3.81 \(\int \text{csch}^7(e+f x) (a+b \sinh ^2(e+f x))^{3/2} \, dx\)

Optimal. Leaf size=199 \[ \frac{(a-b)^2 (5 a+b) \tanh ^{-1}\left (\frac{\sqrt{a} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right )}{16 a^{3/2} f}-\frac{\coth (e+f x) \text{csch}^5(e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{5/2}}{6 a f}+\frac{(5 a+b) \coth (e+f x) \text{csch}^3(e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}}{24 a f}-\frac{(a-b) (5 a+b) \coth (e+f x) \text{csch}(e+f x) \sqrt{a+b \cosh ^2(e+f x)-b}}{16 a f} \]

[Out]

((a - b)^2*(5*a + b)*ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(16*a^(3/2)*f) - ((a -
b)*(5*a + b)*Sqrt[a - b + b*Cosh[e + f*x]^2]*Coth[e + f*x]*Csch[e + f*x])/(16*a*f) + ((5*a + b)*(a - b + b*Cos
h[e + f*x]^2)^(3/2)*Coth[e + f*x]*Csch[e + f*x]^3)/(24*a*f) - ((a - b + b*Cosh[e + f*x]^2)^(5/2)*Coth[e + f*x]
*Csch[e + f*x]^5)/(6*a*f)

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Rubi [A]  time = 0.199041, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3186, 382, 378, 377, 206} \[ \frac{(a-b)^2 (5 a+b) \tanh ^{-1}\left (\frac{\sqrt{a} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right )}{16 a^{3/2} f}-\frac{\coth (e+f x) \text{csch}^5(e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{5/2}}{6 a f}+\frac{(5 a+b) \coth (e+f x) \text{csch}^3(e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}}{24 a f}-\frac{(a-b) (5 a+b) \coth (e+f x) \text{csch}(e+f x) \sqrt{a+b \cosh ^2(e+f x)-b}}{16 a f} \]

Antiderivative was successfully verified.

[In]

Int[Csch[e + f*x]^7*(a + b*Sinh[e + f*x]^2)^(3/2),x]

[Out]

((a - b)^2*(5*a + b)*ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(16*a^(3/2)*f) - ((a -
b)*(5*a + b)*Sqrt[a - b + b*Cosh[e + f*x]^2]*Coth[e + f*x]*Csch[e + f*x])/(16*a*f) + ((5*a + b)*(a - b + b*Cos
h[e + f*x]^2)^(3/2)*Coth[e + f*x]*Csch[e + f*x]^3)/(24*a*f) - ((a - b + b*Cosh[e + f*x]^2)^(5/2)*Coth[e + f*x]
*Csch[e + f*x]^5)/(6*a*f)

Rule 3186

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

Rule 382

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[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d
)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[
n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rule 378

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

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 206

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

Rubi steps

\begin{align*} \int \text{csch}^7(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a-b+b x^2\right )^{3/2}}{\left (1-x^2\right )^4} \, dx,x,\cosh (e+f x)\right )}{f}\\ &=-\frac{\left (a-b+b \cosh ^2(e+f x)\right )^{5/2} \coth (e+f x) \text{csch}^5(e+f x)}{6 a f}+\frac{(5 a+b) \operatorname{Subst}\left (\int \frac{\left (a-b+b x^2\right )^{3/2}}{\left (1-x^2\right )^3} \, dx,x,\cosh (e+f x)\right )}{6 a f}\\ &=\frac{(5 a+b) \left (a-b+b \cosh ^2(e+f x)\right )^{3/2} \coth (e+f x) \text{csch}^3(e+f x)}{24 a f}-\frac{\left (a-b+b \cosh ^2(e+f x)\right )^{5/2} \coth (e+f x) \text{csch}^5(e+f x)}{6 a f}+\frac{((a-b) (5 a+b)) \operatorname{Subst}\left (\int \frac{\sqrt{a-b+b x^2}}{\left (1-x^2\right )^2} \, dx,x,\cosh (e+f x)\right )}{8 a f}\\ &=-\frac{(a-b) (5 a+b) \sqrt{a-b+b \cosh ^2(e+f x)} \coth (e+f x) \text{csch}(e+f x)}{16 a f}+\frac{(5 a+b) \left (a-b+b \cosh ^2(e+f x)\right )^{3/2} \coth (e+f x) \text{csch}^3(e+f x)}{24 a f}-\frac{\left (a-b+b \cosh ^2(e+f x)\right )^{5/2} \coth (e+f x) \text{csch}^5(e+f x)}{6 a f}+\frac{\left ((a-b)^2 (5 a+b)\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{a-b+b x^2}} \, dx,x,\cosh (e+f x)\right )}{16 a f}\\ &=-\frac{(a-b) (5 a+b) \sqrt{a-b+b \cosh ^2(e+f x)} \coth (e+f x) \text{csch}(e+f x)}{16 a f}+\frac{(5 a+b) \left (a-b+b \cosh ^2(e+f x)\right )^{3/2} \coth (e+f x) \text{csch}^3(e+f x)}{24 a f}-\frac{\left (a-b+b \cosh ^2(e+f x)\right )^{5/2} \coth (e+f x) \text{csch}^5(e+f x)}{6 a f}+\frac{\left ((a-b)^2 (5 a+b)\right ) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{\cosh (e+f x)}{\sqrt{a-b+b \cosh ^2(e+f x)}}\right )}{16 a f}\\ &=\frac{(a-b)^2 (5 a+b) \tanh ^{-1}\left (\frac{\sqrt{a} \cosh (e+f x)}{\sqrt{a-b+b \cosh ^2(e+f x)}}\right )}{16 a^{3/2} f}-\frac{(a-b) (5 a+b) \sqrt{a-b+b \cosh ^2(e+f x)} \coth (e+f x) \text{csch}(e+f x)}{16 a f}+\frac{(5 a+b) \left (a-b+b \cosh ^2(e+f x)\right )^{3/2} \coth (e+f x) \text{csch}^3(e+f x)}{24 a f}-\frac{\left (a-b+b \cosh ^2(e+f x)\right )^{5/2} \coth (e+f x) \text{csch}^5(e+f x)}{6 a f}\\ \end{align*}

Mathematica [A]  time = 1.01245, size = 174, normalized size = 0.87 \[ \frac{\frac{(a-b)^2 (5 a+b) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \cosh (e+f x)}{\sqrt{2 a+b \cosh (2 (e+f x))-b}}\right )}{a^{3/2}}-\frac{\coth (e+f x) \text{csch}^5(e+f x) \sqrt{a+\frac{1}{2} b \cosh (2 (e+f x))-\frac{b}{2}} \left (-4 \left (25 a^2-36 a b+3 b^2\right ) \cosh (2 (e+f x))+\left (15 a^2-22 a b+3 b^2\right ) \cosh (4 (e+f x))+149 a^2-122 a b+9 b^2\right )}{24 a}}{16 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[e + f*x]^7*(a + b*Sinh[e + f*x]^2)^(3/2),x]

[Out]

(((a - b)^2*(5*a + b)*ArcTanh[(Sqrt[2]*Sqrt[a]*Cosh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]])/a^(3/2) -
(Sqrt[a - b/2 + (b*Cosh[2*(e + f*x)])/2]*(149*a^2 - 122*a*b + 9*b^2 - 4*(25*a^2 - 36*a*b + 3*b^2)*Cosh[2*(e +
f*x)] + (15*a^2 - 22*a*b + 3*b^2)*Cosh[4*(e + f*x)])*Coth[e + f*x]*Csch[e + f*x]^5)/(24*a))/(16*f)

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Maple [B]  time = 0.122, size = 569, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(f*x+e)^7*(a+b*sinh(f*x+e)^2)^(3/2),x)

[Out]

-1/96*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*(30*a^(7/2)*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*sinh(f*x
+e)^4-44*b*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*sinh(f*x+e)^4*a^(5/2)-15*a^4*ln(((a+b)*cosh(f*x+e)^2+2*a^
(1/2)*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)+a-b)/sinh(f*x+e)^2)*sinh(f*x+e)^6+27*a^3*b*ln(((a+b)*cosh(f*
x+e)^2+2*a^(1/2)*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)+a-b)/sinh(f*x+e)^2)*sinh(f*x+e)^6-9*b^2*ln(((a+b)
*cosh(f*x+e)^2+2*a^(1/2)*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)+a-b)/sinh(f*x+e)^2)*sinh(f*x+e)^6*a^2-3*b
^3*ln(((a+b)*cosh(f*x+e)^2+2*a^(1/2)*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)+a-b)/sinh(f*x+e)^2)*sinh(f*x+
e)^6*a-20*a^(7/2)*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*sinh(f*x+e)^2+6*b^2*((a+b*sinh(f*x+e)^2)*cosh(f*x+
e)^2)^(1/2)*sinh(f*x+e)^4*a^(3/2)+28*b*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*sinh(f*x+e)^2*a^(5/2)+16*a^(7
/2)*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2))/sinh(f*x+e)^6/a^(5/2)/cosh(f*x+e)/(a+b*sinh(f*x+e)^2)^(1/2)/f

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} \operatorname{csch}\left (f x + e\right )^{7}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)^7*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*sinh(f*x + e)^2 + a)^(3/2)*csch(f*x + e)^7, x)

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Fricas [B]  time = 10.1742, size = 18294, normalized size = 91.93 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)^7*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

[1/96*(3*((5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^12 + 12*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x +
e)*sinh(f*x + e)^11 + (5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*sinh(f*x + e)^12 - 6*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3
)*cosh(f*x + e)^10 - 6*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3 - 11*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^2
)*sinh(f*x + e)^10 + 20*(11*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^3 - 3*(5*a^3 - 9*a^2*b + 3*a*b^2 +
b^3)*cosh(f*x + e))*sinh(f*x + e)^9 + 15*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^8 + 15*(33*(5*a^3 -
9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^4 + 5*a^3 - 9*a^2*b + 3*a*b^2 + b^3 - 18*(5*a^3 - 9*a^2*b + 3*a*b^2 + b
^3)*cosh(f*x + e)^2)*sinh(f*x + e)^8 + 24*(33*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^5 - 30*(5*a^3 -
9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^3 + 5*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e))*sinh(f*x + e)^7
- 20*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^6 + 4*(231*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e
)^6 - 315*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^4 - 25*a^3 + 45*a^2*b - 15*a*b^2 - 5*b^3 + 105*(5*a^
3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^6 + 24*(33*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(
f*x + e)^7 - 63*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^5 + 35*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(
f*x + e)^3 - 5*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e))*sinh(f*x + e)^5 + 15*(5*a^3 - 9*a^2*b + 3*a*b^
2 + b^3)*cosh(f*x + e)^4 + 15*(33*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^8 - 84*(5*a^3 - 9*a^2*b + 3*
a*b^2 + b^3)*cosh(f*x + e)^6 + 70*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^4 + 5*a^3 - 9*a^2*b + 3*a*b^
2 + b^3 - 20*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 20*(11*(5*a^3 - 9*a^2*b + 3*
a*b^2 + b^3)*cosh(f*x + e)^9 - 36*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^7 + 42*(5*a^3 - 9*a^2*b + 3*
a*b^2 + b^3)*cosh(f*x + e)^5 - 20*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^3 + 3*(5*a^3 - 9*a^2*b + 3*a
*b^2 + b^3)*cosh(f*x + e))*sinh(f*x + e)^3 + 5*a^3 - 9*a^2*b + 3*a*b^2 + b^3 - 6*(5*a^3 - 9*a^2*b + 3*a*b^2 +
b^3)*cosh(f*x + e)^2 + 6*(11*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^10 - 45*(5*a^3 - 9*a^2*b + 3*a*b^
2 + b^3)*cosh(f*x + e)^8 + 70*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^6 - 50*(5*a^3 - 9*a^2*b + 3*a*b^
2 + b^3)*cosh(f*x + e)^4 - 5*a^3 + 9*a^2*b - 3*a*b^2 - b^3 + 15*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e
)^2)*sinh(f*x + e)^2 + 12*((5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^11 - 5*(5*a^3 - 9*a^2*b + 3*a*b^2 +
b^3)*cosh(f*x + e)^9 + 10*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^7 - 10*(5*a^3 - 9*a^2*b + 3*a*b^2 +
b^3)*cosh(f*x + e)^5 + 5*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^3 - (5*a^3 - 9*a^2*b + 3*a*b^2 + b^3
)*cosh(f*x + e))*sinh(f*x + e))*sqrt(a)*log(-((a + b)*cosh(f*x + e)^4 + 4*(a + b)*cosh(f*x + e)*sinh(f*x + e)^
3 + (a + b)*sinh(f*x + e)^4 + 2*(3*a - b)*cosh(f*x + e)^2 + 2*(3*(a + b)*cosh(f*x + e)^2 + 3*a - 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)*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 + b)*cosh(f*x + e)^3 + (3*a - b)*cosh(f*x + e))*sinh(f*x + e) + a + 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*(c
osh(f*x + e)^3 - cosh(f*x + e))*sinh(f*x + e) + 1)) - 2*sqrt(2)*((15*a^3 - 22*a^2*b + 3*a*b^2)*cosh(f*x + e)^1
0 + 10*(15*a^3 - 22*a^2*b + 3*a*b^2)*cosh(f*x + e)*sinh(f*x + e)^9 + (15*a^3 - 22*a^2*b + 3*a*b^2)*sinh(f*x +
e)^10 - (85*a^3 - 122*a^2*b + 9*a*b^2)*cosh(f*x + e)^8 - (85*a^3 - 122*a^2*b + 9*a*b^2 - 45*(15*a^3 - 22*a^2*b
+ 3*a*b^2)*cosh(f*x + e)^2)*sinh(f*x + e)^8 + 8*(15*(15*a^3 - 22*a^2*b + 3*a*b^2)*cosh(f*x + e)^3 - (85*a^3 -
122*a^2*b + 9*a*b^2)*cosh(f*x + e))*sinh(f*x + e)^7 + 2*(99*a^3 - 50*a^2*b + 3*a*b^2)*cosh(f*x + e)^6 + 2*(10
5*(15*a^3 - 22*a^2*b + 3*a*b^2)*cosh(f*x + e)^4 + 99*a^3 - 50*a^2*b + 3*a*b^2 - 14*(85*a^3 - 122*a^2*b + 9*a*b
^2)*cosh(f*x + e)^2)*sinh(f*x + e)^6 + 4*(63*(15*a^3 - 22*a^2*b + 3*a*b^2)*cosh(f*x + e)^5 - 14*(85*a^3 - 122*
a^2*b + 9*a*b^2)*cosh(f*x + e)^3 + 3*(99*a^3 - 50*a^2*b + 3*a*b^2)*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(99*a^3
- 50*a^2*b + 3*a*b^2)*cosh(f*x + e)^4 + 2*(105*(15*a^3 - 22*a^2*b + 3*a*b^2)*cosh(f*x + e)^6 - 35*(85*a^3 - 12
2*a^2*b + 9*a*b^2)*cosh(f*x + e)^4 + 99*a^3 - 50*a^2*b + 3*a*b^2 + 15*(99*a^3 - 50*a^2*b + 3*a*b^2)*cosh(f*x +
e)^2)*sinh(f*x + e)^4 + 8*(15*(15*a^3 - 22*a^2*b + 3*a*b^2)*cosh(f*x + e)^7 - 7*(85*a^3 - 122*a^2*b + 9*a*b^2
)*cosh(f*x + e)^5 + 5*(99*a^3 - 50*a^2*b + 3*a*b^2)*cosh(f*x + e)^3 + (99*a^3 - 50*a^2*b + 3*a*b^2)*cosh(f*x +
e))*sinh(f*x + e)^3 + 15*a^3 - 22*a^2*b + 3*a*b^2 - (85*a^3 - 122*a^2*b + 9*a*b^2)*cosh(f*x + e)^2 + (45*(15*
a^3 - 22*a^2*b + 3*a*b^2)*cosh(f*x + e)^8 - 28*(85*a^3 - 122*a^2*b + 9*a*b^2)*cosh(f*x + e)^6 + 30*(99*a^3 - 5
0*a^2*b + 3*a*b^2)*cosh(f*x + e)^4 - 85*a^3 + 122*a^2*b - 9*a*b^2 + 12*(99*a^3 - 50*a^2*b + 3*a*b^2)*cosh(f*x
+ e)^2)*sinh(f*x + e)^2 + 2*(5*(15*a^3 - 22*a^2*b + 3*a*b^2)*cosh(f*x + e)^9 - 4*(85*a^3 - 122*a^2*b + 9*a*b^2
)*cosh(f*x + e)^7 + 6*(99*a^3 - 50*a^2*b + 3*a*b^2)*cosh(f*x + e)^5 + 4*(99*a^3 - 50*a^2*b + 3*a*b^2)*cosh(f*x
+ e)^3 - (85*a^3 - 122*a^2*b + 9*a*b^2)*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^2*f*cosh(f*x + e)^12
+ 12*a^2*f*cosh(f*x + e)*sinh(f*x + e)^11 + a^2*f*sinh(f*x + e)^12 - 6*a^2*f*cosh(f*x + e)^10 + 15*a^2*f*cosh(
f*x + e)^8 + 6*(11*a^2*f*cosh(f*x + e)^2 - a^2*f)*sinh(f*x + e)^10 + 20*(11*a^2*f*cosh(f*x + e)^3 - 3*a^2*f*co
sh(f*x + e))*sinh(f*x + e)^9 - 20*a^2*f*cosh(f*x + e)^6 + 15*(33*a^2*f*cosh(f*x + e)^4 - 18*a^2*f*cosh(f*x + e
)^2 + a^2*f)*sinh(f*x + e)^8 + 24*(33*a^2*f*cosh(f*x + e)^5 - 30*a^2*f*cosh(f*x + e)^3 + 5*a^2*f*cosh(f*x + e)
)*sinh(f*x + e)^7 + 15*a^2*f*cosh(f*x + e)^4 + 4*(231*a^2*f*cosh(f*x + e)^6 - 315*a^2*f*cosh(f*x + e)^4 + 105*
a^2*f*cosh(f*x + e)^2 - 5*a^2*f)*sinh(f*x + e)^6 + 24*(33*a^2*f*cosh(f*x + e)^7 - 63*a^2*f*cosh(f*x + e)^5 + 3
5*a^2*f*cosh(f*x + e)^3 - 5*a^2*f*cosh(f*x + e))*sinh(f*x + e)^5 - 6*a^2*f*cosh(f*x + e)^2 + 15*(33*a^2*f*cosh
(f*x + e)^8 - 84*a^2*f*cosh(f*x + e)^6 + 70*a^2*f*cosh(f*x + e)^4 - 20*a^2*f*cosh(f*x + e)^2 + a^2*f)*sinh(f*x
+ e)^4 + 20*(11*a^2*f*cosh(f*x + e)^9 - 36*a^2*f*cosh(f*x + e)^7 + 42*a^2*f*cosh(f*x + e)^5 - 20*a^2*f*cosh(f
*x + e)^3 + 3*a^2*f*cosh(f*x + e))*sinh(f*x + e)^3 + a^2*f + 6*(11*a^2*f*cosh(f*x + e)^10 - 45*a^2*f*cosh(f*x
+ e)^8 + 70*a^2*f*cosh(f*x + e)^6 - 50*a^2*f*cosh(f*x + e)^4 + 15*a^2*f*cosh(f*x + e)^2 - a^2*f)*sinh(f*x + e)
^2 + 12*(a^2*f*cosh(f*x + e)^11 - 5*a^2*f*cosh(f*x + e)^9 + 10*a^2*f*cosh(f*x + e)^7 - 10*a^2*f*cosh(f*x + e)^
5 + 5*a^2*f*cosh(f*x + e)^3 - a^2*f*cosh(f*x + e))*sinh(f*x + e)), -1/48*(3*((5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)
*cosh(f*x + e)^12 + 12*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)*sinh(f*x + e)^11 + (5*a^3 - 9*a^2*b + 3
*a*b^2 + b^3)*sinh(f*x + e)^12 - 6*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^10 - 6*(5*a^3 - 9*a^2*b + 3
*a*b^2 + b^3 - 11*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^10 + 20*(11*(5*a^3 - 9*a^2*
b + 3*a*b^2 + b^3)*cosh(f*x + e)^3 - 3*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e))*sinh(f*x + e)^9 + 15*(
5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^8 + 15*(33*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^4 +
5*a^3 - 9*a^2*b + 3*a*b^2 + b^3 - 18*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^8 + 24*(
33*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^5 - 30*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^3 +
5*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e))*sinh(f*x + e)^7 - 20*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh
(f*x + e)^6 + 4*(231*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^6 - 315*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)
*cosh(f*x + e)^4 - 25*a^3 + 45*a^2*b - 15*a*b^2 - 5*b^3 + 105*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^
2)*sinh(f*x + e)^6 + 24*(33*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^7 - 63*(5*a^3 - 9*a^2*b + 3*a*b^2
+ b^3)*cosh(f*x + e)^5 + 35*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^3 - 5*(5*a^3 - 9*a^2*b + 3*a*b^2 +
b^3)*cosh(f*x + e))*sinh(f*x + e)^5 + 15*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^4 + 15*(33*(5*a^3 -
9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^8 - 84*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^6 + 70*(5*a^3 -
9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^4 + 5*a^3 - 9*a^2*b + 3*a*b^2 + b^3 - 20*(5*a^3 - 9*a^2*b + 3*a*b^2 + b
^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 20*(11*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^9 - 36*(5*a^3 -
9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^7 + 42*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^5 - 20*(5*a^3 -
9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^3 + 3*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e))*sinh(f*x + e)^3
+ 5*a^3 - 9*a^2*b + 3*a*b^2 + b^3 - 6*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^2 + 6*(11*(5*a^3 - 9*a^2
*b + 3*a*b^2 + b^3)*cosh(f*x + e)^10 - 45*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^8 + 70*(5*a^3 - 9*a^
2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^6 - 50*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^4 - 5*a^3 + 9*a^2*b
- 3*a*b^2 - b^3 + 15*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 12*((5*a^3 - 9*a^2*b
+ 3*a*b^2 + b^3)*cosh(f*x + e)^11 - 5*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^9 + 10*(5*a^3 - 9*a^2*b
+ 3*a*b^2 + b^3)*cosh(f*x + e)^7 - 10*(5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e)^5 + 5*(5*a^3 - 9*a^2*b
+ 3*a*b^2 + b^3)*cosh(f*x + e)^3 - (5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cosh(f*x + e))*sinh(f*x + e))*sqrt(-a)*ar
ctan(sqrt(2)*(cosh(f*x + e)^2 + 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 + 1)*sqrt(-a)*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)*((15*a^3 - 22*a^2*b + 3*a*b^2)*cosh(f*x + e)^10 + 10*(15*a^3 - 22*a^2*b + 3*a*b^2)*cosh(f*x + e
)*sinh(f*x + e)^9 + (15*a^3 - 22*a^2*b + 3*a*b^2)*sinh(f*x + e)^10 - (85*a^3 - 122*a^2*b + 9*a*b^2)*cosh(f*x +
e)^8 - (85*a^3 - 122*a^2*b + 9*a*b^2 - 45*(15*a^3 - 22*a^2*b + 3*a*b^2)*cosh(f*x + e)^2)*sinh(f*x + e)^8 + 8*
(15*(15*a^3 - 22*a^2*b + 3*a*b^2)*cosh(f*x + e)^3 - (85*a^3 - 122*a^2*b + 9*a*b^2)*cosh(f*x + e))*sinh(f*x + e
)^7 + 2*(99*a^3 - 50*a^2*b + 3*a*b^2)*cosh(f*x + e)^6 + 2*(105*(15*a^3 - 22*a^2*b + 3*a*b^2)*cosh(f*x + e)^4 +
99*a^3 - 50*a^2*b + 3*a*b^2 - 14*(85*a^3 - 122*a^2*b + 9*a*b^2)*cosh(f*x + e)^2)*sinh(f*x + e)^6 + 4*(63*(15*
a^3 - 22*a^2*b + 3*a*b^2)*cosh(f*x + e)^5 - 14*(85*a^3 - 122*a^2*b + 9*a*b^2)*cosh(f*x + e)^3 + 3*(99*a^3 - 50
*a^2*b + 3*a*b^2)*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(99*a^3 - 50*a^2*b + 3*a*b^2)*cosh(f*x + e)^4 + 2*(105*(1
5*a^3 - 22*a^2*b + 3*a*b^2)*cosh(f*x + e)^6 - 35*(85*a^3 - 122*a^2*b + 9*a*b^2)*cosh(f*x + e)^4 + 99*a^3 - 50*
a^2*b + 3*a*b^2 + 15*(99*a^3 - 50*a^2*b + 3*a*b^2)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 8*(15*(15*a^3 - 22*a^2*b
+ 3*a*b^2)*cosh(f*x + e)^7 - 7*(85*a^3 - 122*a^2*b + 9*a*b^2)*cosh(f*x + e)^5 + 5*(99*a^3 - 50*a^2*b + 3*a*b^
2)*cosh(f*x + e)^3 + (99*a^3 - 50*a^2*b + 3*a*b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 15*a^3 - 22*a^2*b + 3*a*b^
2 - (85*a^3 - 122*a^2*b + 9*a*b^2)*cosh(f*x + e)^2 + (45*(15*a^3 - 22*a^2*b + 3*a*b^2)*cosh(f*x + e)^8 - 28*(8
5*a^3 - 122*a^2*b + 9*a*b^2)*cosh(f*x + e)^6 + 30*(99*a^3 - 50*a^2*b + 3*a*b^2)*cosh(f*x + e)^4 - 85*a^3 + 122
*a^2*b - 9*a*b^2 + 12*(99*a^3 - 50*a^2*b + 3*a*b^2)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*(5*(15*a^3 - 22*a^2*b
+ 3*a*b^2)*cosh(f*x + e)^9 - 4*(85*a^3 - 122*a^2*b + 9*a*b^2)*cosh(f*x + e)^7 + 6*(99*a^3 - 50*a^2*b + 3*a*b^
2)*cosh(f*x + e)^5 + 4*(99*a^3 - 50*a^2*b + 3*a*b^2)*cosh(f*x + e)^3 - (85*a^3 - 122*a^2*b + 9*a*b^2)*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^2*f*cosh(f*x + e)^12 + 12*a^2*f*cosh(f*x + e)*sinh(f*x + e)^11 + a^2*f
*sinh(f*x + e)^12 - 6*a^2*f*cosh(f*x + e)^10 + 15*a^2*f*cosh(f*x + e)^8 + 6*(11*a^2*f*cosh(f*x + e)^2 - a^2*f)
*sinh(f*x + e)^10 + 20*(11*a^2*f*cosh(f*x + e)^3 - 3*a^2*f*cosh(f*x + e))*sinh(f*x + e)^9 - 20*a^2*f*cosh(f*x
+ e)^6 + 15*(33*a^2*f*cosh(f*x + e)^4 - 18*a^2*f*cosh(f*x + e)^2 + a^2*f)*sinh(f*x + e)^8 + 24*(33*a^2*f*cosh(
f*x + e)^5 - 30*a^2*f*cosh(f*x + e)^3 + 5*a^2*f*cosh(f*x + e))*sinh(f*x + e)^7 + 15*a^2*f*cosh(f*x + e)^4 + 4*
(231*a^2*f*cosh(f*x + e)^6 - 315*a^2*f*cosh(f*x + e)^4 + 105*a^2*f*cosh(f*x + e)^2 - 5*a^2*f)*sinh(f*x + e)^6
+ 24*(33*a^2*f*cosh(f*x + e)^7 - 63*a^2*f*cosh(f*x + e)^5 + 35*a^2*f*cosh(f*x + e)^3 - 5*a^2*f*cosh(f*x + e))*
sinh(f*x + e)^5 - 6*a^2*f*cosh(f*x + e)^2 + 15*(33*a^2*f*cosh(f*x + e)^8 - 84*a^2*f*cosh(f*x + e)^6 + 70*a^2*f
*cosh(f*x + e)^4 - 20*a^2*f*cosh(f*x + e)^2 + a^2*f)*sinh(f*x + e)^4 + 20*(11*a^2*f*cosh(f*x + e)^9 - 36*a^2*f
*cosh(f*x + e)^7 + 42*a^2*f*cosh(f*x + e)^5 - 20*a^2*f*cosh(f*x + e)^3 + 3*a^2*f*cosh(f*x + e))*sinh(f*x + e)^
3 + a^2*f + 6*(11*a^2*f*cosh(f*x + e)^10 - 45*a^2*f*cosh(f*x + e)^8 + 70*a^2*f*cosh(f*x + e)^6 - 50*a^2*f*cosh
(f*x + e)^4 + 15*a^2*f*cosh(f*x + e)^2 - a^2*f)*sinh(f*x + e)^2 + 12*(a^2*f*cosh(f*x + e)^11 - 5*a^2*f*cosh(f*
x + e)^9 + 10*a^2*f*cosh(f*x + e)^7 - 10*a^2*f*cosh(f*x + e)^5 + 5*a^2*f*cosh(f*x + e)^3 - a^2*f*cosh(f*x + e)
)*sinh(f*x + e))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)**7*(a+b*sinh(f*x+e)**2)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} \operatorname{csch}\left (f x + e\right )^{7}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)^7*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

integrate((b*sinh(f*x + e)^2 + a)^(3/2)*csch(f*x + e)^7, x)