3.6.1 \(\int \frac {\tanh ^5(e+f x)}{(a+b \sinh ^2(e+f x))^{5/2}} \, dx\) [501]

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

[Out]

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

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Rubi [A]
time = 0.21, 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, 53, 65, 214} \begin {gather*} \frac {8 a^2+24 a b+3 b^2}{8 f (a-b)^4 \sqrt {a+b \sinh ^2(e+f x)}}+\frac {8 a^2+24 a b+3 b^2}{24 f (a-b)^3 \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\left (8 a^2+24 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{8 f (a-b)^{9/2}}-\frac {\text {sech}^4(e+f x)}{4 f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {(8 a-b) \text {sech}^2(e+f x)}{8 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]^5/(a + b*Sinh[e + f*x]^2)^(5/2),x]

[Out]

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

Rule 53

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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 \frac {\tanh ^5(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {x^2}{(1+x)^3 (a+b x)^{5/2}} \, dx,x,\sinh ^2(e+f x)\right )}{2 f}\\ &=-\frac {\text {sech}^4(e+f x)}{4 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {\frac {1}{2} (-4 a-3 b)+2 (a-b) x}{(1+x)^2 (a+b x)^{5/2}} \, dx,x,\sinh ^2(e+f x)\right )}{4 (a-b) f}\\ &=\frac {(8 a-b) \text {sech}^2(e+f x)}{8 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {sech}^4(e+f x)}{4 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\left (8 a^2+24 a b+3 b^2\right ) \text {Subst}\left (\int \frac {1}{(1+x) (a+b x)^{5/2}} \, dx,x,\sinh ^2(e+f x)\right )}{16 (a-b)^2 f}\\ &=\frac {8 a^2+24 a b+3 b^2}{24 (a-b)^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {(8 a-b) \text {sech}^2(e+f x)}{8 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {sech}^4(e+f x)}{4 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\left (8 a^2+24 a b+3 b^2\right ) \text {Subst}\left (\int \frac {1}{(1+x) (a+b x)^{3/2}} \, dx,x,\sinh ^2(e+f x)\right )}{16 (a-b)^3 f}\\ &=\frac {8 a^2+24 a b+3 b^2}{24 (a-b)^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {(8 a-b) \text {sech}^2(e+f x)}{8 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {sech}^4(e+f x)}{4 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {8 a^2+24 a b+3 b^2}{8 (a-b)^4 f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\left (8 a^2+24 a b+3 b^2\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x}} \, dx,x,\sinh ^2(e+f x)\right )}{16 (a-b)^4 f}\\ &=\frac {8 a^2+24 a b+3 b^2}{24 (a-b)^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {(8 a-b) \text {sech}^2(e+f x)}{8 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {sech}^4(e+f x)}{4 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {8 a^2+24 a b+3 b^2}{8 (a-b)^4 f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\left (8 a^2+24 a b+3 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 (a-b)^4 b f}\\ &=-\frac {\left (8 a^2+24 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{8 (a-b)^{9/2} f}+\frac {8 a^2+24 a b+3 b^2}{24 (a-b)^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {(8 a-b) \text {sech}^2(e+f x)}{8 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {sech}^4(e+f x)}{4 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {8 a^2+24 a b+3 b^2}{8 (a-b)^4 f \sqrt {a+b \sinh ^2(e+f x)}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.42, size = 114, normalized size = 0.49 \begin {gather*} \frac {2 \left (8 a^2+24 a b+3 b^2\right ) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {a+b \sinh ^2(e+f x)}{a-b}\right )+3 (a-b) (4 a+3 b+(8 a-b) \cosh (2 (e+f x))) \text {sech}^4(e+f x)}{48 (a-b)^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(8*a^2 + 24*a*b + 3*b^2)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Sinh[e + f*x]^2)/(a - b)] + 3*(a - b)*(4*a
+ 3*b + (8*a - b)*Cosh[2*(e + f*x)])*Sech[e + f*x]^4)/(48*(a - b)^3*f*(a + b*Sinh[e + f*x]^2)^(3/2))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 3.13, size = 213, normalized size = 0.92

method	result	size
default	\(\frac {\mathit {`\,int/indef0`\,}\left (-\frac {\left (\sinh ^{5}\left (f x +e \right )\right ) \left (b^{2} \left (\sinh ^{4}\left (f x +e \right )\right )+2 a b \left (\sinh ^{2}\left (f x +e \right )\right )+a^{2}\right ) \left (\cosh ^{4}\left (f x +e \right )\right )}{\left (-b^{4} \left (\cosh ^{18}\left (f x +e \right )\right )+\left (-4 a \,b^{3}+4 b^{4}\right ) \left (\cosh ^{16}\left (f x +e \right )\right )+\left (-6 a^{2} b^{2}+12 a \,b^{3}-6 b^{4}\right ) \left (\cosh ^{14}\left (f x +e \right )\right )+\left (-4 a^{3} b +12 a^{2} b^{2}-12 a \,b^{3}+4 b^{4}\right ) \left (\cosh ^{12}\left (f x +e \right )\right )+\left (-a^{4}+4 a^{3} b -6 a^{2} b^{2}+4 a \,b^{3}-b^{4}\right ) \left (\cosh ^{10}\left (f x +e \right )\right )\right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f}\)	\(213\)
risch	\(\text {Expression too large to display}\)	\(2629521\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(f*x+e)^5/(a+b*sinh(f*x+e)^2)^(5/2),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)^4/(-b^4*cosh(f*x+e)^18+(-4
*a*b^3+4*b^4)*cosh(f*x+e)^16+(-6*a^2*b^2+12*a*b^3-6*b^4)*cosh(f*x+e)^14+(-4*a^3*b+12*a^2*b^2-12*a*b^3+4*b^4)*c
osh(f*x+e)^12+(-a^4+4*a^3*b-6*a^2*b^2+4*a*b^3-b^4)*cosh(f*x+e)^10)/(a+b*sinh(f*x+e)^2)^(1/2),sinh(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)^5/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(tanh(f*x + e)^5/(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. 10051 vs. \(2 (208) = 416\).
time = 1.38, size = 20298, normalized size = 87.49 \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)^5/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(3*((8*a^2*b^2 + 24*a*b^3 + 3*b^4)*cosh(f*x + e)^16 + 16*(8*a^2*b^2 + 24*a*b^3 + 3*b^4)*cosh(f*x + e)*si
nh(f*x + e)^15 + (8*a^2*b^2 + 24*a*b^3 + 3*b^4)*sinh(f*x + e)^16 + 8*(8*a^3*b + 24*a^2*b^2 + 3*a*b^3)*cosh(f*x
+ e)^14 + 8*(8*a^3*b + 24*a^2*b^2 + 3*a*b^3 + 15*(8*a^2*b^2 + 24*a*b^3 + 3*b^4)*cosh(f*x + e)^2)*sinh(f*x + e
)^14 + 112*(5*(8*a^2*b^2 + 24*a*b^3 + 3*b^4)*cosh(f*x + e)^3 + (8*a^3*b + 24*a^2*b^2 + 3*a*b^3)*cosh(f*x + e))
*sinh(f*x + e)^13 + 4*(32*a^4 + 128*a^3*b + 100*a^2*b^2 - 12*a*b^3 - 3*b^4)*cosh(f*x + e)^12 + 4*(455*(8*a^2*b
^2 + 24*a*b^3 + 3*b^4)*cosh(f*x + e)^4 + 32*a^4 + 128*a^3*b + 100*a^2*b^2 - 12*a*b^3 - 3*b^4 + 182*(8*a^3*b +
24*a^2*b^2 + 3*a*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^12 + 16*(273*(8*a^2*b^2 + 24*a*b^3 + 3*b^4)*cosh(f*x + e)
^5 + 182*(8*a^3*b + 24*a^2*b^2 + 3*a*b^3)*cosh(f*x + e)^3 + 3*(32*a^4 + 128*a^3*b + 100*a^2*b^2 - 12*a*b^3 - 3
*b^4)*cosh(f*x + e))*sinh(f*x + e)^11 + 8*(64*a^4 + 184*a^3*b - 3*a*b^3)*cosh(f*x + e)^10 + 8*(1001*(8*a^2*b^2
+ 24*a*b^ ...

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3690 vs. \(2 (208) = 416\).
time = 21.38, size = 3690, normalized size = 15.91 \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)^5/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="giac")

[Out]

2/3*((3*(a^22*b^3*e^(21*e) - 14*a^21*b^4*e^(21*e) + 88*a^20*b^5*e^(21*e) - 320*a^19*b^6*e^(21*e) + 700*a^18*b^
7*e^(21*e) - 728*a^17*b^8*e^(21*e) - 728*a^16*b^9*e^(21*e) + 4576*a^15*b^10*e^(21*e) - 10010*a^14*b^11*e^(21*e
) + 14300*a^13*b^12*e^(21*e) - 14872*a^12*b^13*e^(21*e) + 11648*a^11*b^14*e^(21*e) - 6916*a^10*b^15*e^(21*e) +
3080*a^9*b^16*e^(21*e) - 1000*a^8*b^17*e^(21*e) + 224*a^7*b^18*e^(21*e) - 31*a^6*b^19*e^(21*e) + 2*a^5*b^20*e
^(21*e))*e^(2*f*x)/(a^24*b^2*e^(16*e) - 20*a^23*b^3*e^(16*e) + 190*a^22*b^4*e^(16*e) - 1140*a^21*b^5*e^(16*e)
+ 4845*a^20*b^6*e^(16*e) - 15504*a^19*b^7*e^(16*e) + 38760*a^18*b^8*e^(16*e) - 77520*a^17*b^9*e^(16*e) + 12597
0*a^16*b^10*e^(16*e) - 167960*a^15*b^11*e^(16*e) + 184756*a^14*b^12*e^(16*e) - 167960*a^13*b^13*e^(16*e) + 125
970*a^12*b^14*e^(16*e) - 77520*a^11*b^15*e^(16*e) + 38760*a^10*b^16*e^(16*e) - 15504*a^9*b^17*e^(16*e) + 4845*
a^8*b^18*e^(16*e) - 1140*a^7*b^19*e^(16*e) + 190*a^6*b^20*e^(16*e) - 20*a^5*b^21*e^(16*e) + a^4*b^22*e^(16*e))
+ 2*(8*a^23*b^2*e^(19*e) - 121*a^22*b^3*e^(19*e) + 842*a^21*b^4*e^(19*e) - 3544*a^20*b^5*e^(19*e) + 9920*a^19
*b^6*e^(19*e) - 18844*a^18*b^7*e^(19*e) + 22568*a^17*b^8*e^(19*e) - 9256*a^16*b^9*e^(19*e) - 25168*a^15*b^10*e
^(19*e) + 67210*a^14*b^11*e^(19*e) - 93236*a^13*b^12*e^(19*e) + 89752*a^12*b^13*e^(19*e) - 64064*a^11*b^14*e^(
19*e) + 34468*a^10*b^15*e^(19*e) - 13880*a^9*b^16*e^(19*e) + 4072*a^8*b^17*e^(19*e) - 824*a^7*b^18*e^(19*e) +
103*a^6*b^19*e^(19*e) - 6*a^5*b^20*e^(19*e))/(a^24*b^2*e^(16*e) - 20*a^23*b^3*e^(16*e) + 190*a^22*b^4*e^(16*e)
- 1140*a^21*b^5*e^(16*e) + 4845*a^20*b^6*e^(16*e) - 15504*a^19*b^7*e^(16*e) + 38760*a^18*b^8*e^(16*e) - 77520
*a^17*b^9*e^(16*e) + 125970*a^16*b^10*e^(16*e) - 167960*a^15*b^11*e^(16*e) + 184756*a^14*b^12*e^(16*e) - 16796
0*a^13*b^13*e^(16*e) + 125970*a^12*b^14*e^(16*e) - 77520*a^11*b^15*e^(16*e) + 38760*a^10*b^16*e^(16*e) - 15504
*a^9*b^17*e^(16*e) + 4845*a^8*b^18*e^(16*e) - 1140*a^7*b^19*e^(16*e) + 190*a^6*b^20*e^(16*e) - 20*a^5*b^21*e^(
16*e) + a^4*b^22*e^(16*e)))*e^(2*f*x) + 3*(a^22*b^3*e^(17*e) - 14*a^21*b^4*e^(17*e) + 88*a^20*b^5*e^(17*e) - 3
20*a^19*b^6*e^(17*e) + 700*a^18*b^7*e^(17*e) - 728*a^17*b^8*e^(17*e) - 728*a^16*b^9*e^(17*e) + 4576*a^15*b^10*
e^(17*e) - 10010*a^14*b^11*e^(17*e) + 14300*a^13*b^12*e^(17*e) - 14872*a^12*b^13*e^(17*e) + 11648*a^11*b^14*e^
(17*e) - 6916*a^10*b^15*e^(17*e) + 3080*a^9*b^16*e^(17*e) - 1000*a^8*b^17*e^(17*e) + 224*a^7*b^18*e^(17*e) - 3
1*a^6*b^19*e^(17*e) + 2*a^5*b^20*e^(17*e))/(a^24*b^2*e^(16*e) - 20*a^23*b^3*e^(16*e) + 190*a^22*b^4*e^(16*e) -
1140*a^21*b^5*e^(16*e) + 4845*a^20*b^6*e^(16*e) - 15504*a^19*b^7*e^(16*e) + 38760*a^18*b^8*e^(16*e) - 77520*a
^17*b^9*e^(16*e) + 125970*a^16*b^10*e^(16*e) - 167960*a^15*b^11*e^(16*e) + 184756*a^14*b^12*e^(16*e) - 167960*
a^13*b^13*e^(16*e) + 125970*a^12*b^14*e^(16*e) - 77520*a^11*b^15*e^(16*e) + 38760*a^10*b^16*e^(16*e) - 15504*a
^9*b^17*e^(16*e) + 4845*a^8*b^18*e^(16*e) - 1140*a^7*b^19*e^(16*e) + 190*a^6*b^20*e^(16*e) - 20*a^5*b^21*e^(16
*e) + a^4*b^22*e^(16*e)))*e^(f*x)/((b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b)^(3/2)*f
) + 1/12*(15*(3*a^2*e^e + 4*a*b*e^e)*arctan(-1/2*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*
f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b) + sqrt(b))/sqrt(a - b))/((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*sqr
t(a - b)) - 24*(a^2*e^e + 2*a*b*e^e)*arctan(-(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x
+ 2*e) - 2*b*e^(2*f*x + 2*e) + b))/sqrt(-b))/((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*sqrt(-b)) - 2*(21*(s
qrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^7*a^2*e^e +
12*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^7*a*b*e
^e + 243*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^6
*a^2*sqrt(b)*e^e - 12*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x +
2*e) + b))^6*a*b^(3/2)*e^e + 436*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*
b*e^(2*f*x + 2*e) + b))^5*a^3*e^e + 117*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e
) - 2*b*e^(2*f*x + 2*e) + b))^5*a^2*b*e^e + 396*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f
*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^5*a*b^2*e^e - 256*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*
a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^5*b^3*e^e + 1796*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*
e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^4*a^3*sqrt(b)*e^e + 363*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b
*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^4*a^2*b^(3/2)*e^e - 1644*(sqrt(b)*e^(2*f*x
+ 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^4*a*b^(5/2)*e^e + 640*(sqrt(
b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^4*b^(7/2)*e^e +
1840*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x...

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \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)^(5/2),x)

[Out]

\text{Hanged}

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