e+f x))-12 a b+b^2\right )}{2 \sqrt {2}}+8 i a (a-b) \sqrt {\frac {2 a+b \cosh (2 (e+f x))-b}{a}} F\left (i (e+f x)\left |\frac {b}{a}\right .\right )-2 i a (7 a-8 b) \sqrt {\frac {2 a+b \cosh (2 (e+f x))-b}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )}{6 f (a-b) \sqrt {2 a+b \cosh (2 (e+f x))-b}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x]^4,x]

[Out]

((-2*I)*a*(7*a - 8*b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (8*I)*a*(a - b)*Sq
rt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] - ((8*a^2 - 12*a*b + b^2 + 4*(4*a^2 - 6*a*b
+ b^2)*Cosh[2*(e + f*x)] + (4*a - 5*b)*b*Cosh[4*(e + f*x)])*Sech[e + f*x]^2*Tanh[e + f*x])/(2*Sqrt[2]))/(6*(a
- b)*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])

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fricas [F]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \sinh \left (f x + e\right )^{2} + a} \tanh \left (f x + e\right )^{4}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)^4,x, algorithm="fricas")

[Out]

integral(sqrt(b*sinh(f*x + e)^2 + a)*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)^(1/2)*tanh(f*x+e)^4,x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Evaluation time:
1.36Unable to divide, perhaps due to rounding error%%%{65536,[8,11,11]%%%}+%%%{%%%{-327680,[1]%%%},[8,11,10]%%
%}+%%%{%%%{655360,[2]%%%},[8,11,9]%%%}+%%%{%%%{-655360,[3]%%%},[8,11,8]%%%}+%%%{%%%{327680,[4]%%%},[8,11,7]%%%
}+%%%{%%%{-65536,[5]%%%},[8,11,6]%%%}+%%%{%%{[-524288,0]:[1,0,%%%{-1,[1]%%%}]%%},[7,11,11]%%%}+%%%{%%{[%%%{262
1440,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,11,10]%%%}+%%%{%%{[%%%{-5242880,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},
[7,11,9]%%%}+%%%{%%{[%%%{5242880,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,11,8]%%%}+%%%{%%{[%%%{-2621440,[4]%%%},
0]:[1,0,%%%{-1,[1]%%%}]%%},[7,11,7]%%%}+%%%{%%{[%%%{524288,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,11,6]%%%}+%%%
{1048576,[6,11,12]%%%}+%%%{%%%{-4456448,[1]%%%},[6,11,11]%%%}+%%%{%%%{6553600,[2]%%%},[6,11,10]%%%}+%%%{%%%{-2
621440,[3]%%%},[6,11,9]%%%}+%%%{%%%{-2621440,[4]%%%},[6,11,8]%%%}+%%%{%%%{2883584,[5]%%%},[6,11,7]%%%}+%%%{%%%
{-786432,[6]%%%},[6,11,6]%%%}+%%%{%%{[-6291456,0]:[1,0,%%%{-1,[1]%%%}]%%},[5,11,12]%%%}+%%%{%%{[%%%{34078720,[
1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,11,11]%%%}+%%%{%%{[%%%{-76021760,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,11
,10]%%%}+%%%{%%{[%%%{89128960,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,11,9]%%%}+%%%{%%{[%%%{-57671680,[4]%%%},0]
:[1,0,%%%{-1,[1]%%%}]%%},[5,11,8]%%%}+%%%{%%{[%%%{19398656,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,11,7]%%%}+%%%
{%%{[%%%{-2621440,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,11,6]%%%}+%%%{6291456,[4,11,13]%%%}+%%%{%%%{-28311552,
[1]%%%},[4,11,12]%%%}+%%%{%%%{42336256,[2]%%%},[4,11,11]%%%}+%%%{%%%{-7208960,[3]%%%},[4,11,10]%%%}+%%%{%%%{-4
8496640,[4]%%%},[4,11,9]%%%}+%%%{%%%{57933824,[5]%%%},[4,11,8]%%%}+%%%{%%%{-27394048,[6]%%%},[4,11,7]%%%}+%%%{
%%%{4849664,[7]%%%},[4,11,6]%%%}+%%%{%%{[-25165824,0]:[1,0,%%%{-1,[1]%%%}]%%},[3,11,13]%%%}+%%%{%%{[%%%{155189
248,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,11,12]%%%}+%%%{%%{[%%%{-406323200,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%}
,[3,11,11]%%%}+%%%{%%{[%%%{584581120,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,11,10]%%%}+%%%{%%{[%%%{-498073600,[
4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,11,9]%%%}+%%%{%%{[%%%{250609664,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,11,
8]%%%}+%%%{%%{[%%%{-68681728,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,11,7]%%%}+%%%{%%{[%%%{7864320,[7]%%%},0]:[1
,0,%%%{-1,[1]%%%}]%%},[3,11,6]%%%}+%%%{16777216,[2,11,14]%%%}+%%%{%%%{-96468992,[1]%%%},[2,11,13]%%%}+%%%{%%%{
221249536,[2]%%%},[2,11,12]%%%}+%%%{%%%{-239337472,[3]%%%},[2,11,11]%%%}+%%%{%%%{79953920,[4]%%%},[2,11,10]%%%
}+%%%{%%%{85458944,[5]%%%},[2,11,9]%%%}+%%%{%%%{-105381888,[6]%%%},[2,11,8]%%%}+%%%{%%%{44826624,[7]%%%},[2,11
,7]%%%}+%%%{%%%{-7077888,[8]%%%},[2,11,6]%%%}+%%%{%%{[-33554432,0]:[1,0,%%%{-1,[1]%%%}]%%},[1,11,14]%%%}+%%%{%
%{[%%%{243269632,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,11,13]%%%}+%%%{%%{[%%%{-769654784,[2]%%%},0]:[1,0,%%%{-
1,[1]%%%}]%%},[1,11,12]%%%}+%%%{%%{[%%%{1387790336,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,11,11]%%%}+%%%{%%{[%%
%{-1559756800,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,11,10]%%%}+%%%{%%{[%%%{1118830592,[5]%%%},0]:[1,0,%%%{-1,[
1]%%%}]%%},[1,11,9]%%%}+%%%{%%{[%%%{-500170752,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,11,8]%%%}+%%%{%%{[%%%{127
401984,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,11,7]%%%}+%%%{%%{[%%%{-14155776,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%
},[1,11,6]%%%}+%%%{16777216,[0,11,15]%%%}+%%%{%%%{-134217728,[1]%%%},[0,11,14]%%%}+%%%{%%%{476053504,[2]%%%},[
0,11,13]%%%}+%%%{%%%{-982515712,[3]%%%},[0,11,12]%%%}+%%%{%%%{1300299776,[4]%%%},[0,11,11]%%%}+%%%{%%%{-114432
4096,[5]%%%},[0,11,10]%%%}+%%%{%%%{669646848,[6]%%%},[0,11,9]%%%}+%%%{%%%{-251265024,[7]%%%},[0,11,8]%%%}+%%%{
%%%{54853632,[8]%%%},[0,11,7]%%%}+%%%{%%%{-5308416,[9]%%%},[0,11,6]%%%} / %%%{%%%{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 = 369, normalized size = 1.26 \[ -\frac {\left (4 \sqrt {-\frac {b}{a}}\, a b -5 \sqrt {-\frac {b}{a}}\, b^{2}\right ) \sinh \left (f x +e \right ) \left (\cosh ^{4}\left (f x +e \right )\right )+\left (4 \sqrt {-\frac {b}{a}}\, a^{2}-10 \sqrt {-\frac {b}{a}}\, a b +6 \sqrt {-\frac {b}{a}}\, b^{2}\right ) \left (\cosh ^{2}\left (f x +e \right )\right ) \sinh \left (f x +e \right )+\left (-\sqrt {-\frac {b}{a}}\, a^{2}+2 \sqrt {-\frac {b}{a}}\, a b -\sqrt {-\frac {b}{a}}\, b^{2}\right ) \sinh \left (f x +e \right )-\sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, \left (3 \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2}-11 \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b +8 \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}+7 \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b -8 \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}{3 \cosh \left (f x +e \right )^{3} \left (a -b \right ) \sqrt {-\frac {b}{a}}\, \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)^4,x)

[Out]

-1/3*((4*(-1/a*b)^(1/2)*a*b-5*(-1/a*b)^(1/2)*b^2)*sinh(f*x+e)*cosh(f*x+e)^4+(4*(-1/a*b)^(1/2)*a^2-10*(-1/a*b)^
(1/2)*a*b+6*(-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)-(cosh(f*x+e)^2)^(1/2)*(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(3*EllipticF(sinh(f*x+e)*(-1/a*b
)^(1/2),(a/b)^(1/2))*a^2-11*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a*b+8*EllipticF(sinh(f*x+e)*(-1/
a*b)^(1/2),(a/b)^(1/2))*b^2+7*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a*b-8*EllipticE(sinh(f*x+e)*(-
1/a*b)^(1/2),(a/b)^(1/2))*b^2)*cosh(f*x+e)^2)/cosh(f*x+e)^3/(a-b)/(-1/a*b)^(1/2)/(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 \[ \int \sqrt {b \sinh \left (f x + e\right )^{2} + a} \tanh \left (f x + e\right )^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)^4,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(e + f*x)^4*(a + b*sinh(e + f*x)^2)^(1/2),x)

[Out]

int(tanh(e + f*x)^4*(a + b*sinh(e + f*x)^2)^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sinh(f*x+e)**2)**(1/2)*tanh(f*x+e)**4,x)

[Out]

Integral(sqrt(a + b*sinh(e + f*x)**2)*tanh(e + f*x)**4, x)

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