3.6.2 \(\int \frac {\tanh ^3(e+f x)}{(a+b \sinh ^2(e+f x))^{5/2}} \, dx\) [502]

Optimal. Leaf size=163 \[ -\frac {(2 a+3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{2 (a-b)^{7/2} f}+\frac {2 a+3 b}{6 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\text {sech}^2(e+f x)}{2 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {2 a+3 b}{2 (a-b)^3 f \sqrt {a+b \sinh ^2(e+f x)}} \]

[Out]

-1/2*(2*a+3*b)*arctanh((a+b*sinh(f*x+e)^2)^(1/2)/(a-b)^(1/2))/(a-b)^(7/2)/f+1/6*(2*a+3*b)/(a-b)^2/f/(a+b*sinh(
f*x+e)^2)^(3/2)+1/2*sech(f*x+e)^2/(a-b)/f/(a+b*sinh(f*x+e)^2)^(3/2)+1/2*(2*a+3*b)/(a-b)^3/f/(a+b*sinh(f*x+e)^2
)^(1/2)

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Rubi [A]
time = 0.11, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3273, 79, 53, 65, 214} \begin {gather*} \frac {2 a+3 b}{2 f (a-b)^3 \sqrt {a+b \sinh ^2(e+f x)}}+\frac {2 a+3 b}{6 f (a-b)^2 \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {(2 a+3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{2 f (a-b)^{7/2}}+\frac {\text {sech}^2(e+f x)}{2 f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((2*a + 3*b)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]])/((a - b)^(7/2)*f) + (2*a + 3*b)/(6*(a - b)
^2*f*(a + b*Sinh[e + f*x]^2)^(3/2)) + Sech[e + f*x]^2/(2*(a - b)*f*(a + b*Sinh[e + f*x]^2)^(3/2)) + (2*a + 3*b
)/(2*(a - b)^3*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 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 ^3(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {x}{(1+x)^2 (a+b x)^{5/2}} \, dx,x,\sinh ^2(e+f x)\right )}{2 f}\\ &=\frac {\text {sech}^2(e+f x)}{2 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {(2 a+3 b) \text {Subst}\left (\int \frac {1}{(1+x) (a+b x)^{5/2}} \, dx,x,\sinh ^2(e+f x)\right )}{4 (a-b) f}\\ &=\frac {2 a+3 b}{6 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\text {sech}^2(e+f x)}{2 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {(2 a+3 b) \text {Subst}\left (\int \frac {1}{(1+x) (a+b x)^{3/2}} \, dx,x,\sinh ^2(e+f x)\right )}{4 (a-b)^2 f}\\ &=\frac {2 a+3 b}{6 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\text {sech}^2(e+f x)}{2 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {2 a+3 b}{2 (a-b)^3 f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {(2 a+3 b) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x}} \, dx,x,\sinh ^2(e+f x)\right )}{4 (a-b)^3 f}\\ &=\frac {2 a+3 b}{6 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\text {sech}^2(e+f x)}{2 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {2 a+3 b}{2 (a-b)^3 f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {(2 a+3 b) \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 )}{2 (a-b)^3 b f}\\ &=-\frac {(2 a+3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{2 (a-b)^{7/2} f}+\frac {2 a+3 b}{6 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\text {sech}^2(e+f x)}{2 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {2 a+3 b}{2 (a-b)^3 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 = 0.09, size = 82, normalized size = 0.50 \begin {gather*} \frac {(2 a+3 b) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {a+b \sinh ^2(e+f x)}{a-b}\right )+3 (a-b) \text {sech}^2(e+f x)}{6 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*a + 3*b)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Sinh[e + f*x]^2)/(a - b)] + 3*(a - b)*Sech[e + f*x]^2)/(6
*(a - b)^2*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 = 2.08, size = 213, normalized size = 1.31

method	result	size
default	\(\frac {\mathit {`\,int/indef0`\,}\left (-\frac {\left (\sinh ^{3}\left (f x +e \right )\right ) \left (b^{2} \left (\sinh ^{4}\left (f x +e \right )\right )+2 a b \left (\sinh ^{2}\left (f x +e \right )\right )+a^{2}\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}{\left (-b^{4} \left (\cosh ^{14}\left (f x +e \right )\right )+\left (-4 a \,b^{3}+4 b^{4}\right ) \left (\cosh ^{12}\left (f x +e \right )\right )+\left (-6 a^{2} b^{2}+12 a \,b^{3}-6 b^{4}\right ) \left (\cosh ^{10}\left (f x +e \right )\right )+\left (-4 a^{3} b +12 a^{2} b^{2}-12 a \,b^{3}+4 b^{4}\right ) \left (\cosh ^{8}\left (f x +e \right )\right )+\left (-a^{4}+4 a^{3} b -6 a^{2} b^{2}+4 a \,b^{3}-b^{4}\right ) \left (\cosh ^{6}\left (f x +e \right )\right )\right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f}\)	\(213\)
risch	\(\text {Expression too large to display}\)	\(309501\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

`int/indef0`(-sinh(f*x+e)^3*(b^2*sinh(f*x+e)^4+2*a*b*sinh(f*x+e)^2+a^2)*cosh(f*x+e)^2/(-b^4*cosh(f*x+e)^14+(-4
*a*b^3+4*b^4)*cosh(f*x+e)^12+(-6*a^2*b^2+12*a*b^3-6*b^4)*cosh(f*x+e)^10+(-4*a^3*b+12*a^2*b^2-12*a*b^3+4*b^4)*c
osh(f*x+e)^8+(-a^4+4*a^3*b-6*a^2*b^2+4*a*b^3-b^4)*cosh(f*x+e)^6)/(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)^3/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(tanh(f*x + e)^3/(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. 5155 vs. \(2 (143) = 286\).
time = 1.01, size = 10506, normalized size = 64.45 \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)^3/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="fricas")

[Out]

[-1/12*(3*((2*a*b^2 + 3*b^3)*cosh(f*x + e)^12 + 12*(2*a*b^2 + 3*b^3)*cosh(f*x + e)*sinh(f*x + e)^11 + (2*a*b^2
+ 3*b^3)*sinh(f*x + e)^12 + 2*(8*a^2*b + 10*a*b^2 - 3*b^3)*cosh(f*x + e)^10 + 2*(8*a^2*b + 10*a*b^2 - 3*b^3 +
33*(2*a*b^2 + 3*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^10 + 20*(11*(2*a*b^2 + 3*b^3)*cosh(f*x + e)^3 + (8*a^2*b
+ 10*a*b^2 - 3*b^3)*cosh(f*x + e))*sinh(f*x + e)^9 + (32*a^3 + 48*a^2*b - 2*a*b^2 - 3*b^3)*cosh(f*x + e)^8 + (
495*(2*a*b^2 + 3*b^3)*cosh(f*x + e)^4 + 32*a^3 + 48*a^2*b - 2*a*b^2 - 3*b^3 + 90*(8*a^2*b + 10*a*b^2 - 3*b^3)*
cosh(f*x + e)^2)*sinh(f*x + e)^8 + 8*(99*(2*a*b^2 + 3*b^3)*cosh(f*x + e)^5 + 30*(8*a^2*b + 10*a*b^2 - 3*b^3)*c
osh(f*x + e)^3 + (32*a^3 + 48*a^2*b - 2*a*b^2 - 3*b^3)*cosh(f*x + e))*sinh(f*x + e)^7 + 4*(16*a^3 + 16*a^2*b -
10*a*b^2 + 3*b^3)*cosh(f*x + e)^6 + 4*(231*(2*a*b^2 + 3*b^3)*cosh(f*x + e)^6 + 105*(8*a^2*b + 10*a*b^2 - 3*b^
3)*cosh(f*x + e)^4 + 16*a^3 + 16*a^2*b - 10*a*b^2 + 3*b^3 + 7*(32*a^3 + 48*a^2*b - 2*a*b^2 - 3*b^3)*cosh(f*x +
e)^2)*sinh(f*x + e)^6 + 8*(99*(2*a*b^2 + 3*b^3)*cosh(f*x + e)^7 + 63*(8*a^2*b + 10*a*b^2 - 3*b^3)*cosh(f*x +
e)^5 + 7*(32*a^3 + 48*a^2*b - 2*a*b^2 - 3*b^3)*cosh(f*x + e)^3 + 3*(16*a^3 + 16*a^2*b - 10*a*b^2 + 3*b^3)*cosh
(f*x + e))*sinh(f*x + e)^5 + (32*a^3 + 48*a^2*b - 2*a*b^2 - 3*b^3)*cosh(f*x + e)^4 + (495*(2*a*b^2 + 3*b^3)*co
sh(f*x + e)^8 + 420*(8*a^2*b + 10*a*b^2 - 3*b^3)*cosh(f*x + e)^6 + 70*(32*a^3 + 48*a^2*b - 2*a*b^2 - 3*b^3)*co
sh(f*x + e)^4 + 32*a^3 + 48*a^2*b - 2*a*b^2 - 3*b^3 + 60*(16*a^3 + 16*a^2*b - 10*a*b^2 + 3*b^3)*cosh(f*x + e)^
2)*sinh(f*x + e)^4 + 4*(55*(2*a*b^2 + 3*b^3)*cosh(f*x + e)^9 + 60*(8*a^2*b + 10*a*b^2 - 3*b^3)*cosh(f*x + e)^7
+ 14*(32*a^3 + 48*a^2*b - 2*a*b^2 - 3*b^3)*cosh(f*x + e)^5 + 20*(16*a^3 + 16*a^2*b - 10*a*b^2 + 3*b^3)*cosh(f
*x + e)^3 + (32*a^3 + 48*a^2*b - 2*a*b^2 - 3*b^3)*cosh(f*x + e))*sinh(f*x + e)^3 + 2*a*b^2 + 3*b^3 + 2*(8*a^2*
b + 10*a*b^2 - 3*b^3)*cosh(f*x + e)^2 + 2*(33*(2*a*b^2 + 3*b^3)*cosh(f*x + e)^10 + 45*(8*a^2*b + 10*a*b^2 - 3*
b^3)*cosh(f*x + e)^8 + 14*(32*a^3 + 48*a^2*b - 2*a*b^2 - 3*b^3)*cosh(f*x + e)^6 + 30*(16*a^3 + 16*a^2*b - 10*a
*b^2 + 3*b^3)*cosh(f*x + e)^4 + 8*a^2*b + 10*a*b^2 - 3*b^3 + 3*(32*a^3 + 48*a^2*b - 2*a*b^2 - 3*b^3)*cosh(f*x
+ e)^2)*sinh(f*x + e)^2 + 4*(3*(2*a*b^2 + 3*b^3)*cosh(f*x + e)^11 + 5*(8*a^2*b + 10*a*b^2 - 3*b^3)*cosh(f*x +
e)^9 + 2*(32*a^3 + 48*a^2*b - 2*a*b^2 - 3*b^3)*cosh(f*x + e)^7 + 6*(16*a^3 + 16*a^2*b - 10*a*b^2 + 3*b^3)*cosh
(f*x + e)^5 + (32*a^3 + 48*a^2*b - 2*a*b^2 - 3*b^3)*cosh(f*x + e)^3 + (8*a^2*b + 10*a*b^2 - 3*b^3)*cosh(f*x +
e))*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)*cosh(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))*(cosh(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*cosh(f*x + e)^2 + 4*(cosh(f*x + e)^3 + cosh(f*x + e))*sinh(f*x + e) + 1)) - 4*sqrt(2)*(3*(2*a^2*
b + a*b^2 - 3*b^3)*cosh(f*x + e)^9 + 27*(2*a^2*b + a*b^2 - 3*b^3)*cosh(f*x + e)*sinh(f*x + e)^8 + 3*(2*a^2*b +
a*b^2 - 3*b^3)*sinh(f*x + e)^9 + 4*(8*a^3 + 2*a^2*b - 13*a*b^2 + 3*b^3)*cosh(f*x + e)^7 + 4*(8*a^3 + 2*a^2*b
- 13*a*b^2 + 3*b^3 + 27*(2*a^2*b + a*b^2 - 3*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^7 + 28*(9*(2*a^2*b + a*b^2 -
3*b^3)*cosh(f*x + e)^3 + (8*a^3 + 2*a^2*b - 13*a*b^2 + 3*b^3)*cosh(f*x + e))*sinh(f*x + e)^6 + 2*(56*a^3 - 70*
a^2*b + 17*a*b^2 - 3*b^3)*cosh(f*x + e)^5 + 2*(189*(2*a^2*b + a*b^2 - 3*b^3)*cosh(f*x + e)^4 + 56*a^3 - 70*a^2
*b + 17*a*b^2 - 3*b^3 + 42*(8*a^3 + 2*a^2*b - 13*a*b^2 + 3*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^5 + 2*(189*(2*a
^2*b + a*b^2 - 3*b^3)*cosh(f*x + e)^5 + 70*(8*a^3 + 2*a^2*b - 13*a*b^2 + 3*b^3)*cosh(f*x + e)^3 + 5*(56*a^3 -
70*a^2*b + 17*a*b^2 - 3*b^3)*cosh(f*x + e))*sinh(f*x + e)^4 + 4*(8*a^3 + 2*a^2*b - 13*a*b^2 + 3*b^3)*cosh(f*x
+ e)^3 + 4*(63*(2*a^2*b + a*b^2 - 3*b^3)*cosh(f*x + e)^6 + 35*(8*a^3 + 2*a^2*b - 13*a*b^2 + 3*b^3)*cosh(f*x +
e)^4 + 8*a^3 + 2*a^2*b - 13*a*b^2 + 3*b^3 + 5*(56*a^3 - 70*a^2*b + 17*a*b^2 - 3*b^3)*cosh(f*x + e)^2)*sinh(f*x
+ e)^3 + 4*(27*(2*a^2*b + a*b^2 - 3*b^3)*cosh(f*x + e)^7 + 21*(8*a^3 + 2*a^2*b - 13*a*b^2 + 3*b^3)*cosh(f*x +
e)^5 + 5*(56*a^3 - 70*a^2*b + 17*a*b^2 - 3*b^3)*cosh(f*x + e)^3 + 3*(8*a^3 + 2*a^2*b - 13*a*b^2 + 3*b^3)*cosh
(f*x + e))*sinh(f*x + e)^2 + 3*(2*a^2*b + a*b^2 - 3*b^3)*cosh(f*x + e) + (27*(2*a^2*b + a*b^2 - 3*b^3)*cosh(f*
x + e)^8 + 28*(8*a^3 + 2*a^2*b - 13*a*b^2 + 3*b^3)*cosh(f*x + e)^6 + 10*(56*a^3 - 70*a^2*b + 17*a*b^2 - 3*b^3)
*cosh(f*x + e)^4 + 6*a^2*b + 3*a*b^2 - 9*b^3 + 12*(8*a^3 + 2*a^2*b - 13*a*b^2 + 3*b^3)*cosh(f*x + e)^2)*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^4*b^2 - 4*a^3*b^3 +...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh ^{3}{\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)**3/(a+b*sinh(f*x+e)**2)**(5/2),x)

[Out]

Integral(tanh(e + f*x)**3/(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. 1981 vs. \(2 (143) = 286\).
time = 7.36, size = 1981, normalized size = 12.15 \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)^3/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="giac")

[Out]

2/3*((3*(a^18*b^3*e^(21*e) - 12*a^17*b^4*e^(21*e) + 65*a^16*b^5*e^(21*e) - 208*a^15*b^6*e^(21*e) + 429*a^14*b^
7*e^(21*e) - 572*a^13*b^8*e^(21*e) + 429*a^12*b^9*e^(21*e) - 429*a^10*b^11*e^(21*e) + 572*a^9*b^12*e^(21*e) -
429*a^8*b^13*e^(21*e) + 208*a^7*b^14*e^(21*e) - 65*a^6*b^15*e^(21*e) + 12*a^5*b^16*e^(21*e) - a^4*b^17*e^(21*e
))*e^(2*f*x)/(a^20*b^2*e^(16*e) - 16*a^19*b^3*e^(16*e) + 120*a^18*b^4*e^(16*e) - 560*a^17*b^5*e^(16*e) + 1820*
a^16*b^6*e^(16*e) - 4368*a^15*b^7*e^(16*e) + 8008*a^14*b^8*e^(16*e) - 11440*a^13*b^9*e^(16*e) + 12870*a^12*b^1
0*e^(16*e) - 11440*a^11*b^11*e^(16*e) + 8008*a^10*b^12*e^(16*e) - 4368*a^9*b^13*e^(16*e) + 1820*a^8*b^14*e^(16
*e) - 560*a^7*b^15*e^(16*e) + 120*a^6*b^16*e^(16*e) - 16*a^5*b^17*e^(16*e) + a^4*b^18*e^(16*e)) + 2*(8*a^19*b^
2*e^(19*e) - 103*a^18*b^3*e^(19*e) + 608*a^17*b^4*e^(19*e) - 2171*a^16*b^5*e^(19*e) + 5200*a^15*b^6*e^(19*e) -
8723*a^14*b^7*e^(19*e) + 10296*a^13*b^8*e^(19*e) - 8151*a^12*b^9*e^(19*e) + 3432*a^11*b^10*e^(19*e) + 715*a^1
0*b^11*e^(19*e) - 2288*a^9*b^12*e^(19*e) + 1807*a^8*b^13*e^(19*e) - 832*a^7*b^14*e^(19*e) + 239*a^6*b^15*e^(19
*e) - 40*a^5*b^16*e^(19*e) + 3*a^4*b^17*e^(19*e))/(a^20*b^2*e^(16*e) - 16*a^19*b^3*e^(16*e) + 120*a^18*b^4*e^(
16*e) - 560*a^17*b^5*e^(16*e) + 1820*a^16*b^6*e^(16*e) - 4368*a^15*b^7*e^(16*e) + 8008*a^14*b^8*e^(16*e) - 114
40*a^13*b^9*e^(16*e) + 12870*a^12*b^10*e^(16*e) - 11440*a^11*b^11*e^(16*e) + 8008*a^10*b^12*e^(16*e) - 4368*a^
9*b^13*e^(16*e) + 1820*a^8*b^14*e^(16*e) - 560*a^7*b^15*e^(16*e) + 120*a^6*b^16*e^(16*e) - 16*a^5*b^17*e^(16*e
) + a^4*b^18*e^(16*e)))*e^(2*f*x) + 3*(a^18*b^3*e^(17*e) - 12*a^17*b^4*e^(17*e) + 65*a^16*b^5*e^(17*e) - 208*a
^15*b^6*e^(17*e) + 429*a^14*b^7*e^(17*e) - 572*a^13*b^8*e^(17*e) + 429*a^12*b^9*e^(17*e) - 429*a^10*b^11*e^(17
*e) + 572*a^9*b^12*e^(17*e) - 429*a^8*b^13*e^(17*e) + 208*a^7*b^14*e^(17*e) - 65*a^6*b^15*e^(17*e) + 12*a^5*b^
16*e^(17*e) - a^4*b^17*e^(17*e))/(a^20*b^2*e^(16*e) - 16*a^19*b^3*e^(16*e) + 120*a^18*b^4*e^(16*e) - 560*a^17*
b^5*e^(16*e) + 1820*a^16*b^6*e^(16*e) - 4368*a^15*b^7*e^(16*e) + 8008*a^14*b^8*e^(16*e) - 11440*a^13*b^9*e^(16
*e) + 12870*a^12*b^10*e^(16*e) - 11440*a^11*b^11*e^(16*e) + 8008*a^10*b^12*e^(16*e) - 4368*a^9*b^13*e^(16*e) +
1820*a^8*b^14*e^(16*e) - 560*a^7*b^15*e^(16*e) + 120*a^6*b^16*e^(16*e) - 16*a^5*b^17*e^(16*e) + a^4*b^18*e^(1
6*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) + ((3*a*e^e + 2*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^3 - 3*a^2*b + 3*a*b^2 - b^3)*sqrt(a - b)) - 2*(a*e^e + 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^3
- 3*a^2*b + 3*a*b^2 - b^3)*sqrt(-b)) - 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))^3*a*e^e + 7*(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))^2*a*sqrt(b)*e^e - 4*(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))^2*b^(3/2)*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))*a^2*e^e - 17*(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))*a*b*e^e + 8*(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))*b^2*e^e - 4*a^2*sqrt(b)*e^e + 9*a*b^(3/2)*e^e - 4*b
^(5/2)*e^e)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*((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))^2 + 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) + 4*a - 3*b)^2))/f^2

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {tanh}\left (e+f\,x\right )}^3}{{\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)^3/(a + b*sinh(e + f*x)^2)^(5/2),x)

[Out]

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

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