3.6.8 \(\int \frac {\tanh ^2(e+f x)}{(a+b \sinh ^2(e+f x))^{5/2}} \, dx\) [508]

Optimal. Leaf size=274 \[ -\frac {4 b \cosh (e+f x) \sinh (e+f x)}{3 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\sqrt {b} (7 a+b) \cosh (e+f x) E\left (\text {ArcTan}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{3 \sqrt {a} (a-b)^3 f \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}+\frac {(3 a+5 b) F\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a (a-b)^3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {\tanh (e+f x)}{(a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]

[Out]

-4/3*b*cosh(f*x+e)*sinh(f*x+e)/(a-b)^2/f/(a+b*sinh(f*x+e)^2)^(3/2)-1/3*(7*a+b)*cosh(f*x+e)*(1/(1+b*sinh(f*x+e)
^2/a))^(1/2)*(1+b*sinh(f*x+e)^2/a)^(1/2)*EllipticE(sinh(f*x+e)*b^(1/2)/a^(1/2)/(1+b*sinh(f*x+e)^2/a)^(1/2),(1-
a/b)^(1/2))*b^(1/2)/(a-b)^3/f/a^(1/2)/(a*cosh(f*x+e)^2/(a+b*sinh(f*x+e)^2))^(1/2)/(a+b*sinh(f*x+e)^2)^(1/2)+1/
3*(3*a+5*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/(a-b)^3/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)-t
anh(f*x+e)/(a-b)/f/(a+b*sinh(f*x+e)^2)^(3/2)

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Rubi [A]
time = 0.20, antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3275, 482, 541, 539, 429, 422} \begin {gather*} -\frac {\sqrt {b} (7 a+b) \cosh (e+f x) E\left (\text {ArcTan}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{3 \sqrt {a} f (a-b)^3 \sqrt {a+b \sinh ^2(e+f x)} \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}}}+\frac {(3 a+5 b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} F\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{3 a f (a-b)^3 \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {\tanh (e+f x)}{f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {4 b \sinh (e+f x) \cosh (e+f x)}{3 f (a-b)^2 \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*b*Cosh[e + f*x]*Sinh[e + f*x])/(3*(a - b)^2*f*(a + b*Sinh[e + f*x]^2)^(3/2)) - (Sqrt[b]*(7*a + b)*Cosh[e +
f*x]*EllipticE[ArcTan[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]], 1 - a/b])/(3*Sqrt[a]*(a - b)^3*f*Sqrt[(a*Cosh[e + f*x
]^2)/(a + b*Sinh[e + f*x]^2)]*Sqrt[a + b*Sinh[e + f*x]^2]) + ((3*a + 5*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 -
b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*(a - b)^3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2
))/a]) - Tanh[e + f*x]/((a - b)*f*(a + b*Sinh[e + f*x]^2)^(3/2))

Rule 422

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

Rule 429

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

Rule 482

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 + 1)/(n*(b*c - a*d)*(p + 1))), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 539

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^(3/2)), x_Symbol] :> Dist[(b*e - a*
f)/(b*c - a*d), Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[Sqrt[a + b
*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] && PosQ[d/c]

Rule 541

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

Rule 3275

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

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Mathematica [C] Result contains complex when optimal does not.
time = 2.14, size = 215, normalized size = 0.78 \begin {gather*} \frac {-2 i a^2 (7 a+b) \left (\frac {2 a-b+b \cosh (2 (e+f x))}{a}\right )^{3/2} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )+8 i a^2 (a-b) \left (\frac {2 a-b+b \cosh (2 (e+f x))}{a}\right )^{3/2} F\left (i (e+f x)\left |\frac {b}{a}\right .\right )-\frac {\left (24 a^3-4 a^2 b+5 a b^2-b^3+4 a (11 a-3 b) b \cosh (2 (e+f x))+b^2 (7 a+b) \cosh (4 (e+f x))\right ) \tanh (e+f x)}{\sqrt {2}}}{6 a (a-b)^3 f (2 a-b+b \cosh (2 (e+f x)))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*I)*a^2*(7*a + b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticE[I*(e + f*x), b/a] + (8*I)*a^2*(a - b
)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticF[I*(e + f*x), b/a] - ((24*a^3 - 4*a^2*b + 5*a*b^2 - b^3 +
4*a*(11*a - 3*b)*b*Cosh[2*(e + f*x)] + b^2*(7*a + b)*Cosh[4*(e + f*x)])*Tanh[e + f*x])/Sqrt[2])/(6*a*(a - b)^
3*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(798\) vs. \(2(342)=684\).
time = 2.19, size = 799, normalized size = 2.92

method	result	size
default	\(\frac {-7 \sqrt {-\frac {b}{a}}\, a \,b^{2} \left (\sinh ^{5}\left (f x +e \right )\right )-\sqrt {-\frac {b}{a}}\, b^{3} \left (\sinh ^{5}\left (f x +e \right )\right )+3 \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2} b \left (\sinh ^{2}\left (f x +e \right )\right )-2 \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \,b^{2} \left (\sinh ^{2}\left (f x +e \right )\right )-\sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{3} \left (\sinh ^{2}\left (f x +e \right )\right )+7 \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \,b^{2} \left (\sinh ^{2}\left (f x +e \right )\right )+\sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{3} \left (\sinh ^{2}\left (f x +e \right )\right )-11 \sqrt {-\frac {b}{a}}\, a^{2} b \left (\sinh ^{3}\left (f x +e \right )\right )-4 \sqrt {-\frac {b}{a}}\, a \,b^{2} \left (\sinh ^{3}\left (f x +e \right )\right )-\sqrt {-\frac {b}{a}}\, b^{3} \left (\sinh ^{3}\left (f x +e \right )\right )+3 \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{3}-2 \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2} b -\sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \,b^{2}+7 \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2} b +\sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \,b^{2}-3 \sqrt {-\frac {b}{a}}\, a^{3} \sinh \left (f x +e \right )-5 \sinh \left (f x +e \right ) b \,a^{2} \sqrt {-\frac {b}{a}}}{3 \sqrt {-\frac {b}{a}}\, \left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}} \left (a -b \right )^{3} a \cosh \left (f x +e \right ) f}\)	\(799\)
risch	\(\text {Expression too large to display}\)	\(74496\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(-7*(-1/a*b)^(1/2)*a*b^2*sinh(f*x+e)^5-(-1/a*b)^(1/2)*b^3*sinh(f*x+e)^5+3*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(c
osh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a^2*b*sinh(f*x+e)^2-2*((a+b*sinh(f*x+e)^
2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a*b^2*sinh(f*x+e)^2-((a+b*
sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*b^3*sinh(f*x+e
)^2+7*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a*
b^2*sinh(f*x+e)^2+((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/
b)^(1/2))*b^3*sinh(f*x+e)^2-11*(-1/a*b)^(1/2)*a^2*b*sinh(f*x+e)^3-4*(-1/a*b)^(1/2)*a*b^2*sinh(f*x+e)^3-(-1/a*b
)^(1/2)*b^3*sinh(f*x+e)^3+3*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-1/a*b)
^(1/2),(a/b)^(1/2))*a^3-2*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-1/a*b)^(
1/2),(a/b)^(1/2))*a^2*b-((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/
2),(a/b)^(1/2))*a*b^2+7*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/
2),(a/b)^(1/2))*a^2*b+((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2)
,(a/b)^(1/2))*a*b^2-3*(-1/a*b)^(1/2)*a^3*sinh(f*x+e)-5*sinh(f*x+e)*b*a^2*(-1/a*b)^(1/2))/(-1/a*b)^(1/2)/(a+b*s
inh(f*x+e)^2)^(3/2)/(a-b)^3/a/cosh(f*x+e)/f

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 8226 vs. \(2 (280) = 560\).
time = 0.29, size = 8226, normalized size = 30.02 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(((14*a^2*b^3 - 5*a*b^4 - b^5)*cosh(f*x + e)^10 + 10*(14*a^2*b^3 - 5*a*b^4 - b^5)*cosh(f*x + e)*sinh(f*x +
e)^9 + (14*a^2*b^3 - 5*a*b^4 - b^5)*sinh(f*x + e)^10 + (112*a^3*b^2 - 82*a^2*b^3 + 7*a*b^4 + 3*b^5)*cosh(f*x
+ e)^8 + (112*a^3*b^2 - 82*a^2*b^3 + 7*a*b^4 + 3*b^5 + 45*(14*a^2*b^3 - 5*a*b^4 - b^5)*cosh(f*x + e)^2)*sinh(f
*x + e)^8 + 8*(15*(14*a^2*b^3 - 5*a*b^4 - b^5)*cosh(f*x + e)^3 + (112*a^3*b^2 - 82*a^2*b^3 + 7*a*b^4 + 3*b^5)*
cosh(f*x + e))*sinh(f*x + e)^7 + 2*(112*a^4*b - 96*a^3*b^2 + 26*a^2*b^3 - a*b^4 - b^5)*cosh(f*x + e)^6 + 2*(11
2*a^4*b - 96*a^3*b^2 + 26*a^2*b^3 - a*b^4 - b^5 + 105*(14*a^2*b^3 - 5*a*b^4 - b^5)*cosh(f*x + e)^4 + 14*(112*a
^3*b^2 - 82*a^2*b^3 + 7*a*b^4 + 3*b^5)*cosh(f*x + e)^2)*sinh(f*x + e)^6 + 4*(63*(14*a^2*b^3 - 5*a*b^4 - b^5)*c
osh(f*x + e)^5 + 14*(112*a^3*b^2 - 82*a^2*b^3 + 7*a*b^4 + 3*b^5)*cosh(f*x + e)^3 + 3*(112*a^4*b - 96*a^3*b^2 +
26*a^2*b^3 - a*b^4 - b^5)*cosh(f*x + e))*sinh(f*x + e)^5 + 14*a^2*b^3 - 5*a*b^4 - b^5 + 2*(112*a^4*b - 96*a^3
*b^2 + 26*a^2*b^3 - a*b^4 - b^5)*cosh(f*x + e)^4 + 2*(105*(14*a^2*b^3 - 5*a*b^4 - b^5)*cosh(f*x + e)^6 + 112*a
^4*b - 96*a^3*b^2 + 26*a^2*b^3 - a*b^4 - b^5 + 35*(112*a^3*b^2 - 82*a^2*b^3 + 7*a*b^4 + 3*b^5)*cosh(f*x + e)^4
+ 15*(112*a^4*b - 96*a^3*b^2 + 26*a^2*b^3 - a*b^4 - b^5)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 8*(15*(14*a^2*b^3
- 5*a*b^4 - b^5)*cosh(f*x + e)^7 + 7*(112*a^3*b^2 - 82*a^2*b^3 + 7*a*b^4 + 3*b^5)*cosh(f*x + e)^5 + 5*(112*a^
4*b - 96*a^3*b^2 + 26*a^2*b^3 - a*b^4 - b^5)*cosh(f*x + e)^3 + (112*a^4*b - 96*a^3*b^2 + 26*a^2*b^3 - a*b^4 -
b^5)*cosh(f*x + e))*sinh(f*x + e)^3 + (112*a^3*b^2 - 82*a^2*b^3 + 7*a*b^4 + 3*b^5)*cosh(f*x + e)^2 + (45*(14*a
^2*b^3 - 5*a*b^4 - b^5)*cosh(f*x + e)^8 + 28*(112*a^3*b^2 - 82*a^2*b^3 + 7*a*b^4 + 3*b^5)*cosh(f*x + e)^6 + 11
2*a^3*b^2 - 82*a^2*b^3 + 7*a*b^4 + 3*b^5 + 30*(112*a^4*b - 96*a^3*b^2 + 26*a^2*b^3 - a*b^4 - b^5)*cosh(f*x + e
)^4 + 12*(112*a^4*b - 96*a^3*b^2 + 26*a^2*b^3 - a*b^4 - b^5)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*(5*(14*a^2*b
^3 - 5*a*b^4 - b^5)*cosh(f*x + e)^9 + 4*(112*a^3*b^2 - 82*a^2*b^3 + 7*a*b^4 + 3*b^5)*cosh(f*x + e)^7 + 6*(112*
a^4*b - 96*a^3*b^2 + 26*a^2*b^3 - a*b^4 - b^5)*cosh(f*x + e)^5 + 4*(112*a^4*b - 96*a^3*b^2 + 26*a^2*b^3 - a*b^
4 - b^5)*cosh(f*x + e)^3 + (112*a^3*b^2 - 82*a^2*b^3 + 7*a*b^4 + 3*b^5)*cosh(f*x + e))*sinh(f*x + e) - 2*((7*a
*b^4 + b^5)*cosh(f*x + e)^10 + 10*(7*a*b^4 + b^5)*cosh(f*x + e)*sinh(f*x + e)^9 + (7*a*b^4 + b^5)*sinh(f*x + e
)^10 + (56*a^2*b^3 - 13*a*b^4 - 3*b^5)*cosh(f*x + e)^8 + (56*a^2*b^3 - 13*a*b^4 - 3*b^5 + 45*(7*a*b^4 + b^5)*c
osh(f*x + e)^2)*sinh(f*x + e)^8 + 8*(15*(7*a*b^4 + b^5)*cosh(f*x + e)^3 + (56*a^2*b^3 - 13*a*b^4 - 3*b^5)*cosh
(f*x + e))*sinh(f*x + e)^7 + 2*(56*a^3*b^2 - 20*a^2*b^3 + 3*a*b^4 + b^5)*cosh(f*x + e)^6 + 2*(56*a^3*b^2 - 20*
a^2*b^3 + 3*a*b^4 + b^5 + 105*(7*a*b^4 + b^5)*cosh(f*x + e)^4 + 14*(56*a^2*b^3 - 13*a*b^4 - 3*b^5)*cosh(f*x +
e)^2)*sinh(f*x + e)^6 + 4*(63*(7*a*b^4 + b^5)*cosh(f*x + e)^5 + 14*(56*a^2*b^3 - 13*a*b^4 - 3*b^5)*cosh(f*x +
e)^3 + 3*(56*a^3*b^2 - 20*a^2*b^3 + 3*a*b^4 + b^5)*cosh(f*x + e))*sinh(f*x + e)^5 + 7*a*b^4 + b^5 + 2*(56*a^3*
b^2 - 20*a^2*b^3 + 3*a*b^4 + b^5)*cosh(f*x + e)^4 + 2*(105*(7*a*b^4 + b^5)*cosh(f*x + e)^6 + 56*a^3*b^2 - 20*a
^2*b^3 + 3*a*b^4 + b^5 + 35*(56*a^2*b^3 - 13*a*b^4 - 3*b^5)*cosh(f*x + e)^4 + 15*(56*a^3*b^2 - 20*a^2*b^3 + 3*
a*b^4 + b^5)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 8*(15*(7*a*b^4 + b^5)*cosh(f*x + e)^7 + 7*(56*a^2*b^3 - 13*a*b
^4 - 3*b^5)*cosh(f*x + e)^5 + 5*(56*a^3*b^2 - 20*a^2*b^3 + 3*a*b^4 + b^5)*cosh(f*x + e)^3 + (56*a^3*b^2 - 20*a
^2*b^3 + 3*a*b^4 + b^5)*cosh(f*x + e))*sinh(f*x + e)^3 + (56*a^2*b^3 - 13*a*b^4 - 3*b^5)*cosh(f*x + e)^2 + (45
*(7*a*b^4 + b^5)*cosh(f*x + e)^8 + 28*(56*a^2*b^3 - 13*a*b^4 - 3*b^5)*cosh(f*x + e)^6 + 56*a^2*b^3 - 13*a*b^4
- 3*b^5 + 30*(56*a^3*b^2 - 20*a^2*b^3 + 3*a*b^4 + b^5)*cosh(f*x + e)^4 + 12*(56*a^3*b^2 - 20*a^2*b^3 + 3*a*b^4
+ b^5)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*(5*(7*a*b^4 + b^5)*cosh(f*x + e)^9 + 4*(56*a^2*b^3 - 13*a*b^4 - 3
*b^5)*cosh(f*x + e)^7 + 6*(56*a^3*b^2 - 20*a^2*b^3 + 3*a*b^4 + b^5)*cosh(f*x + e)^5 + 4*(56*a^3*b^2 - 20*a^2*b
^3 + 3*a*b^4 + b^5)*cosh(f*x + e)^3 + (56*a^2*b^3 - 13*a*b^4 - 3*b^5)*cosh(f*x + e))*sinh(f*x + e))*sqrt((a^2
- a*b)/b^2))*sqrt(b)*sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*elliptic_e(arcsin(sqrt((2*b*sqrt((a^2 - a*b
)/b^2) - 2*a + b)/b)*(cosh(f*x + e) + sinh(f*x + e))), (8*a^2 - 8*a*b + b^2 + 4*(2*a*b - b^2)*sqrt((a^2 - a*b)
/b^2))/b^2) - 2*((6*a^3*b^2 + 7*a^2*b^3 - 5*a*b^4)*cosh(f*x + e)^10 + 10*(6*a^3*b^2 + 7*a^2*b^3 - 5*a*b^4)*cos
h(f*x + e)*sinh(f*x + e)^9 + (6*a^3*b^2 + 7*a^2*b^3 - 5*a*b^4)*sinh(f*x + e)^10 + (48*a^4*b + 38*a^3*b^2 - 61*
a^2*b^3 + 15*a*b^4)*cosh(f*x + e)^8 + (48*a^4*b + 38*a^3*b^2 - 61*a^2*b^3 + 15*a*b^4 + 45*(6*a^3*b^2 + 7*a^2*b
^3 - 5*a*b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^8 + 8*(15*(6*a^3*b^2 + 7*a^2*b^3 - 5*a*b^4)*cosh(f*x + e)^3 + (48
*a^4*b + 38*a^3*b^2 - 61*a^2*b^3 + 15*a*b^4)*cosh(f*x + e))*sinh(f*x + e)^7 + 2*(48*a^5 + 32*a^4*b - 62*a^3*b^
2 + 27*a^2*b^3 - 5*a*b^4)*cosh(f*x + e)^6 + 2*(...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh ^{2}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Evaluation time:
1.86Error: Bad Argument Type

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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