3.491 \(\int \frac{\tanh ^3(e+f x)}{(a+b \sinh ^2(e+f x))^{3/2}} \, dx\)
Optimal. Leaf size=122 \[ \frac{2 a+b}{2 f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}-\frac{(2 a+b) \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right )}{2 f (a-b)^{5/2}}+\frac{\text{sech}^2(e+f x)}{2 f (a-b) \sqrt{a+b \sinh ^2(e+f x)}} \]
[Out]
-((2*a + b)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]])/(2*(a - b)^(5/2)*f) + (2*a + b)/(2*(a - b)^2*f*S
qrt[a + b*Sinh[e + f*x]^2]) + Sech[e + f*x]^2/(2*(a - b)*f*Sqrt[a + b*Sinh[e + f*x]^2])
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Rubi [A] time = 0.139628, antiderivative size = 122, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used =
{3194, 78, 51, 63, 208} \[ \frac{2 a+b}{2 f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}-\frac{(2 a+b) \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right )}{2 f (a-b)^{5/2}}+\frac{\text{sech}^2(e+f x)}{2 f (a-b) \sqrt{a+b \sinh ^2(e+f x)}} \]
Antiderivative was successfully verified.
[In]
Int[Tanh[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(3/2),x]
[Out]
-((2*a + b)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]])/(2*(a - b)^(5/2)*f) + (2*a + b)/(2*(a - b)^2*f*S
qrt[a + b*Sinh[e + f*x]^2]) + Sech[e + f*x]^2/(2*(a - b)*f*Sqrt[a + b*Sinh[e + f*x]^2])
Rule 3194
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]
Rule 78
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] || LtQ
[p, n]))))
Rule 51
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 63
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 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rubi steps
\begin{align*} \int \frac{\tanh ^3(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{(1+x)^2 (a+b x)^{3/2}} \, dx,x,\sinh ^2(e+f x)\right )}{2 f}\\ &=\frac{\text{sech}^2(e+f x)}{2 (a-b) f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{(2 a+b) \operatorname{Subst}\left (\int \frac{1}{(1+x) (a+b x)^{3/2}} \, dx,x,\sinh ^2(e+f x)\right )}{4 (a-b) f}\\ &=\frac{2 a+b}{2 (a-b)^2 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\text{sech}^2(e+f x)}{2 (a-b) f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{(2 a+b) \operatorname{Subst}\left (\int \frac{1}{(1+x) \sqrt{a+b x}} \, dx,x,\sinh ^2(e+f x)\right )}{4 (a-b)^2 f}\\ &=\frac{2 a+b}{2 (a-b)^2 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\text{sech}^2(e+f x)}{2 (a-b) f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{(2 a+b) \operatorname{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)^2 b f}\\ &=-\frac{(2 a+b) \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right )}{2 (a-b)^{5/2} f}+\frac{2 a+b}{2 (a-b)^2 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\text{sech}^2(e+f x)}{2 (a-b) f \sqrt{a+b \sinh ^2(e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.117849, size = 79, normalized size = 0.65 \[ \frac{(2 a+b) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b \sinh ^2(e+f x)+a}{a-b}\right )+(a-b) \text{sech}^2(e+f x)}{2 f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[Tanh[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(3/2),x]
[Out]
((2*a + b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Sinh[e + f*x]^2)/(a - b)] + (a - b)*Sech[e + f*x]^2)/(2*(a -
b)^2*f*Sqrt[a + b*Sinh[e + f*x]^2])
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Maple [C] time = 0.195, size = 103, normalized size = 0.8 \begin{align*}{\frac{1}{f}\mbox{{\tt ` int/indef0`}} \left ( -{\frac{ \left ( \sinh \left ( fx+e \right ) \right ) ^{3} \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{-{b}^{2} \left ( \cosh \left ( fx+e \right ) \right ) ^{10}+ \left ( -2\,ab+2\,{b}^{2} \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{8}+ \left ( -{a}^{2}+2\,ab-{b}^{2} \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{6}}\sqrt{a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2}}},\sinh \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tanh(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(3/2),x)
[Out]
`int/indef0`(-sinh(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(1/2)*cosh(f*x+e)^2/(-b^2*cosh(f*x+e)^10+(-2*a*b+2*b^2)*cosh(f
*x+e)^8+(-a^2+2*a*b-b^2)*cosh(f*x+e)^6),sinh(f*x+e))/f
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh \left (f x + e\right )^{3}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")
[Out]
integrate(tanh(f*x + e)^3/(b*sinh(f*x + e)^2 + a)^(3/2), x)
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Fricas [B] time = 3.54302, size = 9603, normalized size = 78.71 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")
[Out]
[1/4*(((2*a*b + b^2)*cosh(f*x + e)^8 + 8*(2*a*b + b^2)*cosh(f*x + e)*sinh(f*x + e)^7 + (2*a*b + b^2)*sinh(f*x
+ e)^8 + 4*(2*a^2 + a*b)*cosh(f*x + e)^6 + 4*(7*(2*a*b + b^2)*cosh(f*x + e)^2 + 2*a^2 + a*b)*sinh(f*x + e)^6 +
8*(7*(2*a*b + b^2)*cosh(f*x + e)^3 + 3*(2*a^2 + a*b)*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(8*a^2 + 2*a*b - b^2)
*cosh(f*x + e)^4 + 2*(35*(2*a*b + b^2)*cosh(f*x + e)^4 + 30*(2*a^2 + a*b)*cosh(f*x + e)^2 + 8*a^2 + 2*a*b - b^
2)*sinh(f*x + e)^4 + 8*(7*(2*a*b + b^2)*cosh(f*x + e)^5 + 10*(2*a^2 + a*b)*cosh(f*x + e)^3 + (8*a^2 + 2*a*b -
b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(2*a^2 + a*b)*cosh(f*x + e)^2 + 4*(7*(2*a*b + b^2)*cosh(f*x + e)^6 + 1
5*(2*a^2 + a*b)*cosh(f*x + e)^4 + 3*(8*a^2 + 2*a*b - b^2)*cosh(f*x + e)^2 + 2*a^2 + a*b)*sinh(f*x + e)^2 + 2*a
*b + b^2 + 8*((2*a*b + b^2)*cosh(f*x + e)^7 + 3*(2*a^2 + a*b)*cosh(f*x + e)^5 + (8*a^2 + 2*a*b - b^2)*cosh(f*x
+ e)^3 + (2*a^2 + a*b)*cosh(f*x + e))*sinh(f*x + e))*sqrt(a - b)*log((b*cosh(f*x + e)^4 + 4*b*cosh(f*x + e)*s
inh(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)*((2*a^2 - a*b - b^2)*cosh(f*x + e)^5 + 5*(2*a^2 - a*b - b^2)*cosh(f*x + e)*sinh(f*x +
e)^4 + (2*a^2 - a*b - b^2)*sinh(f*x + e)^5 + 2*(4*a^2 - 5*a*b + b^2)*cosh(f*x + e)^3 + 2*(5*(2*a^2 - a*b - b^
2)*cosh(f*x + e)^2 + 4*a^2 - 5*a*b + b^2)*sinh(f*x + e)^3 + 2*(5*(2*a^2 - a*b - b^2)*cosh(f*x + e)^3 + 3*(4*a^
2 - 5*a*b + b^2)*cosh(f*x + e))*sinh(f*x + e)^2 + (2*a^2 - a*b - b^2)*cosh(f*x + e) + (5*(2*a^2 - a*b - b^2)*c
osh(f*x + e)^4 + 6*(4*a^2 - 5*a*b + b^2)*cosh(f*x + e)^2 + 2*a^2 - a*b - b^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^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*f*cosh(f*x + e)^8 + 8*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*f*cosh(f*x + e)*s
inh(f*x + e)^7 + (a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*f*sinh(f*x + e)^8 + 4*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)
*f*cosh(f*x + e)^6 + 4*(7*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*f*cosh(f*x + e)^2 + (a^4 - 3*a^3*b + 3*a^2*b^2 -
a*b^3)*f)*sinh(f*x + e)^6 + 2*(4*a^4 - 13*a^3*b + 15*a^2*b^2 - 7*a*b^3 + b^4)*f*cosh(f*x + e)^4 + 8*(7*(a^3*b
- 3*a^2*b^2 + 3*a*b^3 - b^4)*f*cosh(f*x + e)^3 + 3*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*f*cosh(f*x + e))*sinh(
f*x + e)^5 + 2*(35*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*f*cosh(f*x + e)^4 + 30*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b
^3)*f*cosh(f*x + e)^2 + (4*a^4 - 13*a^3*b + 15*a^2*b^2 - 7*a*b^3 + b^4)*f)*sinh(f*x + e)^4 + 4*(a^4 - 3*a^3*b
+ 3*a^2*b^2 - a*b^3)*f*cosh(f*x + e)^2 + 8*(7*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*f*cosh(f*x + e)^5 + 10*(a^4
- 3*a^3*b + 3*a^2*b^2 - a*b^3)*f*cosh(f*x + e)^3 + (4*a^4 - 13*a^3*b + 15*a^2*b^2 - 7*a*b^3 + b^4)*f*cosh(f*x
+ e))*sinh(f*x + e)^3 + 4*(7*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*f*cosh(f*x + e)^6 + 15*(a^4 - 3*a^3*b + 3*a^2
*b^2 - a*b^3)*f*cosh(f*x + e)^4 + 3*(4*a^4 - 13*a^3*b + 15*a^2*b^2 - 7*a*b^3 + b^4)*f*cosh(f*x + e)^2 + (a^4 -
3*a^3*b + 3*a^2*b^2 - a*b^3)*f)*sinh(f*x + e)^2 + (a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*f + 8*((a^3*b - 3*a^2*b
^2 + 3*a*b^3 - b^4)*f*cosh(f*x + e)^7 + 3*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*f*cosh(f*x + e)^5 + (4*a^4 - 13*
a^3*b + 15*a^2*b^2 - 7*a*b^3 + b^4)*f*cosh(f*x + e)^3 + (a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*f*cosh(f*x + e))*s
inh(f*x + e)), -1/2*(((2*a*b + b^2)*cosh(f*x + e)^8 + 8*(2*a*b + b^2)*cosh(f*x + e)*sinh(f*x + e)^7 + (2*a*b +
b^2)*sinh(f*x + e)^8 + 4*(2*a^2 + a*b)*cosh(f*x + e)^6 + 4*(7*(2*a*b + b^2)*cosh(f*x + e)^2 + 2*a^2 + a*b)*si
nh(f*x + e)^6 + 8*(7*(2*a*b + b^2)*cosh(f*x + e)^3 + 3*(2*a^2 + a*b)*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(8*a^2
+ 2*a*b - b^2)*cosh(f*x + e)^4 + 2*(35*(2*a*b + b^2)*cosh(f*x + e)^4 + 30*(2*a^2 + a*b)*cosh(f*x + e)^2 + 8*a
^2 + 2*a*b - b^2)*sinh(f*x + e)^4 + 8*(7*(2*a*b + b^2)*cosh(f*x + e)^5 + 10*(2*a^2 + a*b)*cosh(f*x + e)^3 + (8
*a^2 + 2*a*b - b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(2*a^2 + a*b)*cosh(f*x + e)^2 + 4*(7*(2*a*b + b^2)*cosh
(f*x + e)^6 + 15*(2*a^2 + a*b)*cosh(f*x + e)^4 + 3*(8*a^2 + 2*a*b - b^2)*cosh(f*x + e)^2 + 2*a^2 + a*b)*sinh(f
*x + e)^2 + 2*a*b + b^2 + 8*((2*a*b + b^2)*cosh(f*x + e)^7 + 3*(2*a^2 + a*b)*cosh(f*x + e)^5 + (8*a^2 + 2*a*b
- b^2)*cosh(f*x + e)^3 + (2*a^2 + a*b)*cosh(f*x + e))*sinh(f*x + e))*sqrt(-a + b)*arctan(-1/2*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))/((a - b)*cosh(f*x + e) + (a - b)*sinh(f*x + e))) - 2*sqrt(2)*((2*a^2 - a*b - b^2)*cosh(f*x
+ e)^5 + 5*(2*a^2 - a*b - b^2)*cosh(f*x + e)*sinh(f*x + e)^4 + (2*a^2 - a*b - b^2)*sinh(f*x + e)^5 + 2*(4*a^2
- 5*a*b + b^2)*cosh(f*x + e)^3 + 2*(5*(2*a^2 - a*b - b^2)*cosh(f*x + e)^2 + 4*a^2 - 5*a*b + b^2)*sinh(f*x + e)
^3 + 2*(5*(2*a^2 - a*b - b^2)*cosh(f*x + e)^3 + 3*(4*a^2 - 5*a*b + b^2)*cosh(f*x + e))*sinh(f*x + e)^2 + (2*a^
2 - a*b - b^2)*cosh(f*x + e) + (5*(2*a^2 - a*b - b^2)*cosh(f*x + e)^4 + 6*(4*a^2 - 5*a*b + b^2)*cosh(f*x + e)^
2 + 2*a^2 - a*b - b^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^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*f*cosh(f*x + e)^8 +
8*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*f*cosh(f*x + e)*sinh(f*x + e)^7 + (a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*f*
sinh(f*x + e)^8 + 4*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*f*cosh(f*x + e)^6 + 4*(7*(a^3*b - 3*a^2*b^2 + 3*a*b^3
- b^4)*f*cosh(f*x + e)^2 + (a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*f)*sinh(f*x + e)^6 + 2*(4*a^4 - 13*a^3*b + 15*a
^2*b^2 - 7*a*b^3 + b^4)*f*cosh(f*x + e)^4 + 8*(7*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*f*cosh(f*x + e)^3 + 3*(a^
4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*f*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(35*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*
f*cosh(f*x + e)^4 + 30*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*f*cosh(f*x + e)^2 + (4*a^4 - 13*a^3*b + 15*a^2*b^2
- 7*a*b^3 + b^4)*f)*sinh(f*x + e)^4 + 4*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*f*cosh(f*x + e)^2 + 8*(7*(a^3*b -
3*a^2*b^2 + 3*a*b^3 - b^4)*f*cosh(f*x + e)^5 + 10*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*f*cosh(f*x + e)^3 + (4*a
^4 - 13*a^3*b + 15*a^2*b^2 - 7*a*b^3 + b^4)*f*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(7*(a^3*b - 3*a^2*b^2 + 3*a*b
^3 - b^4)*f*cosh(f*x + e)^6 + 15*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*f*cosh(f*x + e)^4 + 3*(4*a^4 - 13*a^3*b +
15*a^2*b^2 - 7*a*b^3 + b^4)*f*cosh(f*x + e)^2 + (a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*f)*sinh(f*x + e)^2 + (a^3
*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*f + 8*((a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*f*cosh(f*x + e)^7 + 3*(a^4 - 3*a^3*
b + 3*a^2*b^2 - a*b^3)*f*cosh(f*x + e)^5 + (4*a^4 - 13*a^3*b + 15*a^2*b^2 - 7*a*b^3 + b^4)*f*cosh(f*x + e)^3 +
(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*f*cosh(f*x + e))*sinh(f*x + e))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh ^{3}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(f*x+e)**3/(a+b*sinh(f*x+e)**2)**(3/2),x)
[Out]
Integral(tanh(e + f*x)**3/(a + b*sinh(e + f*x)**2)**(3/2), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh \left (f x + e\right )^{3}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")
[Out]
integrate(tanh(f*x + e)^3/(b*sinh(f*x + e)^2 + a)^(3/2), x)