3.200 \(\int \text{csch}(c+d x) (a+b \sinh ^4(c+d x))^2 \, dx\)
Optimal. Leaf size=92 \[ -\frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b (2 a+3 b) \cosh ^3(c+d x)}{3 d}-\frac{b (2 a+b) \cosh (c+d x)}{d}+\frac{b^2 \cosh ^7(c+d x)}{7 d}-\frac{3 b^2 \cosh ^5(c+d x)}{5 d} \]
[Out]
-((a^2*ArcTanh[Cosh[c + d*x]])/d) - (b*(2*a + b)*Cosh[c + d*x])/d + (b*(2*a + 3*b)*Cosh[c + d*x]^3)/(3*d) - (3
*b^2*Cosh[c + d*x]^5)/(5*d) + (b^2*Cosh[c + d*x]^7)/(7*d)
________________________________________________________________________________________
Rubi [A] time = 0.092699, antiderivative size = 92, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used =
{3215, 1153, 206} \[ -\frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b (2 a+3 b) \cosh ^3(c+d x)}{3 d}-\frac{b (2 a+b) \cosh (c+d x)}{d}+\frac{b^2 \cosh ^7(c+d x)}{7 d}-\frac{3 b^2 \cosh ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
[In]
Int[Csch[c + d*x]*(a + b*Sinh[c + d*x]^4)^2,x]
[Out]
-((a^2*ArcTanh[Cosh[c + d*x]])/d) - (b*(2*a + b)*Cosh[c + d*x])/d + (b*(2*a + 3*b)*Cosh[c + d*x]^3)/(3*d) - (3
*b^2*Cosh[c + d*x]^5)/(5*d) + (b^2*Cosh[c + d*x]^7)/(7*d)
Rule 3215
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(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 - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Rule 1153
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
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}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (a+b-2 b x^2+b x^4\right )^2}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (b (2 a+b)-b (2 a+3 b) x^2+3 b^2 x^4-b^2 x^6+\frac{a^2}{1-x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{b (2 a+b) \cosh (c+d x)}{d}+\frac{b (2 a+3 b) \cosh ^3(c+d x)}{3 d}-\frac{3 b^2 \cosh ^5(c+d x)}{5 d}+\frac{b^2 \cosh ^7(c+d x)}{7 d}-\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{b (2 a+b) \cosh (c+d x)}{d}+\frac{b (2 a+3 b) \cosh ^3(c+d x)}{3 d}-\frac{3 b^2 \cosh ^5(c+d x)}{5 d}+\frac{b^2 \cosh ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.0411324, size = 146, normalized size = 1.59 \[ \frac{a^2 \log \left (\sinh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{a^2 \log \left (\cosh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{3 a b \cosh (c+d x)}{2 d}+\frac{a b \cosh (3 (c+d x))}{6 d}-\frac{35 b^2 \cosh (c+d x)}{64 d}+\frac{7 b^2 \cosh (3 (c+d x))}{64 d}-\frac{7 b^2 \cosh (5 (c+d x))}{320 d}+\frac{b^2 \cosh (7 (c+d x))}{448 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Csch[c + d*x]*(a + b*Sinh[c + d*x]^4)^2,x]
[Out]
(-3*a*b*Cosh[c + d*x])/(2*d) - (35*b^2*Cosh[c + d*x])/(64*d) + (a*b*Cosh[3*(c + d*x)])/(6*d) + (7*b^2*Cosh[3*(
c + d*x)])/(64*d) - (7*b^2*Cosh[5*(c + d*x)])/(320*d) + (b^2*Cosh[7*(c + d*x)])/(448*d) - (a^2*Log[Cosh[c/2 +
(d*x)/2]])/d + (a^2*Log[Sinh[c/2 + (d*x)/2]])/d
________________________________________________________________________________________
Maple [A] time = 0.039, size = 82, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ( -2\,{a}^{2}{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) +2\,ab \left ( -2/3+1/3\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2} \right ) \cosh \left ( dx+c \right ) +{b}^{2} \left ( -{\frac{16}{35}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}{7}}-{\frac{6\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{35}}+{\frac{8\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{35}} \right ) \cosh \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(d*x+c)*(a+b*sinh(d*x+c)^4)^2,x)
[Out]
1/d*(-2*a^2*arctanh(exp(d*x+c))+2*a*b*(-2/3+1/3*sinh(d*x+c)^2)*cosh(d*x+c)+b^2*(-16/35+1/7*sinh(d*x+c)^6-6/35*
sinh(d*x+c)^4+8/35*sinh(d*x+c)^2)*cosh(d*x+c))
________________________________________________________________________________________
Maxima [B] time = 1.05125, size = 239, normalized size = 2.6 \begin{align*} -\frac{1}{4480} \, b^{2}{\left (\frac{{\left (49 \, e^{\left (-2 \, d x - 2 \, c\right )} - 245 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1225 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac{1225 \, e^{\left (-d x - c\right )} - 245 \, e^{\left (-3 \, d x - 3 \, c\right )} + 49 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac{1}{12} \, a b{\left (\frac{e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac{9 \, e^{\left (d x + c\right )}}{d} - \frac{9 \, e^{\left (-d x - c\right )}}{d} + \frac{e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac{a^{2} \log \left (\tanh \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)*(a+b*sinh(d*x+c)^4)^2,x, algorithm="maxima")
[Out]
-1/4480*b^2*((49*e^(-2*d*x - 2*c) - 245*e^(-4*d*x - 4*c) + 1225*e^(-6*d*x - 6*c) - 5)*e^(7*d*x + 7*c)/d + (122
5*e^(-d*x - c) - 245*e^(-3*d*x - 3*c) + 49*e^(-5*d*x - 5*c) - 5*e^(-7*d*x - 7*c))/d) + 1/12*a*b*(e^(3*d*x + 3*
c)/d - 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d + e^(-3*d*x - 3*c)/d) + a^2*log(tanh(1/2*d*x + 1/2*c))/d
________________________________________________________________________________________
Fricas [B] time = 1.92082, size = 4263, normalized size = 46.34 \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)*(a+b*sinh(d*x+c)^4)^2,x, algorithm="fricas")
[Out]
1/13440*(15*b^2*cosh(d*x + c)^14 + 210*b^2*cosh(d*x + c)*sinh(d*x + c)^13 + 15*b^2*sinh(d*x + c)^14 - 147*b^2*
cosh(d*x + c)^12 + 21*(65*b^2*cosh(d*x + c)^2 - 7*b^2)*sinh(d*x + c)^12 + 84*(65*b^2*cosh(d*x + c)^3 - 21*b^2*
cosh(d*x + c))*sinh(d*x + c)^11 + 35*(32*a*b + 21*b^2)*cosh(d*x + c)^10 + 7*(2145*b^2*cosh(d*x + c)^4 - 1386*b
^2*cosh(d*x + c)^2 + 160*a*b + 105*b^2)*sinh(d*x + c)^10 + 70*(429*b^2*cosh(d*x + c)^5 - 462*b^2*cosh(d*x + c)
^3 + 5*(32*a*b + 21*b^2)*cosh(d*x + c))*sinh(d*x + c)^9 - 105*(96*a*b + 35*b^2)*cosh(d*x + c)^8 + 105*(429*b^2
*cosh(d*x + c)^6 - 693*b^2*cosh(d*x + c)^4 + 15*(32*a*b + 21*b^2)*cosh(d*x + c)^2 - 96*a*b - 35*b^2)*sinh(d*x
+ c)^8 + 24*(2145*b^2*cosh(d*x + c)^7 - 4851*b^2*cosh(d*x + c)^5 + 175*(32*a*b + 21*b^2)*cosh(d*x + c)^3 - 35*
(96*a*b + 35*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 105*(96*a*b + 35*b^2)*cosh(d*x + c)^6 + 21*(2145*b^2*cosh(d
*x + c)^8 - 6468*b^2*cosh(d*x + c)^6 + 350*(32*a*b + 21*b^2)*cosh(d*x + c)^4 - 140*(96*a*b + 35*b^2)*cosh(d*x
+ c)^2 - 480*a*b - 175*b^2)*sinh(d*x + c)^6 + 42*(715*b^2*cosh(d*x + c)^9 - 2772*b^2*cosh(d*x + c)^7 + 210*(32
*a*b + 21*b^2)*cosh(d*x + c)^5 - 140*(96*a*b + 35*b^2)*cosh(d*x + c)^3 - 15*(96*a*b + 35*b^2)*cosh(d*x + c))*s
inh(d*x + c)^5 + 35*(32*a*b + 21*b^2)*cosh(d*x + c)^4 + 35*(429*b^2*cosh(d*x + c)^10 - 2079*b^2*cosh(d*x + c)^
8 + 210*(32*a*b + 21*b^2)*cosh(d*x + c)^6 - 210*(96*a*b + 35*b^2)*cosh(d*x + c)^4 - 45*(96*a*b + 35*b^2)*cosh(
d*x + c)^2 + 32*a*b + 21*b^2)*sinh(d*x + c)^4 - 147*b^2*cosh(d*x + c)^2 + 140*(39*b^2*cosh(d*x + c)^11 - 231*b
^2*cosh(d*x + c)^9 + 30*(32*a*b + 21*b^2)*cosh(d*x + c)^7 - 42*(96*a*b + 35*b^2)*cosh(d*x + c)^5 - 15*(96*a*b
+ 35*b^2)*cosh(d*x + c)^3 + (32*a*b + 21*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 21*(65*b^2*cosh(d*x + c)^12 - 4
62*b^2*cosh(d*x + c)^10 + 75*(32*a*b + 21*b^2)*cosh(d*x + c)^8 - 140*(96*a*b + 35*b^2)*cosh(d*x + c)^6 - 75*(9
6*a*b + 35*b^2)*cosh(d*x + c)^4 + 10*(32*a*b + 21*b^2)*cosh(d*x + c)^2 - 7*b^2)*sinh(d*x + c)^2 + 15*b^2 - 134
40*(a^2*cosh(d*x + c)^7 + 7*a^2*cosh(d*x + c)^6*sinh(d*x + c) + 21*a^2*cosh(d*x + c)^5*sinh(d*x + c)^2 + 35*a^
2*cosh(d*x + c)^4*sinh(d*x + c)^3 + 35*a^2*cosh(d*x + c)^3*sinh(d*x + c)^4 + 21*a^2*cosh(d*x + c)^2*sinh(d*x +
c)^5 + 7*a^2*cosh(d*x + c)*sinh(d*x + c)^6 + a^2*sinh(d*x + c)^7)*log(cosh(d*x + c) + sinh(d*x + c) + 1) + 13
440*(a^2*cosh(d*x + c)^7 + 7*a^2*cosh(d*x + c)^6*sinh(d*x + c) + 21*a^2*cosh(d*x + c)^5*sinh(d*x + c)^2 + 35*a
^2*cosh(d*x + c)^4*sinh(d*x + c)^3 + 35*a^2*cosh(d*x + c)^3*sinh(d*x + c)^4 + 21*a^2*cosh(d*x + c)^2*sinh(d*x
+ c)^5 + 7*a^2*cosh(d*x + c)*sinh(d*x + c)^6 + a^2*sinh(d*x + c)^7)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 1
4*(15*b^2*cosh(d*x + c)^13 - 126*b^2*cosh(d*x + c)^11 + 25*(32*a*b + 21*b^2)*cosh(d*x + c)^9 - 60*(96*a*b + 35
*b^2)*cosh(d*x + c)^7 - 45*(96*a*b + 35*b^2)*cosh(d*x + c)^5 + 10*(32*a*b + 21*b^2)*cosh(d*x + c)^3 - 21*b^2*c
osh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^7 + 7*d*cosh(d*x + c)^6*sinh(d*x + c) + 21*d*cosh(d*x + c)^5*sin
h(d*x + c)^2 + 35*d*cosh(d*x + c)^4*sinh(d*x + c)^3 + 35*d*cosh(d*x + c)^3*sinh(d*x + c)^4 + 21*d*cosh(d*x + c
)^2*sinh(d*x + c)^5 + 7*d*cosh(d*x + c)*sinh(d*x + c)^6 + d*sinh(d*x + c)^7)
________________________________________________________________________________________
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)*(a+b*sinh(d*x+c)**4)**2,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.30251, size = 301, normalized size = 3.27 \begin{align*} -\frac{a^{2} \log \left (e^{\left (d x + c\right )} + 1\right )}{d} + \frac{a^{2} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{d} - \frac{{\left (10080 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 3675 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 1120 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 735 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 147 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 15 \, b^{2}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{13440 \, d} + \frac{15 \, b^{2} d^{6} e^{\left (7 \, d x + 7 \, c\right )} - 147 \, b^{2} d^{6} e^{\left (5 \, d x + 5 \, c\right )} + 1120 \, a b d^{6} e^{\left (3 \, d x + 3 \, c\right )} + 735 \, b^{2} d^{6} e^{\left (3 \, d x + 3 \, c\right )} - 10080 \, a b d^{6} e^{\left (d x + c\right )} - 3675 \, b^{2} d^{6} e^{\left (d x + c\right )}}{13440 \, d^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)*(a+b*sinh(d*x+c)^4)^2,x, algorithm="giac")
[Out]
-a^2*log(e^(d*x + c) + 1)/d + a^2*log(abs(e^(d*x + c) - 1))/d - 1/13440*(10080*a*b*e^(6*d*x + 6*c) + 3675*b^2*
e^(6*d*x + 6*c) - 1120*a*b*e^(4*d*x + 4*c) - 735*b^2*e^(4*d*x + 4*c) + 147*b^2*e^(2*d*x + 2*c) - 15*b^2)*e^(-7
*d*x - 7*c)/d + 1/13440*(15*b^2*d^6*e^(7*d*x + 7*c) - 147*b^2*d^6*e^(5*d*x + 5*c) + 1120*a*b*d^6*e^(3*d*x + 3*
c) + 735*b^2*d^6*e^(3*d*x + 3*c) - 10080*a*b*d^6*e^(d*x + c) - 3675*b^2*d^6*e^(d*x + c))/d^7