3.202 \(\int \text{csch}^3(c+d x) (a+b \sinh ^4(c+d x))^2 \, dx\)

Optimal. Leaf size=92 \[ \frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a^2 \coth (c+d x) \text{csch}(c+d x)}{2 d}+\frac{b (2 a+b) \cosh (c+d x)}{d}+\frac{b^2 \cosh ^5(c+d x)}{5 d}-\frac{2 b^2 \cosh ^3(c+d x)}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.134862, antiderivative size = 92, 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 = {3215, 1157, 1810, 206} \[ \frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a^2 \coth (c+d x) \text{csch}(c+d x)}{2 d}+\frac{b (2 a+b) \cosh (c+d x)}{d}+\frac{b^2 \cosh ^5(c+d x)}{5 d}-\frac{2 b^2 \cosh ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^4)^2,x]

[Out]

(a^2*ArcTanh[Cosh[c + d*x]])/(2*d) + (b*(2*a + b)*Cosh[c + d*x])/d - (2*b^2*Cosh[c + d*x]^3)/(3*d) + (b^2*Cosh
[c + d*x]^5)/(5*d) - (a^2*Coth[c + d*x]*Csch[c + d*x])/(2*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 1157

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, -Simp[(R*x*(d + e*x^2)^(q + 1))/(2*d*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*
ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N
eQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 1810

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -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}^3(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}{\left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{a^2 \coth (c+d x) \text{csch}(c+d x)}{2 d}-\frac{\operatorname{Subst}\left (\int \frac{-a^2-4 a b-2 b^2+2 b (2 a+3 b) x^2-6 b^2 x^4+2 b^2 x^6}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{2 d}\\ &=-\frac{a^2 \coth (c+d x) \text{csch}(c+d x)}{2 d}-\frac{\operatorname{Subst}\left (\int \left (-2 b (2 a+b)+4 b^2 x^2-2 b^2 x^4-\frac{a^2}{1-x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{2 d}\\ &=\frac{b (2 a+b) \cosh (c+d x)}{d}-\frac{2 b^2 \cosh ^3(c+d x)}{3 d}+\frac{b^2 \cosh ^5(c+d x)}{5 d}-\frac{a^2 \coth (c+d x) \text{csch}(c+d x)}{2 d}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{2 d}\\ &=\frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{2 d}+\frac{b (2 a+b) \cosh (c+d x)}{d}-\frac{2 b^2 \cosh ^3(c+d x)}{3 d}+\frac{b^2 \cosh ^5(c+d x)}{5 d}-\frac{a^2 \coth (c+d x) \text{csch}(c+d x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.0457128, size = 144, normalized size = 1.57 \[ -\frac{a^2 \text{csch}^2\left (\frac{1}{2} (c+d x)\right )}{8 d}-\frac{a^2 \text{sech}^2\left (\frac{1}{2} (c+d x)\right )}{8 d}-\frac{a^2 \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}+\frac{2 a b \sinh (c) \sinh (d x)}{d}+\frac{2 a b \cosh (c) \cosh (d x)}{d}+\frac{5 b^2 \cosh (c+d x)}{8 d}-\frac{5 b^2 \cosh (3 (c+d x))}{48 d}+\frac{b^2 \cosh (5 (c+d x))}{80 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^4)^2,x]

[Out]

(2*a*b*Cosh[c]*Cosh[d*x])/d + (5*b^2*Cosh[c + d*x])/(8*d) - (5*b^2*Cosh[3*(c + d*x)])/(48*d) + (b^2*Cosh[5*(c
+ d*x)])/(80*d) - (a^2*Csch[(c + d*x)/2]^2)/(8*d) - (a^2*Log[Tanh[(c + d*x)/2]])/(2*d) - (a^2*Sech[(c + d*x)/2
]^2)/(8*d) + (2*a*b*Sinh[c]*Sinh[d*x])/d

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Maple [A]  time = 0.048, size = 74, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( -{\frac{{\rm csch} \left (dx+c\right ){\rm coth} \left (dx+c\right )}{2}}+{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \right ) +2\,ab\cosh \left ( dx+c \right ) +{b}^{2} \left ({\frac{8}{15}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{15}} \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)^3*(a+b*sinh(d*x+c)^4)^2,x)

[Out]

1/d*(a^2*(-1/2*csch(d*x+c)*coth(d*x+c)+arctanh(exp(d*x+c)))+2*a*b*cosh(d*x+c)+b^2*(8/15+1/5*sinh(d*x+c)^4-4/15
*sinh(d*x+c)^2)*cosh(d*x+c))

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Maxima [B]  time = 1.1179, size = 275, normalized size = 2.99 \begin{align*} \frac{1}{480} \, b^{2}{\left (\frac{3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} - \frac{25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac{150 \, e^{\left (d x + c\right )}}{d} + \frac{150 \, e^{\left (-d x - c\right )}}{d} - \frac{25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac{3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} + a b{\left (\frac{e^{\left (d x + c\right )}}{d} + \frac{e^{\left (-d x - c\right )}}{d}\right )} + \frac{1}{2} \, a^{2}{\left (\frac{\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac{\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac{2 \,{\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^3*(a+b*sinh(d*x+c)^4)^2,x, algorithm="maxima")

[Out]

1/480*b^2*(3*e^(5*d*x + 5*c)/d - 25*e^(3*d*x + 3*c)/d + 150*e^(d*x + c)/d + 150*e^(-d*x - c)/d - 25*e^(-3*d*x
- 3*c)/d + 3*e^(-5*d*x - 5*c)/d) + a*b*(e^(d*x + c)/d + e^(-d*x - c)/d) + 1/2*a^2*(log(e^(-d*x - c) + 1)/d - l
og(e^(-d*x - c) - 1)/d + 2*(e^(-d*x - c) + e^(-3*d*x - 3*c))/(d*(2*e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c) - 1)))

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Fricas [B]  time = 1.98212, size = 6047, normalized size = 65.73 \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)^3*(a+b*sinh(d*x+c)^4)^2,x, algorithm="fricas")

[Out]

1/480*(3*b^2*cosh(d*x + c)^14 + 42*b^2*cosh(d*x + c)*sinh(d*x + c)^13 + 3*b^2*sinh(d*x + c)^14 - 31*b^2*cosh(d
*x + c)^12 + (273*b^2*cosh(d*x + c)^2 - 31*b^2)*sinh(d*x + c)^12 + 12*(91*b^2*cosh(d*x + c)^3 - 31*b^2*cosh(d*
x + c))*sinh(d*x + c)^11 + (480*a*b + 203*b^2)*cosh(d*x + c)^10 + (3003*b^2*cosh(d*x + c)^4 - 2046*b^2*cosh(d*
x + c)^2 + 480*a*b + 203*b^2)*sinh(d*x + c)^10 + 2*(3003*b^2*cosh(d*x + c)^5 - 3410*b^2*cosh(d*x + c)^3 + 5*(4
80*a*b + 203*b^2)*cosh(d*x + c))*sinh(d*x + c)^9 - 5*(96*a^2 + 96*a*b + 35*b^2)*cosh(d*x + c)^8 + (9009*b^2*co
sh(d*x + c)^6 - 15345*b^2*cosh(d*x + c)^4 + 45*(480*a*b + 203*b^2)*cosh(d*x + c)^2 - 480*a^2 - 480*a*b - 175*b
^2)*sinh(d*x + c)^8 + 8*(1287*b^2*cosh(d*x + c)^7 - 3069*b^2*cosh(d*x + c)^5 + 15*(480*a*b + 203*b^2)*cosh(d*x
+ c)^3 - 5*(96*a^2 + 96*a*b + 35*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 5*(96*a^2 + 96*a*b + 35*b^2)*cosh(d*x
+ c)^6 + (9009*b^2*cosh(d*x + c)^8 - 28644*b^2*cosh(d*x + c)^6 + 210*(480*a*b + 203*b^2)*cosh(d*x + c)^4 - 140
*(96*a^2 + 96*a*b + 35*b^2)*cosh(d*x + c)^2 - 480*a^2 - 480*a*b - 175*b^2)*sinh(d*x + c)^6 + 2*(3003*b^2*cosh(
d*x + c)^9 - 12276*b^2*cosh(d*x + c)^7 + 126*(480*a*b + 203*b^2)*cosh(d*x + c)^5 - 140*(96*a^2 + 96*a*b + 35*b
^2)*cosh(d*x + c)^3 - 15*(96*a^2 + 96*a*b + 35*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + (480*a*b + 203*b^2)*cosh(
d*x + c)^4 + (3003*b^2*cosh(d*x + c)^10 - 15345*b^2*cosh(d*x + c)^8 + 210*(480*a*b + 203*b^2)*cosh(d*x + c)^6
- 350*(96*a^2 + 96*a*b + 35*b^2)*cosh(d*x + c)^4 - 75*(96*a^2 + 96*a*b + 35*b^2)*cosh(d*x + c)^2 + 480*a*b + 2
03*b^2)*sinh(d*x + c)^4 - 31*b^2*cosh(d*x + c)^2 + 4*(273*b^2*cosh(d*x + c)^11 - 1705*b^2*cosh(d*x + c)^9 + 30
*(480*a*b + 203*b^2)*cosh(d*x + c)^7 - 70*(96*a^2 + 96*a*b + 35*b^2)*cosh(d*x + c)^5 - 25*(96*a^2 + 96*a*b + 3
5*b^2)*cosh(d*x + c)^3 + (480*a*b + 203*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + (273*b^2*cosh(d*x + c)^12 - 2046
*b^2*cosh(d*x + c)^10 + 45*(480*a*b + 203*b^2)*cosh(d*x + c)^8 - 140*(96*a^2 + 96*a*b + 35*b^2)*cosh(d*x + c)^
6 - 75*(96*a^2 + 96*a*b + 35*b^2)*cosh(d*x + c)^4 + 6*(480*a*b + 203*b^2)*cosh(d*x + c)^2 - 31*b^2)*sinh(d*x +
c)^2 + 3*b^2 + 240*(a^2*cosh(d*x + c)^9 + 9*a^2*cosh(d*x + c)*sinh(d*x + c)^8 + a^2*sinh(d*x + c)^9 - 2*a^2*c
osh(d*x + c)^7 + 2*(18*a^2*cosh(d*x + c)^2 - a^2)*sinh(d*x + c)^7 + a^2*cosh(d*x + c)^5 + 14*(6*a^2*cosh(d*x +
c)^3 - a^2*cosh(d*x + c))*sinh(d*x + c)^6 + (126*a^2*cosh(d*x + c)^4 - 42*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*x
+ c)^5 + (126*a^2*cosh(d*x + c)^5 - 70*a^2*cosh(d*x + c)^3 + 5*a^2*cosh(d*x + c))*sinh(d*x + c)^4 + 2*(42*a^2
*cosh(d*x + c)^6 - 35*a^2*cosh(d*x + c)^4 + 5*a^2*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(18*a^2*cosh(d*x + c)^7
- 21*a^2*cosh(d*x + c)^5 + 5*a^2*cosh(d*x + c)^3)*sinh(d*x + c)^2 + (9*a^2*cosh(d*x + c)^8 - 14*a^2*cosh(d*x
+ c)^6 + 5*a^2*cosh(d*x + c)^4)*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) - 240*(a^2*cosh(d*x + c)
^9 + 9*a^2*cosh(d*x + c)*sinh(d*x + c)^8 + a^2*sinh(d*x + c)^9 - 2*a^2*cosh(d*x + c)^7 + 2*(18*a^2*cosh(d*x +
c)^2 - a^2)*sinh(d*x + c)^7 + a^2*cosh(d*x + c)^5 + 14*(6*a^2*cosh(d*x + c)^3 - a^2*cosh(d*x + c))*sinh(d*x +
c)^6 + (126*a^2*cosh(d*x + c)^4 - 42*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*x + c)^5 + (126*a^2*cosh(d*x + c)^5 - 7
0*a^2*cosh(d*x + c)^3 + 5*a^2*cosh(d*x + c))*sinh(d*x + c)^4 + 2*(42*a^2*cosh(d*x + c)^6 - 35*a^2*cosh(d*x + c
)^4 + 5*a^2*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(18*a^2*cosh(d*x + c)^7 - 21*a^2*cosh(d*x + c)^5 + 5*a^2*cosh
(d*x + c)^3)*sinh(d*x + c)^2 + (9*a^2*cosh(d*x + c)^8 - 14*a^2*cosh(d*x + c)^6 + 5*a^2*cosh(d*x + c)^4)*sinh(d
*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 2*(21*b^2*cosh(d*x + c)^13 - 186*b^2*cosh(d*x + c)^11 + 5*(4
80*a*b + 203*b^2)*cosh(d*x + c)^9 - 20*(96*a^2 + 96*a*b + 35*b^2)*cosh(d*x + c)^7 - 15*(96*a^2 + 96*a*b + 35*b
^2)*cosh(d*x + c)^5 + 2*(480*a*b + 203*b^2)*cosh(d*x + c)^3 - 31*b^2*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x
+ c)^9 + 9*d*cosh(d*x + c)*sinh(d*x + c)^8 + d*sinh(d*x + c)^9 - 2*d*cosh(d*x + c)^7 + 2*(18*d*cosh(d*x + c)^
2 - d)*sinh(d*x + c)^7 + 14*(6*d*cosh(d*x + c)^3 - d*cosh(d*x + c))*sinh(d*x + c)^6 + d*cosh(d*x + c)^5 + (126
*d*cosh(d*x + c)^4 - 42*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^5 + (126*d*cosh(d*x + c)^5 - 70*d*cosh(d*x + c)^3
+ 5*d*cosh(d*x + c))*sinh(d*x + c)^4 + 2*(42*d*cosh(d*x + c)^6 - 35*d*cosh(d*x + c)^4 + 5*d*cosh(d*x + c)^2)*
sinh(d*x + c)^3 + 2*(18*d*cosh(d*x + c)^7 - 21*d*cosh(d*x + c)^5 + 5*d*cosh(d*x + c)^3)*sinh(d*x + c)^2 + (9*d
*cosh(d*x + c)^8 - 14*d*cosh(d*x + c)^6 + 5*d*cosh(d*x + c)^4)*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)**3*(a+b*sinh(d*x+c)**4)**2,x)

[Out]

Timed out

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Giac [B]  time = 1.28386, size = 275, normalized size = 2.99 \begin{align*} \frac{a^{2} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right )}{4 \, d} - \frac{a^{2} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right )}{4 \, d} - \frac{a^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4\right )} d} + \frac{3 \, b^{2} d^{4}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 40 \, b^{2} d^{4}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 480 \, a b d^{4}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 240 \, b^{2} d^{4}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{480 \, d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^3*(a+b*sinh(d*x+c)^4)^2,x, algorithm="giac")

[Out]

1/4*a^2*log(e^(d*x + c) + e^(-d*x - c) + 2)/d - 1/4*a^2*log(e^(d*x + c) + e^(-d*x - c) - 2)/d - a^2*(e^(d*x +
c) + e^(-d*x - c))/(((e^(d*x + c) + e^(-d*x - c))^2 - 4)*d) + 1/480*(3*b^2*d^4*(e^(d*x + c) + e^(-d*x - c))^5
- 40*b^2*d^4*(e^(d*x + c) + e^(-d*x - c))^3 + 480*a*b*d^4*(e^(d*x + c) + e^(-d*x - c)) + 240*b^2*d^4*(e^(d*x +
c) + e^(-d*x - c)))/d^5