3.14 \(\int \frac{1}{(a+b \text{csch}^2(c+d x))^{5/2}} \, dx\)

Optimal. Leaf size=135 \[ \frac{b (5 a-3 b) \coth (c+d x)}{3 a^2 d (a-b)^2 \sqrt{a+b \coth ^2(c+d x)-b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a+b \coth ^2(c+d x)-b}}\right )}{a^{5/2} d}+\frac{b \coth (c+d x)}{3 a d (a-b) \left (a+b \coth ^2(c+d x)-b\right )^{3/2}} \]

[Out]

ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a - b + b*Coth[c + d*x]^2]]/(a^(5/2)*d) + (b*Coth[c + d*x])/(3*a*(a - b)*
d*(a - b + b*Coth[c + d*x]^2)^(3/2)) + ((5*a - 3*b)*b*Coth[c + d*x])/(3*a^2*(a - b)^2*d*Sqrt[a - b + b*Coth[c
+ d*x]^2])

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Rubi [A]  time = 0.108533, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4128, 414, 527, 12, 377, 206} \[ \frac{b (5 a-3 b) \coth (c+d x)}{3 a^2 d (a-b)^2 \sqrt{a+b \coth ^2(c+d x)-b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a+b \coth ^2(c+d x)-b}}\right )}{a^{5/2} d}+\frac{b \coth (c+d x)}{3 a d (a-b) \left (a+b \coth ^2(c+d x)-b\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Csch[c + d*x]^2)^(-5/2),x]

[Out]

ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a - b + b*Coth[c + d*x]^2]]/(a^(5/2)*d) + (b*Coth[c + d*x])/(3*a*(a - b)*
d*(a - b + b*Coth[c + d*x]^2)^(3/2)) + ((5*a - 3*b)*b*Coth[c + d*x])/(3*a^2*(a - b)^2*d*Sqrt[a - b + b*Coth[c
+ d*x]^2])

Rule 4128

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]

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 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 \frac{1}{\left (a+b \text{csch}^2(c+d x)\right )^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \left (a-b+b x^2\right )^{5/2}} \, dx,x,\coth (c+d x)\right )}{d}\\ &=\frac{b \coth (c+d x)}{3 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{-3 a+b+2 b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )^{3/2}} \, dx,x,\coth (c+d x)\right )}{3 a (a-b) d}\\ &=\frac{b \coth (c+d x)}{3 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^{3/2}}+\frac{(5 a-3 b) b \coth (c+d x)}{3 a^2 (a-b)^2 d \sqrt{a-b+b \coth ^2(c+d x)}}+\frac{\operatorname{Subst}\left (\int \frac{3 (a-b)^2}{\left (1-x^2\right ) \sqrt{a-b+b x^2}} \, dx,x,\coth (c+d x)\right )}{3 a^2 (a-b)^2 d}\\ &=\frac{b \coth (c+d x)}{3 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^{3/2}}+\frac{(5 a-3 b) b \coth (c+d x)}{3 a^2 (a-b)^2 d \sqrt{a-b+b \coth ^2(c+d x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{a-b+b x^2}} \, dx,x,\coth (c+d x)\right )}{a^2 d}\\ &=\frac{b \coth (c+d x)}{3 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^{3/2}}+\frac{(5 a-3 b) b \coth (c+d x)}{3 a^2 (a-b)^2 d \sqrt{a-b+b \coth ^2(c+d x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{\coth (c+d x)}{\sqrt{a-b+b \coth ^2(c+d x)}}\right )}{a^2 d}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a-b+b \coth ^2(c+d x)}}\right )}{a^{5/2} d}+\frac{b \coth (c+d x)}{3 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^{3/2}}+\frac{(5 a-3 b) b \coth (c+d x)}{3 a^2 (a-b)^2 d \sqrt{a-b+b \coth ^2(c+d x)}}\\ \end{align*}

Mathematica [A]  time = 1.6458, size = 174, normalized size = 1.29 \[ \frac{\text{csch}^5(c+d x) \left (\frac{\sqrt{2} (a \cosh (2 (c+d x))-a+2 b)^{5/2} \log \left (\sqrt{a \cosh (2 (c+d x))-a+2 b}+\sqrt{2} \sqrt{a} \cosh (c+d x)\right )}{a^{5/2}}-\frac{4 b \cosh (c+d x) (a \cosh (2 (c+d x))-a+2 b) \left (3 a^2+a (2 b-3 a) \cosh (2 (c+d x))-7 a b+3 b^2\right )}{3 a^2 (a-b)^2}\right )}{8 d \left (a+b \text{csch}^2(c+d x)\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Csch[c + d*x]^2)^(-5/2),x]

[Out]

(Csch[c + d*x]^5*((-4*b*Cosh[c + d*x]*(-a + 2*b + a*Cosh[2*(c + d*x)])*(3*a^2 - 7*a*b + 3*b^2 + a*(-3*a + 2*b)
*Cosh[2*(c + d*x)]))/(3*a^2*(a - b)^2) + (Sqrt[2]*(-a + 2*b + a*Cosh[2*(c + d*x)])^(5/2)*Log[Sqrt[2]*Sqrt[a]*C
osh[c + d*x] + Sqrt[-a + 2*b + a*Cosh[2*(c + d*x)]]])/a^(5/2)))/(8*d*(a + b*Csch[c + d*x]^2)^(5/2))

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Maple [F]  time = 0.14, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ({\rm csch} \left (dx+c\right ) \right ) ^{2} \right ) ^{-{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*csch(d*x+c)^2)^(5/2),x)

[Out]

int(1/(a+b*csch(d*x+c)^2)^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{csch}\left (d x + c\right )^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*csch(d*x+c)^2)^(5/2),x, algorithm="maxima")

[Out]

integrate((b*csch(d*x + c)^2 + a)^(-5/2), x)

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Fricas [B]  time = 6.75727, size = 18590, normalized size = 137.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*csch(d*x+c)^2)^(5/2),x, algorithm="fricas")

[Out]

[1/12*(3*((a^4 - 2*a^3*b + a^2*b^2)*cosh(d*x + c)^8 + 8*(a^4 - 2*a^3*b + a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^
7 + (a^4 - 2*a^3*b + a^2*b^2)*sinh(d*x + c)^8 - 4*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^6 - 4*(a
^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3 - 7*(a^4 - 2*a^3*b + a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a^4
- 2*a^3*b + a^2*b^2)*cosh(d*x + c)^3 - 3*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5
+ 2*(3*a^4 - 14*a^3*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4)*cosh(d*x + c)^4 + 2*(35*(a^4 - 2*a^3*b + a^2*b^2)*cosh(
d*x + c)^4 + 3*a^4 - 14*a^3*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4 - 30*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*cosh(
d*x + c)^2)*sinh(d*x + c)^4 + a^4 - 2*a^3*b + a^2*b^2 + 8*(7*(a^4 - 2*a^3*b + a^2*b^2)*cosh(d*x + c)^5 - 10*(a
^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^3 + (3*a^4 - 14*a^3*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4)*cosh(
d*x + c))*sinh(d*x + c)^3 - 4*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^2 + 4*(7*(a^4 - 2*a^3*b + a^
2*b^2)*cosh(d*x + c)^6 - 15*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^4 - a^4 + 4*a^3*b - 5*a^2*b^2
+ 2*a*b^3 + 3*(3*a^4 - 14*a^3*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^4 -
2*a^3*b + a^2*b^2)*cosh(d*x + c)^7 - 3*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^5 + (3*a^4 - 14*a^3
*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4)*cosh(d*x + c)^3 - (a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*cosh(d*x + c))*sin
h(d*x + c))*sqrt(a)*log((a*b^2*cosh(d*x + c)^8 + 8*a*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + a*b^2*sinh(d*x + c)^8
+ 2*(a*b^2 + b^3)*cosh(d*x + c)^6 + 2*(14*a*b^2*cosh(d*x + c)^2 + a*b^2 + b^3)*sinh(d*x + c)^6 + 4*(14*a*b^2*
cosh(d*x + c)^3 + 3*(a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + (a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c)^4 +
(70*a*b^2*cosh(d*x + c)^4 + a^3 - 4*a^2*b + 9*a*b^2 + 30*(a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(
14*a*b^2*cosh(d*x + c)^5 + 10*(a*b^2 + b^3)*cosh(d*x + c)^3 + (a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c))*sinh(d*
x + c)^3 + a^3 - 2*(a^3 - 3*a^2*b)*cosh(d*x + c)^2 + 2*(14*a*b^2*cosh(d*x + c)^6 + 15*(a*b^2 + b^3)*cosh(d*x +
c)^4 - a^3 + 3*a^2*b + 3*(a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + sqrt(2)*(b^2*cosh(d*x +
c)^6 + 6*b^2*cosh(d*x + c)*sinh(d*x + c)^5 + b^2*sinh(d*x + c)^6 + 3*b^2*cosh(d*x + c)^4 + 3*(5*b^2*cosh(d*x
+ c)^2 + b^2)*sinh(d*x + c)^4 + 4*(5*b^2*cosh(d*x + c)^3 + 3*b^2*cosh(d*x + c))*sinh(d*x + c)^3 - (a^2 - 4*a*b
)*cosh(d*x + c)^2 + (15*b^2*cosh(d*x + c)^4 + 18*b^2*cosh(d*x + c)^2 - a^2 + 4*a*b)*sinh(d*x + c)^2 + a^2 + 2*
(3*b^2*cosh(d*x + c)^5 + 6*b^2*cosh(d*x + c)^3 - (a^2 - 4*a*b)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a)*sqrt((a*c
osh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)
^2)) + 4*(2*a*b^2*cosh(d*x + c)^7 + 3*(a*b^2 + b^3)*cosh(d*x + c)^5 + (a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c)^
3 - (a^3 - 3*a^2*b)*cosh(d*x + c))*sinh(d*x + c))/(cosh(d*x + c)^6 + 6*cosh(d*x + c)^5*sinh(d*x + c) + 15*cosh
(d*x + c)^4*sinh(d*x + c)^2 + 20*cosh(d*x + c)^3*sinh(d*x + c)^3 + 15*cosh(d*x + c)^2*sinh(d*x + c)^4 + 6*cosh
(d*x + c)*sinh(d*x + c)^5 + sinh(d*x + c)^6)) + 3*((a^4 - 2*a^3*b + a^2*b^2)*cosh(d*x + c)^8 + 8*(a^4 - 2*a^3*
b + a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^4 - 2*a^3*b + a^2*b^2)*sinh(d*x + c)^8 - 4*(a^4 - 4*a^3*b + 5*
a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^6 - 4*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3 - 7*(a^4 - 2*a^3*b + a^2*b^2)*cosh
(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a^4 - 2*a^3*b + a^2*b^2)*cosh(d*x + c)^3 - 3*(a^4 - 4*a^3*b + 5*a^2*b^2 -
2*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a^4 - 14*a^3*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4)*cosh(d*x + c)^
4 + 2*(35*(a^4 - 2*a^3*b + a^2*b^2)*cosh(d*x + c)^4 + 3*a^4 - 14*a^3*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4 - 30*(a
^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + a^4 - 2*a^3*b + a^2*b^2 + 8*(7*(a^4 - 2
*a^3*b + a^2*b^2)*cosh(d*x + c)^5 - 10*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^3 + (3*a^4 - 14*a^3
*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*c
osh(d*x + c)^2 + 4*(7*(a^4 - 2*a^3*b + a^2*b^2)*cosh(d*x + c)^6 - 15*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*cos
h(d*x + c)^4 - a^4 + 4*a^3*b - 5*a^2*b^2 + 2*a*b^3 + 3*(3*a^4 - 14*a^3*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4)*cosh
(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^4 - 2*a^3*b + a^2*b^2)*cosh(d*x + c)^7 - 3*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2
*a*b^3)*cosh(d*x + c)^5 + (3*a^4 - 14*a^3*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4)*cosh(d*x + c)^3 - (a^4 - 4*a^3*b
+ 5*a^2*b^2 - 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a)*log(-(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(
d*x + c)^3 + a*sinh(d*x + c)^4 - 2*(a - b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 - a + b)*sinh(d*x + c)^2 +
sqrt(2)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*sqrt(a)*sqrt((a*cosh(d*x + c)
^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) + 4*(a*
cosh(d*x + c)^3 - (a - b)*cosh(d*x + c))*sinh(d*x + c) + a)/(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) +
sinh(d*x + c)^2)) + 8*sqrt(2)*((3*a^3*b - 2*a^2*b^2)*cosh(d*x + c)^6 + 6*(3*a^3*b - 2*a^2*b^2)*cosh(d*x + c)*
sinh(d*x + c)^5 + (3*a^3*b - 2*a^2*b^2)*sinh(d*x + c)^6 - 3*(a^3*b - 4*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^4 - 3*
(a^3*b - 4*a^2*b^2 + 2*a*b^3 - 5*(3*a^3*b - 2*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 3*a^3*b - 2*a^2*b^2
+ 4*(5*(3*a^3*b - 2*a^2*b^2)*cosh(d*x + c)^3 - 3*(a^3*b - 4*a^2*b^2 + 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^3
- 3*(a^3*b - 4*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^2 + 3*(5*(3*a^3*b - 2*a^2*b^2)*cosh(d*x + c)^4 - a^3*b + 4*a^2
*b^2 - 2*a*b^3 - 6*(a^3*b - 4*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 6*((3*a^3*b - 2*a^2*b^2)*c
osh(d*x + c)^5 - 2*(a^3*b - 4*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^3 - (a^3*b - 4*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)
)*sinh(d*x + c))*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sin
h(d*x + c) + sinh(d*x + c)^2)))/((a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^8 + 8*(a^7 - 2*a^6*b + a^5*b^2)*d*c
osh(d*x + c)*sinh(d*x + c)^7 + (a^7 - 2*a^6*b + a^5*b^2)*d*sinh(d*x + c)^8 - 4*(a^7 - 4*a^6*b + 5*a^5*b^2 - 2*
a^4*b^3)*d*cosh(d*x + c)^6 + 4*(7*(a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^2 - (a^7 - 4*a^6*b + 5*a^5*b^2 - 2
*a^4*b^3)*d)*sinh(d*x + c)^6 + 2*(3*a^7 - 14*a^6*b + 27*a^5*b^2 - 24*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x + c)^4 +
8*(7*(a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^3 - 3*(a^7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d*cosh(d*x + c))*
sinh(d*x + c)^5 + 2*(35*(a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^4 - 30*(a^7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^
3)*d*cosh(d*x + c)^2 + (3*a^7 - 14*a^6*b + 27*a^5*b^2 - 24*a^4*b^3 + 8*a^3*b^4)*d)*sinh(d*x + c)^4 - 4*(a^7 -
4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d*cosh(d*x + c)^2 + 8*(7*(a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^5 - 10*(a^
7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d*cosh(d*x + c)^3 + (3*a^7 - 14*a^6*b + 27*a^5*b^2 - 24*a^4*b^3 + 8*a^3*b
^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^6 - 15*(a^7 - 4*a^6*b +
5*a^5*b^2 - 2*a^4*b^3)*d*cosh(d*x + c)^4 + 3*(3*a^7 - 14*a^6*b + 27*a^5*b^2 - 24*a^4*b^3 + 8*a^3*b^4)*d*cosh(d
*x + c)^2 - (a^7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d)*sinh(d*x + c)^2 + (a^7 - 2*a^6*b + a^5*b^2)*d + 8*((a^7
- 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^7 - 3*(a^7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d*cosh(d*x + c)^5 + (3*a^7
- 14*a^6*b + 27*a^5*b^2 - 24*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x + c)^3 - (a^7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)
*d*cosh(d*x + c))*sinh(d*x + c)), -1/6*(3*((a^4 - 2*a^3*b + a^2*b^2)*cosh(d*x + c)^8 + 8*(a^4 - 2*a^3*b + a^2*
b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^4 - 2*a^3*b + a^2*b^2)*sinh(d*x + c)^8 - 4*(a^4 - 4*a^3*b + 5*a^2*b^2
- 2*a*b^3)*cosh(d*x + c)^6 - 4*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3 - 7*(a^4 - 2*a^3*b + a^2*b^2)*cosh(d*x + c
)^2)*sinh(d*x + c)^6 + 8*(7*(a^4 - 2*a^3*b + a^2*b^2)*cosh(d*x + c)^3 - 3*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3
)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a^4 - 14*a^3*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4)*cosh(d*x + c)^4 + 2*(3
5*(a^4 - 2*a^3*b + a^2*b^2)*cosh(d*x + c)^4 + 3*a^4 - 14*a^3*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4 - 30*(a^4 - 4*a
^3*b + 5*a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + a^4 - 2*a^3*b + a^2*b^2 + 8*(7*(a^4 - 2*a^3*b +
a^2*b^2)*cosh(d*x + c)^5 - 10*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^3 + (3*a^4 - 14*a^3*b + 27*
a^2*b^2 - 24*a*b^3 + 8*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*cosh(d*x
+ c)^2 + 4*(7*(a^4 - 2*a^3*b + a^2*b^2)*cosh(d*x + c)^6 - 15*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*cosh(d*x +
c)^4 - a^4 + 4*a^3*b - 5*a^2*b^2 + 2*a*b^3 + 3*(3*a^4 - 14*a^3*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4)*cosh(d*x + c
)^2)*sinh(d*x + c)^2 + 8*((a^4 - 2*a^3*b + a^2*b^2)*cosh(d*x + c)^7 - 3*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*
cosh(d*x + c)^5 + (3*a^4 - 14*a^3*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4)*cosh(d*x + c)^3 - (a^4 - 4*a^3*b + 5*a^2*
b^2 - 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a)*arctan(sqrt(2)*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*si
nh(d*x + c) + b*sinh(d*x + c)^2 + a)*sqrt(-a)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x
+ c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2))/(a*b*cosh(d*x + c)^4 + 4*a*b*cosh(d*x + c)*sinh(d*
x + c)^3 + a*b*sinh(d*x + c)^4 - (a^2 - 3*a*b)*cosh(d*x + c)^2 + (6*a*b*cosh(d*x + c)^2 - a^2 + 3*a*b)*sinh(d*
x + c)^2 + a^2 + 2*(2*a*b*cosh(d*x + c)^3 - (a^2 - 3*a*b)*cosh(d*x + c))*sinh(d*x + c))) + 3*((a^4 - 2*a^3*b +
a^2*b^2)*cosh(d*x + c)^8 + 8*(a^4 - 2*a^3*b + a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^4 - 2*a^3*b + a^2*b
^2)*sinh(d*x + c)^8 - 4*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^6 - 4*(a^4 - 4*a^3*b + 5*a^2*b^2 -
2*a*b^3 - 7*(a^4 - 2*a^3*b + a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a^4 - 2*a^3*b + a^2*b^2)*cosh(
d*x + c)^3 - 3*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a^4 - 14*a^3*b + 27
*a^2*b^2 - 24*a*b^3 + 8*b^4)*cosh(d*x + c)^4 + 2*(35*(a^4 - 2*a^3*b + a^2*b^2)*cosh(d*x + c)^4 + 3*a^4 - 14*a^
3*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4 - 30*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^
4 + a^4 - 2*a^3*b + a^2*b^2 + 8*(7*(a^4 - 2*a^3*b + a^2*b^2)*cosh(d*x + c)^5 - 10*(a^4 - 4*a^3*b + 5*a^2*b^2 -
2*a*b^3)*cosh(d*x + c)^3 + (3*a^4 - 14*a^3*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4)*cosh(d*x + c))*sinh(d*x + c)^3
- 4*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^2 + 4*(7*(a^4 - 2*a^3*b + a^2*b^2)*cosh(d*x + c)^6 - 1
5*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^4 - a^4 + 4*a^3*b - 5*a^2*b^2 + 2*a*b^3 + 3*(3*a^4 - 14*
a^3*b + 27*a^2*b^2 - 24*a*b^3 + 8*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^4 - 2*a^3*b + a^2*b^2)*cosh(d*
x + c)^7 - 3*(a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^5 + (3*a^4 - 14*a^3*b + 27*a^2*b^2 - 24*a*b^3
+ 8*b^4)*cosh(d*x + c)^3 - (a^4 - 4*a^3*b + 5*a^2*b^2 - 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a)*arcta
n(sqrt(2)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*sqrt(-a)*sqrt((a*cosh(d*x +
c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2))/(a*co
sh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 - 2*(a - 2*b)*cosh(d*x + c)^2 + 2*(3*a*c
osh(d*x + c)^2 - a + 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 - (a - 2*b)*cosh(d*x + c))*sinh(d*x + c) + a)
) - 4*sqrt(2)*((3*a^3*b - 2*a^2*b^2)*cosh(d*x + c)^6 + 6*(3*a^3*b - 2*a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^5 +
(3*a^3*b - 2*a^2*b^2)*sinh(d*x + c)^6 - 3*(a^3*b - 4*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^4 - 3*(a^3*b - 4*a^2*b^
2 + 2*a*b^3 - 5*(3*a^3*b - 2*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 3*a^3*b - 2*a^2*b^2 + 4*(5*(3*a^3*b -
2*a^2*b^2)*cosh(d*x + c)^3 - 3*(a^3*b - 4*a^2*b^2 + 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - 3*(a^3*b - 4*a^
2*b^2 + 2*a*b^3)*cosh(d*x + c)^2 + 3*(5*(3*a^3*b - 2*a^2*b^2)*cosh(d*x + c)^4 - a^3*b + 4*a^2*b^2 - 2*a*b^3 -
6*(a^3*b - 4*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 6*((3*a^3*b - 2*a^2*b^2)*cosh(d*x + c)^5 -
2*(a^3*b - 4*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^3 - (a^3*b - 4*a^2*b^2 + 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*
sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh
(d*x + c)^2)))/((a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^8 + 8*(a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)*sinh
(d*x + c)^7 + (a^7 - 2*a^6*b + a^5*b^2)*d*sinh(d*x + c)^8 - 4*(a^7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d*cosh(d
*x + c)^6 + 4*(7*(a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^2 - (a^7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d)*sinh
(d*x + c)^6 + 2*(3*a^7 - 14*a^6*b + 27*a^5*b^2 - 24*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x + c)^4 + 8*(7*(a^7 - 2*a^6
*b + a^5*b^2)*d*cosh(d*x + c)^3 - 3*(a^7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^5 +
2*(35*(a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^4 - 30*(a^7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d*cosh(d*x + c
)^2 + (3*a^7 - 14*a^6*b + 27*a^5*b^2 - 24*a^4*b^3 + 8*a^3*b^4)*d)*sinh(d*x + c)^4 - 4*(a^7 - 4*a^6*b + 5*a^5*b
^2 - 2*a^4*b^3)*d*cosh(d*x + c)^2 + 8*(7*(a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^5 - 10*(a^7 - 4*a^6*b + 5*a
^5*b^2 - 2*a^4*b^3)*d*cosh(d*x + c)^3 + (3*a^7 - 14*a^6*b + 27*a^5*b^2 - 24*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x +
c))*sinh(d*x + c)^3 + 4*(7*(a^7 - 2*a^6*b + a^5*b^2)*d*cosh(d*x + c)^6 - 15*(a^7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4
*b^3)*d*cosh(d*x + c)^4 + 3*(3*a^7 - 14*a^6*b + 27*a^5*b^2 - 24*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x + c)^2 - (a^7
- 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d)*sinh(d*x + c)^2 + (a^7 - 2*a^6*b + a^5*b^2)*d + 8*((a^7 - 2*a^6*b + a^5*
b^2)*d*cosh(d*x + c)^7 - 3*(a^7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d*cosh(d*x + c)^5 + (3*a^7 - 14*a^6*b + 27*
a^5*b^2 - 24*a^4*b^3 + 8*a^3*b^4)*d*cosh(d*x + c)^3 - (a^7 - 4*a^6*b + 5*a^5*b^2 - 2*a^4*b^3)*d*cosh(d*x + c))
*sinh(d*x + c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \operatorname{csch}^{2}{\left (c + d x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*csch(d*x+c)**2)**(5/2),x)

[Out]

Integral((a + b*csch(c + d*x)**2)**(-5/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{csch}\left (d x + c\right )^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*csch(d*x+c)^2)^(5/2),x, algorithm="giac")

[Out]

integrate((b*csch(d*x + c)^2 + a)^(-5/2), x)