3.39 \(\int \frac{\text{csch}^4(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\)
Optimal. Leaf size=78 \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{a^{5/2} d \sqrt{a-b}}+\frac{(a+b) \coth (c+d x)}{a^2 d}-\frac{\coth ^3(c+d x)}{3 a d} \]
[Out]
(b^2*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a^(5/2)*Sqrt[a - b]*d) + ((a + b)*Coth[c + d*x])/(a^2*d) -
Coth[c + d*x]^3/(3*a*d)
________________________________________________________________________________________
Rubi [A] time = 0.120393, antiderivative size = 78, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used =
{3187, 461, 208} \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{a^{5/2} d \sqrt{a-b}}+\frac{(a+b) \coth (c+d x)}{a^2 d}-\frac{\coth ^3(c+d x)}{3 a d} \]
Antiderivative was successfully verified.
[In]
Int[Csch[c + d*x]^4/(a + b*Sinh[c + d*x]^2),x]
[Out]
(b^2*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a^(5/2)*Sqrt[a - b]*d) + ((a + b)*Coth[c + d*x])/(a^2*d) -
Coth[c + d*x]^3/(3*a*d)
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 461
Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[((e*x)^m*(a + b*x^n)^p)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] || !RationalQ[m])
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)}{a+b \sinh ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{x^4 \left (a-(a-b) x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a x^4}+\frac{-a-b}{a^2 x^2}+\frac{b^2}{a^2 \left (a-(a-b) x^2\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{(a+b) \coth (c+d x)}{a^2 d}-\frac{\coth ^3(c+d x)}{3 a d}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{a-(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{a^2 d}\\ &=\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{a^{5/2} \sqrt{a-b} d}+\frac{(a+b) \coth (c+d x)}{a^2 d}-\frac{\coth ^3(c+d x)}{3 a d}\\ \end{align*}
Mathematica [A] time = 0.693455, size = 126, normalized size = 1.62 \[ -\frac{\text{csch}^2(c+d x) (2 a+b \cosh (2 (c+d x))-b) \left (\sqrt{a} \sqrt{a-b} \coth (c+d x) \left (a \text{csch}^2(c+d x)-2 a-3 b\right )-3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )\right )}{6 a^{5/2} d \sqrt{a-b} \left (a \text{csch}^2(c+d x)+b\right )} \]
Antiderivative was successfully verified.
[In]
Integrate[Csch[c + d*x]^4/(a + b*Sinh[c + d*x]^2),x]
[Out]
-((2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^2*(-3*b^2*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]] + Sqrt[
a]*Sqrt[a - b]*Coth[c + d*x]*(-2*a - 3*b + a*Csch[c + d*x]^2)))/(6*a^(5/2)*Sqrt[a - b]*d*(b + a*Csch[c + d*x]^
2))
________________________________________________________________________________________
Maple [B] time = 0.071, size = 401, normalized size = 5.1 \begin{align*} -{\frac{1}{24\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{3}{8\,da}\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{b}{2\,d{a}^{2}}\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{1}{24\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}+{\frac{3}{8\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}+{\frac{b}{2\,d{a}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}+{\frac{{b}^{2}}{d{a}^{2}}{\it Artanh} \left ({a\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{-b \left ( a-b \right ) }+a-2\,b \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{-b \left ( a-b \right ) }+a-2\,b \right ) a}}}}-{\frac{{b}^{3}}{d{a}^{2}}{\it Artanh} \left ({a\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{-b \left ( a-b \right ) }+a-2\,b \right ) a}}}} \right ){\frac{1}{\sqrt{-b \left ( a-b \right ) }}}{\frac{1}{\sqrt{ \left ( 2\,\sqrt{-b \left ( a-b \right ) }+a-2\,b \right ) a}}}}-{\frac{{b}^{2}}{d{a}^{2}}\arctan \left ({a\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{-b \left ( a-b \right ) }-a+2\,b \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{-b \left ( a-b \right ) }-a+2\,b \right ) a}}}}-{\frac{{b}^{3}}{d{a}^{2}}\arctan \left ({a\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{-b \left ( a-b \right ) }-a+2\,b \right ) a}}}} \right ){\frac{1}{\sqrt{-b \left ( a-b \right ) }}}{\frac{1}{\sqrt{ \left ( 2\,\sqrt{-b \left ( a-b \right ) }-a+2\,b \right ) a}}}} \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),x)
[Out]
-1/24/d/a*tanh(1/2*d*x+1/2*c)^3+3/8/d/a*tanh(1/2*d*x+1/2*c)+1/2/d/a^2*tanh(1/2*d*x+1/2*c)*b-1/24/d/a/tanh(1/2*
d*x+1/2*c)^3+3/8/d/a/tanh(1/2*d*x+1/2*c)+1/2/d*b/a^2/tanh(1/2*d*x+1/2*c)+1/d*b^2/a^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))-1/d*b^3/a^2/(-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))-1/d*b^
2/a^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))-
1/d*b^3/a^2/(-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))
________________________________________________________________________________________
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),x, algorithm="maxima")
[Out]
Exception raised: ValueError
________________________________________________________________________________________
Fricas [B] time = 2.3784, size = 4710, normalized size = 60.38 \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),x, algorithm="fricas")
[Out]
[1/6*(12*(a^2*b - a*b^2)*cosh(d*x + c)^4 + 48*(a^2*b - a*b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + 12*(a^2*b - a*b^
2)*sinh(d*x + c)^4 + 8*a^3 + 4*a^2*b - 12*a*b^2 - 24*(a^3 - a*b^2)*cosh(d*x + c)^2 - 24*(a^3 - a*b^2 - 3*(a^2*
b - a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 3*(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 + 3*b^2*cosh(d*x
+ c)^2 + 4*(5*b^2*cosh(d*x + c)^3 - 3*b^2*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*b^2*cosh(d*x + c)^4 - 6*b^2*co
sh(d*x + c)^2 + b^2)*sinh(d*x + c)^2 - b^2 + 6*(b^2*cosh(d*x + c)^5 - 2*b^2*cosh(d*x + c)^3 + b^2*cosh(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)*s
inh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + (2*a - b)*cosh(d*x + c))*sinh(d*x + c) + b)) + 48*((a^2*b - a*b^2)*cos
h(d*x + c)^3 - (a^3 - a*b^2)*cosh(d*x + c))*sinh(d*x + c))/((a^4 - a^3*b)*d*cosh(d*x + c)^6 + 6*(a^4 - a^3*b)*
d*cosh(d*x + c)*sinh(d*x + c)^5 + (a^4 - a^3*b)*d*sinh(d*x + c)^6 - 3*(a^4 - a^3*b)*d*cosh(d*x + c)^4 + 3*(5*(
a^4 - a^3*b)*d*cosh(d*x + c)^2 - (a^4 - a^3*b)*d)*sinh(d*x + c)^4 + 3*(a^4 - a^3*b)*d*cosh(d*x + c)^2 + 4*(5*(
a^4 - a^3*b)*d*cosh(d*x + c)^3 - 3*(a^4 - a^3*b)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*(a^4 - a^3*b)*d*cosh(
d*x + c)^4 - 6*(a^4 - a^3*b)*d*cosh(d*x + c)^2 + (a^4 - a^3*b)*d)*sinh(d*x + c)^2 - (a^4 - a^3*b)*d + 6*((a^4
- a^3*b)*d*cosh(d*x + c)^5 - 2*(a^4 - a^3*b)*d*cosh(d*x + c)^3 + (a^4 - a^3*b)*d*cosh(d*x + c))*sinh(d*x + c))
, 1/3*(6*(a^2*b - a*b^2)*cosh(d*x + c)^4 + 24*(a^2*b - a*b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + 6*(a^2*b - a*b^2
)*sinh(d*x + c)^4 + 4*a^3 + 2*a^2*b - 6*a*b^2 - 12*(a^3 - a*b^2)*cosh(d*x + c)^2 - 12*(a^3 - a*b^2 - 3*(a^2*b
- a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 3*(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 + 3*b^2*cosh(d*x +
c)^2 + 4*(5*b^2*cosh(d*x + c)^3 - 3*b^2*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*b^2*cosh(d*x + c)^4 - 6*b^2*cosh
(d*x + c)^2 + b^2)*sinh(d*x + c)^2 - b^2 + 6*(b^2*cosh(d*x + c)^5 - 2*b^2*cosh(d*x + c)^3 + b^2*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)) + 24*((a^2*b - a*b^2)*cosh(d*x + c)^3 - (a^3 - a*b^2)*cosh(d*
x + c))*sinh(d*x + c))/((a^4 - a^3*b)*d*cosh(d*x + c)^6 + 6*(a^4 - a^3*b)*d*cosh(d*x + c)*sinh(d*x + c)^5 + (a
^4 - a^3*b)*d*sinh(d*x + c)^6 - 3*(a^4 - a^3*b)*d*cosh(d*x + c)^4 + 3*(5*(a^4 - a^3*b)*d*cosh(d*x + c)^2 - (a^
4 - a^3*b)*d)*sinh(d*x + c)^4 + 3*(a^4 - a^3*b)*d*cosh(d*x + c)^2 + 4*(5*(a^4 - a^3*b)*d*cosh(d*x + c)^3 - 3*(
a^4 - a^3*b)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*(a^4 - a^3*b)*d*cosh(d*x + c)^4 - 6*(a^4 - a^3*b)*d*cosh(
d*x + c)^2 + (a^4 - a^3*b)*d)*sinh(d*x + c)^2 - (a^4 - a^3*b)*d + 6*((a^4 - a^3*b)*d*cosh(d*x + c)^5 - 2*(a^4
- a^3*b)*d*cosh(d*x + c)^3 + (a^4 - a^3*b)*d*cosh(d*x + c))*sinh(d*x + c))]
________________________________________________________________________________________
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),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [A] time = 1.42493, size = 159, normalized size = 2.04 \begin{align*} \frac{b^{2} \arctan \left (\frac{b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt{-a^{2} + a b}}\right )}{\sqrt{-a^{2} + a b} a^{2} d} + \frac{2 \,{\left (3 \, b e^{\left (4 \, d x + 4 \, c\right )} - 6 \, a e^{\left (2 \, d x + 2 \, c\right )} - 6 \, b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a + 3 \, b\right )}}{3 \, a^{2} 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),x, algorithm="giac")
[Out]
b^2*arctan(1/2*(b*e^(2*d*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/(sqrt(-a^2 + a*b)*a^2*d) + 2/3*(3*b*e^(4*d*x +
4*c) - 6*a*e^(2*d*x + 2*c) - 6*b*e^(2*d*x + 2*c) + 2*a + 3*b)/(a^2*d*(e^(2*d*x + 2*c) - 1)^3)