3.50 \(\int \frac{\text{csch}^4(c+d x)}{(a+b \sinh ^2(c+d x))^2} \, dx\)

Optimal. Leaf size=174 \[ \frac{b^2 (6 a-5 b) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{2 a^{7/2} d (a-b)^{3/2}}+\frac{\left (2 a^2+a b-5 b^2\right ) \coth (c+d x)}{2 a^3 d (a-b)}-\frac{(2 a-5 b) \coth ^3(c+d x)}{6 a^2 d (a-b)}-\frac{b \text{csch}^3(c+d x) \text{sech}(c+d x)}{2 a d (a-b) \left (a-(a-b) \tanh ^2(c+d x)\right )} \]

[Out]

((6*a - 5*b)*b^2*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(7/2)*(a - b)^(3/2)*d) + ((2*a^2 + a*b - 5
*b^2)*Coth[c + d*x])/(2*a^3*(a - b)*d) - ((2*a - 5*b)*Coth[c + d*x]^3)/(6*a^2*(a - b)*d) - (b*Csch[c + d*x]^3*
Sech[c + d*x])/(2*a*(a - b)*d*(a - (a - b)*Tanh[c + d*x]^2))

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Rubi [A]  time = 0.206368, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3187, 468, 570, 208} \[ \frac{b^2 (6 a-5 b) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{2 a^{7/2} d (a-b)^{3/2}}+\frac{\left (2 a^2+a b-5 b^2\right ) \coth (c+d x)}{2 a^3 d (a-b)}-\frac{(2 a-5 b) \coth ^3(c+d x)}{6 a^2 d (a-b)}-\frac{b \text{csch}^3(c+d x) \text{sech}(c+d x)}{2 a d (a-b) \left (a-(a-b) \tanh ^2(c+d x)\right )} \]

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^2,x]

[Out]

((6*a - 5*b)*b^2*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(7/2)*(a - b)^(3/2)*d) + ((2*a^2 + a*b - 5
*b^2)*Coth[c + d*x])/(2*a^3*(a - b)*d) - ((2*a - 5*b)*Coth[c + d*x]^3)/(6*a^2*(a - b)*d) - (b*Csch[c + d*x]^3*
Sech[c + d*x])/(2*a*(a - b)*d*(a - (a - b)*Tanh[c + d*x]^2))

Rule 3187

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

Rule 468

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[((c*b -
a*d)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1))/(a*b*e*n*(p + 1)), x] + Dist[1/(a*b*n*(p + 1)), I
nt[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c*b - a*d)*(m + 1)) + d*(c*b*n*(p
+ 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 570

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]

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{\text{csch}^4(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^3}{x^4 \left (a-(a-b) x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{b \text{csch}^3(c+d x) \text{sech}(c+d x)}{2 a (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right ) \left (2 a-5 b+(-2 a+b) x^2\right )}{x^4 \left (a+(-a+b) x^2\right )} \, dx,x,\tanh (c+d x)\right )}{2 a (a-b) d}\\ &=-\frac{b \text{csch}^3(c+d x) \text{sech}(c+d x)}{2 a (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \left (\frac{2 a-5 b}{a x^4}+\frac{-2 a^2-a b+5 b^2}{a^2 x^2}+\frac{(6 a-5 b) b^2}{a^2 \left (a-(a-b) x^2\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{2 a (a-b) d}\\ &=\frac{\left (2 a^2+a b-5 b^2\right ) \coth (c+d x)}{2 a^3 (a-b) d}-\frac{(2 a-5 b) \coth ^3(c+d x)}{6 a^2 (a-b) d}-\frac{b \text{csch}^3(c+d x) \text{sech}(c+d x)}{2 a (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac{\left ((6 a-5 b) b^2\right ) \operatorname{Subst}\left (\int \frac{1}{a-(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{2 a^3 (a-b) d}\\ &=\frac{(6 a-5 b) b^2 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{2 a^{7/2} (a-b)^{3/2} d}+\frac{\left (2 a^2+a b-5 b^2\right ) \coth (c+d x)}{2 a^3 (a-b) d}-\frac{(2 a-5 b) \coth ^3(c+d x)}{6 a^2 (a-b) d}-\frac{b \text{csch}^3(c+d x) \text{sech}(c+d x)}{2 a (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 1.25862, size = 210, normalized size = 1.21 \[ \frac{\text{csch}^4(c+d x) (2 a+b \cosh (2 (c+d x))-b) \left (-2 a^{3/2} \coth (c+d x) \text{csch}^2(c+d x) (2 a+b \cosh (2 (c+d x))-b)-\frac{3 \sqrt{a} b^3 \sinh (2 (c+d x))}{a-b}+\frac{3 b^2 (6 a-5 b) (2 a+b \cosh (2 (c+d x))-b) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{(a-b)^{3/2}}+4 \sqrt{a} (a+3 b) \coth (c+d x) (2 a+b \cosh (2 (c+d x))-b)\right )}{24 a^{7/2} d \left (a \text{csch}^2(c+d x)+b\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^2,x]

[Out]

((2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^4*((3*(6*a - 5*b)*b^2*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[
a]]*(2*a - b + b*Cosh[2*(c + d*x)]))/(a - b)^(3/2) + 4*Sqrt[a]*(a + 3*b)*(2*a - b + b*Cosh[2*(c + d*x)])*Coth[
c + d*x] - 2*a^(3/2)*(2*a - b + b*Cosh[2*(c + d*x)])*Coth[c + d*x]*Csch[c + d*x]^2 - (3*Sqrt[a]*b^3*Sinh[2*(c
+ d*x)])/(a - b)))/(24*a^(7/2)*d*(b + a*Csch[c + d*x]^2)^2)

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Maple [B]  time = 0.089, size = 890, normalized size = 5.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^4/(a+b*sinh(d*x+c)^2)^2,x)

[Out]

-1/24/d/a^2*tanh(1/2*d*x+1/2*c)^3+3/8/d/a^2*tanh(1/2*d*x+1/2*c)+1/d/a^3*tanh(1/2*d*x+1/2*c)*b-1/24/d/a^2/tanh(
1/2*d*x+1/2*c)^3+3/8/d/a^2/tanh(1/2*d*x+1/2*c)+1/d/a^3/tanh(1/2*d*x+1/2*c)*b-1/d*b^3/a^3/(tanh(1/2*d*x+1/2*c)^
4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)/(a-b)*tanh(1/2*d*x+1/2*c)^3-1/d*b^3/a^3/(tanh(1/2*d
*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)/(a-b)*tanh(1/2*d*x+1/2*c)+3/d/a^2*b^2/(a-
b)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))-3/
d/a^2*b^3/(a-b)/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a
-b))^(1/2)+a-2*b)*a)^(1/2))-3/d/a^2*b^2/(a-b)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c
)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))-3/d/a^2*b^3/(a-b)/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2
)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))-5/2/d*b^3/a^3/(a-b)/((2*(-b*(a-b))^(1/2)+
a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))+5/2/d*b^4/a^3/(a-b)/(-b*(a
-b))^(1/2)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(
1/2))+5/2/d*b^3/a^3/(a-b)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/
2)-a+2*b)*a)^(1/2))+5/2/d*b^4/a^3/(a-b)/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/
2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^4/(a+b*sinh(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.16269, size = 16662, normalized size = 95.76 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^4/(a+b*sinh(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

[1/12*(12*(6*a^3*b^2 - 11*a^2*b^3 + 5*a*b^4)*cosh(d*x + c)^8 + 96*(6*a^3*b^2 - 11*a^2*b^3 + 5*a*b^4)*cosh(d*x
+ c)*sinh(d*x + c)^7 + 12*(6*a^3*b^2 - 11*a^2*b^3 + 5*a*b^4)*sinh(d*x + c)^8 + 24*(6*a^4*b - 23*a^3*b^2 + 27*a
^2*b^3 - 10*a*b^4)*cosh(d*x + c)^6 + 24*(6*a^4*b - 23*a^3*b^2 + 27*a^2*b^3 - 10*a*b^4 + 14*(6*a^3*b^2 - 11*a^2
*b^3 + 5*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 48*(14*(6*a^3*b^2 - 11*a^2*b^3 + 5*a*b^4)*cosh(d*x + c)^3 +
3*(6*a^4*b - 23*a^3*b^2 + 27*a^2*b^3 - 10*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 16*a^4*b + 16*a^3*b^2 - 92*
a^2*b^3 + 60*a*b^4 - 8*(24*a^5 - 14*a^4*b - 89*a^3*b^2 + 124*a^2*b^3 - 45*a*b^4)*cosh(d*x + c)^4 - 8*(24*a^5 -
14*a^4*b - 89*a^3*b^2 + 124*a^2*b^3 - 45*a*b^4 - 105*(6*a^3*b^2 - 11*a^2*b^3 + 5*a*b^4)*cosh(d*x + c)^4 - 45*
(6*a^4*b - 23*a^3*b^2 + 27*a^2*b^3 - 10*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 32*(21*(6*a^3*b^2 - 11*a^2*b
^3 + 5*a*b^4)*cosh(d*x + c)^5 + 15*(6*a^4*b - 23*a^3*b^2 + 27*a^2*b^3 - 10*a*b^4)*cosh(d*x + c)^3 - (24*a^5 -
14*a^4*b - 89*a^3*b^2 + 124*a^2*b^3 - 45*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 8*(8*a^5 - 2*a^4*b - 47*a^3*b
^2 + 71*a^2*b^3 - 30*a*b^4)*cosh(d*x + c)^2 + 8*(42*(6*a^3*b^2 - 11*a^2*b^3 + 5*a*b^4)*cosh(d*x + c)^6 + 8*a^5
- 2*a^4*b - 47*a^3*b^2 + 71*a^2*b^3 - 30*a*b^4 + 45*(6*a^4*b - 23*a^3*b^2 + 27*a^2*b^3 - 10*a*b^4)*cosh(d*x +
c)^4 - 6*(24*a^5 - 14*a^4*b - 89*a^3*b^2 + 124*a^2*b^3 - 45*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 3*((6*a
*b^3 - 5*b^4)*cosh(d*x + c)^10 + 10*(6*a*b^3 - 5*b^4)*cosh(d*x + c)*sinh(d*x + c)^9 + (6*a*b^3 - 5*b^4)*sinh(d
*x + c)^10 + (24*a^2*b^2 - 50*a*b^3 + 25*b^4)*cosh(d*x + c)^8 + (24*a^2*b^2 - 50*a*b^3 + 25*b^4 + 45*(6*a*b^3
- 5*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(15*(6*a*b^3 - 5*b^4)*cosh(d*x + c)^3 + (24*a^2*b^2 - 50*a*b^3 +
25*b^4)*cosh(d*x + c))*sinh(d*x + c)^7 - 2*(36*a^2*b^2 - 60*a*b^3 + 25*b^4)*cosh(d*x + c)^6 + 2*(105*(6*a*b^3
- 5*b^4)*cosh(d*x + c)^4 - 36*a^2*b^2 + 60*a*b^3 - 25*b^4 + 14*(24*a^2*b^2 - 50*a*b^3 + 25*b^4)*cosh(d*x + c)
^2)*sinh(d*x + c)^6 + 4*(63*(6*a*b^3 - 5*b^4)*cosh(d*x + c)^5 + 14*(24*a^2*b^2 - 50*a*b^3 + 25*b^4)*cosh(d*x +
c)^3 - 3*(36*a^2*b^2 - 60*a*b^3 + 25*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(36*a^2*b^2 - 60*a*b^3 + 25*b^4)
*cosh(d*x + c)^4 + 2*(105*(6*a*b^3 - 5*b^4)*cosh(d*x + c)^6 + 35*(24*a^2*b^2 - 50*a*b^3 + 25*b^4)*cosh(d*x + c
)^4 + 36*a^2*b^2 - 60*a*b^3 + 25*b^4 - 15*(36*a^2*b^2 - 60*a*b^3 + 25*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 -
6*a*b^3 + 5*b^4 + 8*(15*(6*a*b^3 - 5*b^4)*cosh(d*x + c)^7 + 7*(24*a^2*b^2 - 50*a*b^3 + 25*b^4)*cosh(d*x + c)^5
- 5*(36*a^2*b^2 - 60*a*b^3 + 25*b^4)*cosh(d*x + c)^3 + (36*a^2*b^2 - 60*a*b^3 + 25*b^4)*cosh(d*x + c))*sinh(d
*x + c)^3 - (24*a^2*b^2 - 50*a*b^3 + 25*b^4)*cosh(d*x + c)^2 + (45*(6*a*b^3 - 5*b^4)*cosh(d*x + c)^8 + 28*(24*
a^2*b^2 - 50*a*b^3 + 25*b^4)*cosh(d*x + c)^6 - 30*(36*a^2*b^2 - 60*a*b^3 + 25*b^4)*cosh(d*x + c)^4 - 24*a^2*b^
2 + 50*a*b^3 - 25*b^4 + 12*(36*a^2*b^2 - 60*a*b^3 + 25*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(5*(6*a*b^3 -
5*b^4)*cosh(d*x + c)^9 + 4*(24*a^2*b^2 - 50*a*b^3 + 25*b^4)*cosh(d*x + c)^7 - 6*(36*a^2*b^2 - 60*a*b^3 + 25*b
^4)*cosh(d*x + c)^5 + 4*(36*a^2*b^2 - 60*a*b^3 + 25*b^4)*cosh(d*x + c)^3 - (24*a^2*b^2 - 50*a*b^3 + 25*b^4)*co
sh(d*x + c))*sinh(d*x + c))*sqrt(a^2 - a*b)*log((b^2*cosh(d*x + c)^4 + 4*b^2*cosh(d*x + c)*sinh(d*x + c)^3 + b
^2*sinh(d*x + c)^4 + 2*(2*a*b - b^2)*cosh(d*x + c)^2 + 2*(3*b^2*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x + c)^2
+ 8*a^2 - 8*a*b + b^2 + 4*(b^2*cosh(d*x + c)^3 + (2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c) - 4*(b*cosh(d*x +
c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + 2*a - b)*sqrt(a^2 - a*b))/(b*cosh(d*x + c)^4 + 4
*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 + 2*(2*a - b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 +
2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + (2*a - b)*cosh(d*x + c))*sinh(d*x + c) + b)) + 16*(6*(6*a^3*
b^2 - 11*a^2*b^3 + 5*a*b^4)*cosh(d*x + c)^7 + 9*(6*a^4*b - 23*a^3*b^2 + 27*a^2*b^3 - 10*a*b^4)*cosh(d*x + c)^5
- 2*(24*a^5 - 14*a^4*b - 89*a^3*b^2 + 124*a^2*b^3 - 45*a*b^4)*cosh(d*x + c)^3 + (8*a^5 - 2*a^4*b - 47*a^3*b^2
+ 71*a^2*b^3 - 30*a*b^4)*cosh(d*x + c))*sinh(d*x + c))/((a^6*b - 2*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^10 + 10
*(a^6*b - 2*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^6*b - 2*a^5*b^2 + a^4*b^3)*d*sinh(d*x + c)
^10 + (4*a^7 - 13*a^6*b + 14*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^8 + (45*(a^6*b - 2*a^5*b^2 + a^4*b^3)*d*cosh
(d*x + c)^2 + (4*a^7 - 13*a^6*b + 14*a^5*b^2 - 5*a^4*b^3)*d)*sinh(d*x + c)^8 - 2*(6*a^7 - 17*a^6*b + 16*a^5*b^
2 - 5*a^4*b^3)*d*cosh(d*x + c)^6 + 8*(15*(a^6*b - 2*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^3 + (4*a^7 - 13*a^6*b +
14*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(105*(a^6*b - 2*a^5*b^2 + a^4*b^3)*d*cosh(d*x +
c)^4 + 14*(4*a^7 - 13*a^6*b + 14*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^2 - (6*a^7 - 17*a^6*b + 16*a^5*b^2 - 5*a
^4*b^3)*d)*sinh(d*x + c)^6 + 2*(6*a^7 - 17*a^6*b + 16*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^4 + 4*(63*(a^6*b -
2*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^5 + 14*(4*a^7 - 13*a^6*b + 14*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^3 - 3*
(6*a^7 - 17*a^6*b + 16*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*(a^6*b - 2*a^5*b^2 + a^4
*b^3)*d*cosh(d*x + c)^6 + 35*(4*a^7 - 13*a^6*b + 14*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^4 - 15*(6*a^7 - 17*a^
6*b + 16*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^2 + (6*a^7 - 17*a^6*b + 16*a^5*b^2 - 5*a^4*b^3)*d)*sinh(d*x + c)
^4 - (4*a^7 - 13*a^6*b + 14*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^2 + 8*(15*(a^6*b - 2*a^5*b^2 + a^4*b^3)*d*cos
h(d*x + c)^7 + 7*(4*a^7 - 13*a^6*b + 14*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^5 - 5*(6*a^7 - 17*a^6*b + 16*a^5*
b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^3 + (6*a^7 - 17*a^6*b + 16*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c))*sinh(d*x + c
)^3 + (45*(a^6*b - 2*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^8 + 28*(4*a^7 - 13*a^6*b + 14*a^5*b^2 - 5*a^4*b^3)*d*c
osh(d*x + c)^6 - 30*(6*a^7 - 17*a^6*b + 16*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^4 + 12*(6*a^7 - 17*a^6*b + 16*
a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^2 - (4*a^7 - 13*a^6*b + 14*a^5*b^2 - 5*a^4*b^3)*d)*sinh(d*x + c)^2 - (a^6
*b - 2*a^5*b^2 + a^4*b^3)*d + 2*(5*(a^6*b - 2*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^9 + 4*(4*a^7 - 13*a^6*b + 14*
a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^7 - 6*(6*a^7 - 17*a^6*b + 16*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^5 + 4*(
6*a^7 - 17*a^6*b + 16*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^3 - (4*a^7 - 13*a^6*b + 14*a^5*b^2 - 5*a^4*b^3)*d*c
osh(d*x + c))*sinh(d*x + c)), 1/6*(6*(6*a^3*b^2 - 11*a^2*b^3 + 5*a*b^4)*cosh(d*x + c)^8 + 48*(6*a^3*b^2 - 11*a
^2*b^3 + 5*a*b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + 6*(6*a^3*b^2 - 11*a^2*b^3 + 5*a*b^4)*sinh(d*x + c)^8 + 12*(6
*a^4*b - 23*a^3*b^2 + 27*a^2*b^3 - 10*a*b^4)*cosh(d*x + c)^6 + 12*(6*a^4*b - 23*a^3*b^2 + 27*a^2*b^3 - 10*a*b^
4 + 14*(6*a^3*b^2 - 11*a^2*b^3 + 5*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 24*(14*(6*a^3*b^2 - 11*a^2*b^3 +
5*a*b^4)*cosh(d*x + c)^3 + 3*(6*a^4*b - 23*a^3*b^2 + 27*a^2*b^3 - 10*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 8
*a^4*b + 8*a^3*b^2 - 46*a^2*b^3 + 30*a*b^4 - 4*(24*a^5 - 14*a^4*b - 89*a^3*b^2 + 124*a^2*b^3 - 45*a*b^4)*cosh(
d*x + c)^4 - 4*(24*a^5 - 14*a^4*b - 89*a^3*b^2 + 124*a^2*b^3 - 45*a*b^4 - 105*(6*a^3*b^2 - 11*a^2*b^3 + 5*a*b^
4)*cosh(d*x + c)^4 - 45*(6*a^4*b - 23*a^3*b^2 + 27*a^2*b^3 - 10*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*(
21*(6*a^3*b^2 - 11*a^2*b^3 + 5*a*b^4)*cosh(d*x + c)^5 + 15*(6*a^4*b - 23*a^3*b^2 + 27*a^2*b^3 - 10*a*b^4)*cosh
(d*x + c)^3 - (24*a^5 - 14*a^4*b - 89*a^3*b^2 + 124*a^2*b^3 - 45*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(8*
a^5 - 2*a^4*b - 47*a^3*b^2 + 71*a^2*b^3 - 30*a*b^4)*cosh(d*x + c)^2 + 4*(42*(6*a^3*b^2 - 11*a^2*b^3 + 5*a*b^4)
*cosh(d*x + c)^6 + 8*a^5 - 2*a^4*b - 47*a^3*b^2 + 71*a^2*b^3 - 30*a*b^4 + 45*(6*a^4*b - 23*a^3*b^2 + 27*a^2*b^
3 - 10*a*b^4)*cosh(d*x + c)^4 - 6*(24*a^5 - 14*a^4*b - 89*a^3*b^2 + 124*a^2*b^3 - 45*a*b^4)*cosh(d*x + c)^2)*s
inh(d*x + c)^2 - 3*((6*a*b^3 - 5*b^4)*cosh(d*x + c)^10 + 10*(6*a*b^3 - 5*b^4)*cosh(d*x + c)*sinh(d*x + c)^9 +
(6*a*b^3 - 5*b^4)*sinh(d*x + c)^10 + (24*a^2*b^2 - 50*a*b^3 + 25*b^4)*cosh(d*x + c)^8 + (24*a^2*b^2 - 50*a*b^3
+ 25*b^4 + 45*(6*a*b^3 - 5*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(15*(6*a*b^3 - 5*b^4)*cosh(d*x + c)^3 +
(24*a^2*b^2 - 50*a*b^3 + 25*b^4)*cosh(d*x + c))*sinh(d*x + c)^7 - 2*(36*a^2*b^2 - 60*a*b^3 + 25*b^4)*cosh(d*x
+ c)^6 + 2*(105*(6*a*b^3 - 5*b^4)*cosh(d*x + c)^4 - 36*a^2*b^2 + 60*a*b^3 - 25*b^4 + 14*(24*a^2*b^2 - 50*a*b^3
+ 25*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*(6*a*b^3 - 5*b^4)*cosh(d*x + c)^5 + 14*(24*a^2*b^2 - 50*a*
b^3 + 25*b^4)*cosh(d*x + c)^3 - 3*(36*a^2*b^2 - 60*a*b^3 + 25*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(36*a^2*
b^2 - 60*a*b^3 + 25*b^4)*cosh(d*x + c)^4 + 2*(105*(6*a*b^3 - 5*b^4)*cosh(d*x + c)^6 + 35*(24*a^2*b^2 - 50*a*b^
3 + 25*b^4)*cosh(d*x + c)^4 + 36*a^2*b^2 - 60*a*b^3 + 25*b^4 - 15*(36*a^2*b^2 - 60*a*b^3 + 25*b^4)*cosh(d*x +
c)^2)*sinh(d*x + c)^4 - 6*a*b^3 + 5*b^4 + 8*(15*(6*a*b^3 - 5*b^4)*cosh(d*x + c)^7 + 7*(24*a^2*b^2 - 50*a*b^3 +
25*b^4)*cosh(d*x + c)^5 - 5*(36*a^2*b^2 - 60*a*b^3 + 25*b^4)*cosh(d*x + c)^3 + (36*a^2*b^2 - 60*a*b^3 + 25*b^
4)*cosh(d*x + c))*sinh(d*x + c)^3 - (24*a^2*b^2 - 50*a*b^3 + 25*b^4)*cosh(d*x + c)^2 + (45*(6*a*b^3 - 5*b^4)*c
osh(d*x + c)^8 + 28*(24*a^2*b^2 - 50*a*b^3 + 25*b^4)*cosh(d*x + c)^6 - 30*(36*a^2*b^2 - 60*a*b^3 + 25*b^4)*cos
h(d*x + c)^4 - 24*a^2*b^2 + 50*a*b^3 - 25*b^4 + 12*(36*a^2*b^2 - 60*a*b^3 + 25*b^4)*cosh(d*x + c)^2)*sinh(d*x
+ c)^2 + 2*(5*(6*a*b^3 - 5*b^4)*cosh(d*x + c)^9 + 4*(24*a^2*b^2 - 50*a*b^3 + 25*b^4)*cosh(d*x + c)^7 - 6*(36*a
^2*b^2 - 60*a*b^3 + 25*b^4)*cosh(d*x + c)^5 + 4*(36*a^2*b^2 - 60*a*b^3 + 25*b^4)*cosh(d*x + c)^3 - (24*a^2*b^2
- 50*a*b^3 + 25*b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a^2 + a*b)*arctan(-1/2*(b*cosh(d*x + c)^2 + 2*b*cosh
(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + 2*a - b)*sqrt(-a^2 + a*b)/(a^2 - a*b)) + 8*(6*(6*a^3*b^2 - 11*a^
2*b^3 + 5*a*b^4)*cosh(d*x + c)^7 + 9*(6*a^4*b - 23*a^3*b^2 + 27*a^2*b^3 - 10*a*b^4)*cosh(d*x + c)^5 - 2*(24*a^
5 - 14*a^4*b - 89*a^3*b^2 + 124*a^2*b^3 - 45*a*b^4)*cosh(d*x + c)^3 + (8*a^5 - 2*a^4*b - 47*a^3*b^2 + 71*a^2*b
^3 - 30*a*b^4)*cosh(d*x + c))*sinh(d*x + c))/((a^6*b - 2*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^10 + 10*(a^6*b - 2
*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^6*b - 2*a^5*b^2 + a^4*b^3)*d*sinh(d*x + c)^10 + (4*a^
7 - 13*a^6*b + 14*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^8 + (45*(a^6*b - 2*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^2
+ (4*a^7 - 13*a^6*b + 14*a^5*b^2 - 5*a^4*b^3)*d)*sinh(d*x + c)^8 - 2*(6*a^7 - 17*a^6*b + 16*a^5*b^2 - 5*a^4*b
^3)*d*cosh(d*x + c)^6 + 8*(15*(a^6*b - 2*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^3 + (4*a^7 - 13*a^6*b + 14*a^5*b^2
- 5*a^4*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(105*(a^6*b - 2*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^4 + 14*(
4*a^7 - 13*a^6*b + 14*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^2 - (6*a^7 - 17*a^6*b + 16*a^5*b^2 - 5*a^4*b^3)*d)*
sinh(d*x + c)^6 + 2*(6*a^7 - 17*a^6*b + 16*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^4 + 4*(63*(a^6*b - 2*a^5*b^2 +
a^4*b^3)*d*cosh(d*x + c)^5 + 14*(4*a^7 - 13*a^6*b + 14*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^3 - 3*(6*a^7 - 17
*a^6*b + 16*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*(a^6*b - 2*a^5*b^2 + a^4*b^3)*d*cos
h(d*x + c)^6 + 35*(4*a^7 - 13*a^6*b + 14*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^4 - 15*(6*a^7 - 17*a^6*b + 16*a^
5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^2 + (6*a^7 - 17*a^6*b + 16*a^5*b^2 - 5*a^4*b^3)*d)*sinh(d*x + c)^4 - (4*a^7
- 13*a^6*b + 14*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^2 + 8*(15*(a^6*b - 2*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^
7 + 7*(4*a^7 - 13*a^6*b + 14*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^5 - 5*(6*a^7 - 17*a^6*b + 16*a^5*b^2 - 5*a^4
*b^3)*d*cosh(d*x + c)^3 + (6*a^7 - 17*a^6*b + 16*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (45*(
a^6*b - 2*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^8 + 28*(4*a^7 - 13*a^6*b + 14*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c
)^6 - 30*(6*a^7 - 17*a^6*b + 16*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^4 + 12*(6*a^7 - 17*a^6*b + 16*a^5*b^2 - 5
*a^4*b^3)*d*cosh(d*x + c)^2 - (4*a^7 - 13*a^6*b + 14*a^5*b^2 - 5*a^4*b^3)*d)*sinh(d*x + c)^2 - (a^6*b - 2*a^5*
b^2 + a^4*b^3)*d + 2*(5*(a^6*b - 2*a^5*b^2 + a^4*b^3)*d*cosh(d*x + c)^9 + 4*(4*a^7 - 13*a^6*b + 14*a^5*b^2 - 5
*a^4*b^3)*d*cosh(d*x + c)^7 - 6*(6*a^7 - 17*a^6*b + 16*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^5 + 4*(6*a^7 - 17*
a^6*b + 16*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c)^3 - (4*a^7 - 13*a^6*b + 14*a^5*b^2 - 5*a^4*b^3)*d*cosh(d*x + c
))*sinh(d*x + c))]

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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(d*x+c)**4/(a+b*sinh(d*x+c)**2)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.50775, size = 301, normalized size = 1.73 \begin{align*} \frac{{\left (6 \, a b^{2} - 5 \, b^{3}\right )} \arctan \left (\frac{b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt{-a^{2} + a b}}\right )}{2 \,{\left (a^{4} d - a^{3} b d\right )} \sqrt{-a^{2} + a b}} + \frac{2 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - b^{3} e^{\left (2 \, d x + 2 \, c\right )} + b^{3}}{{\left (a^{4} d - a^{3} b d\right )}{\left (b e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + b\right )}} + \frac{4 \,{\left (3 \, b e^{\left (4 \, d x + 4 \, c\right )} - 3 \, a e^{\left (2 \, d x + 2 \, c\right )} - 6 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + 3 \, b\right )}}{3 \, a^{3} d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^4/(a+b*sinh(d*x+c)^2)^2,x, algorithm="giac")

[Out]

1/2*(6*a*b^2 - 5*b^3)*arctan(1/2*(b*e^(2*d*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/((a^4*d - a^3*b*d)*sqrt(-a^2
+ a*b)) + (2*a*b^2*e^(2*d*x + 2*c) - b^3*e^(2*d*x + 2*c) + b^3)/((a^4*d - a^3*b*d)*(b*e^(4*d*x + 4*c) + 4*a*e^
(2*d*x + 2*c) - 2*b*e^(2*d*x + 2*c) + b)) + 4/3*(3*b*e^(4*d*x + 4*c) - 3*a*e^(2*d*x + 2*c) - 6*b*e^(2*d*x + 2*
c) + a + 3*b)/(a^3*d*(e^(2*d*x + 2*c) - 1)^3)