3.385 \(\int \frac{\text{sech}^3(e+f x)}{(a+b \sinh ^2(e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=142 \[ \frac{b (a+2 b) \sinh (e+f x)}{2 a f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}+\frac{(a-4 b) \tan ^{-1}\left (\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right )}{2 f (a-b)^{5/2}}+\frac{\tanh (e+f x) \text{sech}(e+f x)}{2 f (a-b) \sqrt{a+b \sinh ^2(e+f x)}} \]

[Out]

((a - 4*b)*ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(2*(a - b)^(5/2)*f) + (b*(a + 2*b)
*Sinh[e + f*x])/(2*a*(a - b)^2*f*Sqrt[a + b*Sinh[e + f*x]^2]) + (Sech[e + f*x]*Tanh[e + f*x])/(2*(a - b)*f*Sqr
t[a + b*Sinh[e + f*x]^2])

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Rubi [A]  time = 0.164915, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {3190, 414, 527, 12, 377, 203} \[ \frac{b (a+2 b) \sinh (e+f x)}{2 a f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}+\frac{(a-4 b) \tan ^{-1}\left (\frac{\sqrt{a-b} \sinh (e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}}\right )}{2 f (a-b)^{5/2}}+\frac{\tanh (e+f x) \text{sech}(e+f x)}{2 f (a-b) \sqrt{a+b \sinh ^2(e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a - 4*b)*ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(2*(a - b)^(5/2)*f) + (b*(a + 2*b)
*Sinh[e + f*x])/(2*a*(a - b)^2*f*Sqrt[a + b*Sinh[e + f*x]^2]) + (Sech[e + f*x]*Tanh[e + f*x])/(2*(a - b)*f*Sqr
t[a + b*Sinh[e + f*x]^2])

Rule 3190

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e +
f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 527

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [C]  time = 5.36355, size = 231, normalized size = 1.63 \[ \frac{\tanh (e+f x) \text{sech}^5(e+f x) \left (16 (a-b) \sinh ^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^2 \text{HypergeometricPFQ}\left (\{2,2,3\},\left \{1,\frac{9}{2}\right \},\frac{(a-b) \tanh ^2(e+f x)}{a}\right )+16 (a-b) \sinh ^2(e+f x) \left (4 a^2+7 a b \sinh ^2(e+f x)+3 b^2 \sinh ^4(e+f x)\right ) \, _2F_1\left (2,3;\frac{9}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right )+7 a \cosh ^2(e+f x) \left (15 a^2+20 a b \sinh ^2(e+f x)+8 b^2 \sinh ^4(e+f x)\right ) \, _2F_1\left (1,2;\frac{7}{2};\frac{(a-b) \tanh ^2(e+f x)}{a}\right )\right )}{105 a^4 f \sqrt{a+b \sinh ^2(e+f x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sech[e + f*x]^5*(16*(a - b)*HypergeometricPFQ[{2, 2, 3}, {1, 9/2}, ((a - b)*Tanh[e + f*x]^2)/a]*Sinh[e + f*x]
^2*(a + b*Sinh[e + f*x]^2)^2 + 16*(a - b)*Hypergeometric2F1[2, 3, 9/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sinh[e + f
*x]^2*(4*a^2 + 7*a*b*Sinh[e + f*x]^2 + 3*b^2*Sinh[e + f*x]^4) + 7*a*Cosh[e + f*x]^2*Hypergeometric2F1[1, 2, 7/
2, ((a - b)*Tanh[e + f*x]^2)/a]*(15*a^2 + 20*a*b*Sinh[e + f*x]^2 + 8*b^2*Sinh[e + f*x]^4))*Tanh[e + f*x])/(105
*a^4*f*Sqrt[a + b*Sinh[e + f*x]^2])

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Maple [C]  time = 0.149, size = 95, normalized size = 0.7 \begin{align*}{\frac{1}{f}\mbox{{\tt ` int/indef0`}} \left ( -{\frac{ \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(sech(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(3/2),x)

[Out]

`int/indef0`(-(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{\operatorname{sech}\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(sech(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate(sech(f*x + e)^3/(b*sinh(f*x + e)^2 + a)^(3/2), x)

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Fricas [B]  time = 4.65664, size = 11344, normalized size = 79.89 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

[1/4*(((a^2*b - 4*a*b^2)*cosh(f*x + e)^8 + 8*(a^2*b - 4*a*b^2)*cosh(f*x + e)*sinh(f*x + e)^7 + (a^2*b - 4*a*b^
2)*sinh(f*x + e)^8 + 4*(a^3 - 4*a^2*b)*cosh(f*x + e)^6 + 4*(a^3 - 4*a^2*b + 7*(a^2*b - 4*a*b^2)*cosh(f*x + e)^
2)*sinh(f*x + e)^6 + 8*(7*(a^2*b - 4*a*b^2)*cosh(f*x + e)^3 + 3*(a^3 - 4*a^2*b)*cosh(f*x + e))*sinh(f*x + e)^5
+ 2*(4*a^3 - 17*a^2*b + 4*a*b^2)*cosh(f*x + e)^4 + 2*(35*(a^2*b - 4*a*b^2)*cosh(f*x + e)^4 + 4*a^3 - 17*a^2*b
+ 4*a*b^2 + 30*(a^3 - 4*a^2*b)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 8*(7*(a^2*b - 4*a*b^2)*cosh(f*x + e)^5 + 10
*(a^3 - 4*a^2*b)*cosh(f*x + e)^3 + (4*a^3 - 17*a^2*b + 4*a*b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + a^2*b - 4*a*b
^2 + 4*(a^3 - 4*a^2*b)*cosh(f*x + e)^2 + 4*(7*(a^2*b - 4*a*b^2)*cosh(f*x + e)^6 + 15*(a^3 - 4*a^2*b)*cosh(f*x
+ e)^4 + a^3 - 4*a^2*b + 3*(4*a^3 - 17*a^2*b + 4*a*b^2)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 8*((a^2*b - 4*a*b^2
)*cosh(f*x + e)^7 + 3*(a^3 - 4*a^2*b)*cosh(f*x + e)^5 + (4*a^3 - 17*a^2*b + 4*a*b^2)*cosh(f*x + e)^3 + (a^3 -
4*a^2*b)*cosh(f*x + e))*sinh(f*x + e))*sqrt(-a + b)*log(((a - 2*b)*cosh(f*x + e)^4 + 4*(a - 2*b)*cosh(f*x + e)
*sinh(f*x + e)^3 + (a - 2*b)*sinh(f*x + e)^4 - 2*(3*a - 2*b)*cosh(f*x + e)^2 + 2*(3*(a - 2*b)*cosh(f*x + e)^2
- 3*a + 2*b)*sinh(f*x + e)^2 + 2*sqrt(2)*(cosh(f*x + e)^2 + 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 -
1)*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)) + 4*((a - 2*b)*cosh(f*x + e)^3 - (3*a - 2*b)*cosh(f*x + e))*sinh(f*x + e) + a -
2*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)) + 2*sqrt(2)*((a^2*b + a*
b^2 - 2*b^3)*cosh(f*x + e)^6 + 6*(a^2*b + a*b^2 - 2*b^3)*cosh(f*x + e)*sinh(f*x + e)^5 + (a^2*b + a*b^2 - 2*b^
3)*sinh(f*x + e)^6 + (4*a^3 - 7*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e)^4 + (4*a^3 - 7*a^2*b + 5*a*b^2 - 2*b^3
+ 15*(a^2*b + a*b^2 - 2*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 4*(5*(a^2*b + a*b^2 - 2*b^3)*cosh(f*x + e)^3 +
(4*a^3 - 7*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e))*sinh(f*x + e)^3 - a^2*b - a*b^2 + 2*b^3 - (4*a^3 - 7*a^2*b
+ 5*a*b^2 - 2*b^3)*cosh(f*x + e)^2 + (15*(a^2*b + a*b^2 - 2*b^3)*cosh(f*x + e)^4 - 4*a^3 + 7*a^2*b - 5*a*b^2
+ 2*b^3 + 6*(4*a^3 - 7*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*(3*(a^2*b + a*b^2 - 2*b^3
)*cosh(f*x + e)^5 + 2*(4*a^3 - 7*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e)^3 - (4*a^3 - 7*a^2*b + 5*a*b^2 - 2*b^3
)*cosh(f*x + e))*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*co
sh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)))/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*f*cosh(f*x + e)^8 + 8*
(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*f*cosh(f*x + e)*sinh(f*x + e)^7 + (a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b
^4)*f*sinh(f*x + e)^8 + 4*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*f*cosh(f*x + e)^6 + 4*(7*(a^4*b - 3*a^3*b^2 +
3*a^2*b^3 - a*b^4)*f*cosh(f*x + e)^2 + (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*f)*sinh(f*x + e)^6 + 2*(4*a^5 - 1
3*a^4*b + 15*a^3*b^2 - 7*a^2*b^3 + a*b^4)*f*cosh(f*x + e)^4 + 8*(7*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*f*c
osh(f*x + e)^3 + 3*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*f*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(35*(a^4*b - 3*a
^3*b^2 + 3*a^2*b^3 - a*b^4)*f*cosh(f*x + e)^4 + 30*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*f*cosh(f*x + e)^2 + (
4*a^5 - 13*a^4*b + 15*a^3*b^2 - 7*a^2*b^3 + a*b^4)*f)*sinh(f*x + e)^4 + 4*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3
)*f*cosh(f*x + e)^2 + 8*(7*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*f*cosh(f*x + e)^5 + 10*(a^5 - 3*a^4*b + 3*a
^3*b^2 - a^2*b^3)*f*cosh(f*x + e)^3 + (4*a^5 - 13*a^4*b + 15*a^3*b^2 - 7*a^2*b^3 + a*b^4)*f*cosh(f*x + e))*sin
h(f*x + e)^3 + 4*(7*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*f*cosh(f*x + e)^6 + 15*(a^5 - 3*a^4*b + 3*a^3*b^2
- a^2*b^3)*f*cosh(f*x + e)^4 + 3*(4*a^5 - 13*a^4*b + 15*a^3*b^2 - 7*a^2*b^3 + a*b^4)*f*cosh(f*x + e)^2 + (a^5
- 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*f)*sinh(f*x + e)^2 + (a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*f + 8*((a^4*b -
3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*f*cosh(f*x + e)^7 + 3*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*f*cosh(f*x + e)^5 +
(4*a^5 - 13*a^4*b + 15*a^3*b^2 - 7*a^2*b^3 + a*b^4)*f*cosh(f*x + e)^3 + (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)
*f*cosh(f*x + e))*sinh(f*x + e)), 1/2*(((a^2*b - 4*a*b^2)*cosh(f*x + e)^8 + 8*(a^2*b - 4*a*b^2)*cosh(f*x + e)*
sinh(f*x + e)^7 + (a^2*b - 4*a*b^2)*sinh(f*x + e)^8 + 4*(a^3 - 4*a^2*b)*cosh(f*x + e)^6 + 4*(a^3 - 4*a^2*b + 7
*(a^2*b - 4*a*b^2)*cosh(f*x + e)^2)*sinh(f*x + e)^6 + 8*(7*(a^2*b - 4*a*b^2)*cosh(f*x + e)^3 + 3*(a^3 - 4*a^2*
b)*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(4*a^3 - 17*a^2*b + 4*a*b^2)*cosh(f*x + e)^4 + 2*(35*(a^2*b - 4*a*b^2)*c
osh(f*x + e)^4 + 4*a^3 - 17*a^2*b + 4*a*b^2 + 30*(a^3 - 4*a^2*b)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 8*(7*(a^2*
b - 4*a*b^2)*cosh(f*x + e)^5 + 10*(a^3 - 4*a^2*b)*cosh(f*x + e)^3 + (4*a^3 - 17*a^2*b + 4*a*b^2)*cosh(f*x + e)
)*sinh(f*x + e)^3 + a^2*b - 4*a*b^2 + 4*(a^3 - 4*a^2*b)*cosh(f*x + e)^2 + 4*(7*(a^2*b - 4*a*b^2)*cosh(f*x + e)
^6 + 15*(a^3 - 4*a^2*b)*cosh(f*x + e)^4 + a^3 - 4*a^2*b + 3*(4*a^3 - 17*a^2*b + 4*a*b^2)*cosh(f*x + e)^2)*sinh
(f*x + e)^2 + 8*((a^2*b - 4*a*b^2)*cosh(f*x + e)^7 + 3*(a^3 - 4*a^2*b)*cosh(f*x + e)^5 + (4*a^3 - 17*a^2*b + 4
*a*b^2)*cosh(f*x + e)^3 + (a^3 - 4*a^2*b)*cosh(f*x + e))*sinh(f*x + e))*sqrt(a - b)*arctan(sqrt(2)*(cosh(f*x +
e)^2 + 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 - 1)*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))/(b*cosh(f*x + e)^4 + 4*
b*cosh(f*x + e)*sinh(f*x + e)^3 + b*sinh(f*x + e)^4 + 2*(2*a - b)*cosh(f*x + e)^2 + 2*(3*b*cosh(f*x + e)^2 + 2
*a - b)*sinh(f*x + e)^2 + 4*(b*cosh(f*x + e)^3 + (2*a - b)*cosh(f*x + e))*sinh(f*x + e) + b)) + sqrt(2)*((a^2*
b + a*b^2 - 2*b^3)*cosh(f*x + e)^6 + 6*(a^2*b + a*b^2 - 2*b^3)*cosh(f*x + e)*sinh(f*x + e)^5 + (a^2*b + a*b^2
- 2*b^3)*sinh(f*x + e)^6 + (4*a^3 - 7*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e)^4 + (4*a^3 - 7*a^2*b + 5*a*b^2 -
2*b^3 + 15*(a^2*b + a*b^2 - 2*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 4*(5*(a^2*b + a*b^2 - 2*b^3)*cosh(f*x +
e)^3 + (4*a^3 - 7*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e))*sinh(f*x + e)^3 - a^2*b - a*b^2 + 2*b^3 - (4*a^3 - 7
*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e)^2 + (15*(a^2*b + a*b^2 - 2*b^3)*cosh(f*x + e)^4 - 4*a^3 + 7*a^2*b - 5*
a*b^2 + 2*b^3 + 6*(4*a^3 - 7*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*(3*(a^2*b + a*b^2 -
2*b^3)*cosh(f*x + e)^5 + 2*(4*a^3 - 7*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e)^3 - (4*a^3 - 7*a^2*b + 5*a*b^2 -
2*b^3)*cosh(f*x + e))*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 - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*f*cosh(f*x + e)^
8 + 8*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*f*cosh(f*x + e)*sinh(f*x + e)^7 + (a^4*b - 3*a^3*b^2 + 3*a^2*b^3
- a*b^4)*f*sinh(f*x + e)^8 + 4*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*f*cosh(f*x + e)^6 + 4*(7*(a^4*b - 3*a^3*
b^2 + 3*a^2*b^3 - a*b^4)*f*cosh(f*x + e)^2 + (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*f)*sinh(f*x + e)^6 + 2*(4*a
^5 - 13*a^4*b + 15*a^3*b^2 - 7*a^2*b^3 + a*b^4)*f*cosh(f*x + e)^4 + 8*(7*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^
4)*f*cosh(f*x + e)^3 + 3*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*f*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(35*(a^4*b
- 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*f*cosh(f*x + e)^4 + 30*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*f*cosh(f*x + e)
^2 + (4*a^5 - 13*a^4*b + 15*a^3*b^2 - 7*a^2*b^3 + a*b^4)*f)*sinh(f*x + e)^4 + 4*(a^5 - 3*a^4*b + 3*a^3*b^2 - a
^2*b^3)*f*cosh(f*x + e)^2 + 8*(7*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*f*cosh(f*x + e)^5 + 10*(a^5 - 3*a^4*b
+ 3*a^3*b^2 - a^2*b^3)*f*cosh(f*x + e)^3 + (4*a^5 - 13*a^4*b + 15*a^3*b^2 - 7*a^2*b^3 + a*b^4)*f*cosh(f*x + e
))*sinh(f*x + e)^3 + 4*(7*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*f*cosh(f*x + e)^6 + 15*(a^5 - 3*a^4*b + 3*a^
3*b^2 - a^2*b^3)*f*cosh(f*x + e)^4 + 3*(4*a^5 - 13*a^4*b + 15*a^3*b^2 - 7*a^2*b^3 + a*b^4)*f*cosh(f*x + e)^2 +
(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*f)*sinh(f*x + e)^2 + (a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*f + 8*((a^
4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*f*cosh(f*x + e)^7 + 3*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*f*cosh(f*x +
e)^5 + (4*a^5 - 13*a^4*b + 15*a^3*b^2 - 7*a^2*b^3 + a*b^4)*f*cosh(f*x + e)^3 + (a^5 - 3*a^4*b + 3*a^3*b^2 - a^
2*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{\operatorname{sech}^{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(sech(f*x+e)**3/(a+b*sinh(f*x+e)**2)**(3/2),x)

[Out]

Integral(sech(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{\operatorname{sech}\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(sech(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

integrate(sech(f*x + e)^3/(b*sinh(f*x + e)^2 + a)^(3/2), x)