3.206 \(\int \text{csch}^7(c+d x) (a+b \sinh ^4(c+d x))^2 \, dx\)

Optimal. Leaf size=111 \[ -\frac{a^2 \coth (c+d x) \text{csch}^5(c+d x)}{6 d}+\frac{5 a^2 \coth (c+d x) \text{csch}^3(c+d x)}{24 d}+\frac{a (5 a+16 b) \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac{a (5 a+16 b) \coth (c+d x) \text{csch}(c+d x)}{16 d}+\frac{b^2 \cosh (c+d x)}{d} \]

[Out]

(a*(5*a + 16*b)*ArcTanh[Cosh[c + d*x]])/(16*d) + (b^2*Cosh[c + d*x])/d - (a*(5*a + 16*b)*Coth[c + d*x]*Csch[c
+ d*x])/(16*d) + (5*a^2*Coth[c + d*x]*Csch[c + d*x]^3)/(24*d) - (a^2*Coth[c + d*x]*Csch[c + d*x]^5)/(6*d)

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Rubi [A]  time = 0.174123, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3215, 1157, 1814, 388, 206} \[ -\frac{a^2 \coth (c+d x) \text{csch}^5(c+d x)}{6 d}+\frac{5 a^2 \coth (c+d x) \text{csch}^3(c+d x)}{24 d}+\frac{a (5 a+16 b) \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac{a (5 a+16 b) \coth (c+d x) \text{csch}(c+d x)}{16 d}+\frac{b^2 \cosh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^7*(a + b*Sinh[c + d*x]^4)^2,x]

[Out]

(a*(5*a + 16*b)*ArcTanh[Cosh[c + d*x]])/(16*d) + (b^2*Cosh[c + d*x])/d - (a*(5*a + 16*b)*Coth[c + d*x]*Csch[c
+ d*x])/(16*d) + (5*a^2*Coth[c + d*x]*Csch[c + d*x]^3)/(24*d) - (a^2*Coth[c + d*x]*Csch[c + d*x]^5)/(6*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 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

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}^7(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 )^4} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{a^2 \coth (c+d x) \text{csch}^5(c+d x)}{6 d}-\frac{\operatorname{Subst}\left (\int \frac{-5 a^2-12 a b-6 b^2+6 b (2 a+3 b) x^2-18 b^2 x^4+6 b^2 x^6}{\left (1-x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{6 d}\\ &=\frac{5 a^2 \coth (c+d x) \text{csch}^3(c+d x)}{24 d}-\frac{a^2 \coth (c+d x) \text{csch}^5(c+d x)}{6 d}+\frac{\operatorname{Subst}\left (\int \frac{3 \left (5 a^2+16 a b+8 b^2\right )-48 b^2 x^2+24 b^2 x^4}{\left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{24 d}\\ &=-\frac{a (5 a+16 b) \coth (c+d x) \text{csch}(c+d x)}{16 d}+\frac{5 a^2 \coth (c+d x) \text{csch}^3(c+d x)}{24 d}-\frac{a^2 \coth (c+d x) \text{csch}^5(c+d x)}{6 d}-\frac{\operatorname{Subst}\left (\int \frac{-3 \left (5 a^2+16 a b+16 b^2\right )+48 b^2 x^2}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{48 d}\\ &=\frac{b^2 \cosh (c+d x)}{d}-\frac{a (5 a+16 b) \coth (c+d x) \text{csch}(c+d x)}{16 d}+\frac{5 a^2 \coth (c+d x) \text{csch}^3(c+d x)}{24 d}-\frac{a^2 \coth (c+d x) \text{csch}^5(c+d x)}{6 d}+\frac{(a (5 a+16 b)) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{16 d}\\ &=\frac{a (5 a+16 b) \tanh ^{-1}(\cosh (c+d x))}{16 d}+\frac{b^2 \cosh (c+d x)}{d}-\frac{a (5 a+16 b) \coth (c+d x) \text{csch}(c+d x)}{16 d}+\frac{5 a^2 \coth (c+d x) \text{csch}^3(c+d x)}{24 d}-\frac{a^2 \coth (c+d x) \text{csch}^5(c+d x)}{6 d}\\ \end{align*}

Mathematica [B]  time = 0.0355098, size = 240, normalized size = 2.16 \[ -\frac{a^2 \text{csch}^6\left (\frac{1}{2} (c+d x)\right )}{384 d}+\frac{a^2 \text{csch}^4\left (\frac{1}{2} (c+d x)\right )}{64 d}-\frac{5 a^2 \text{csch}^2\left (\frac{1}{2} (c+d x)\right )}{64 d}-\frac{a^2 \text{sech}^6\left (\frac{1}{2} (c+d x)\right )}{384 d}-\frac{a^2 \text{sech}^4\left (\frac{1}{2} (c+d x)\right )}{64 d}-\frac{5 a^2 \text{sech}^2\left (\frac{1}{2} (c+d x)\right )}{64 d}-\frac{5 a^2 \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{16 d}-\frac{a b \text{csch}^2\left (\frac{1}{2} (c+d x)\right )}{4 d}-\frac{a b \text{sech}^2\left (\frac{1}{2} (c+d x)\right )}{4 d}-\frac{a b \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{d}+\frac{b^2 \sinh (c) \sinh (d x)}{d}+\frac{b^2 \cosh (c) \cosh (d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^7*(a + b*Sinh[c + d*x]^4)^2,x]

[Out]

(b^2*Cosh[c]*Cosh[d*x])/d - (5*a^2*Csch[(c + d*x)/2]^2)/(64*d) - (a*b*Csch[(c + d*x)/2]^2)/(4*d) + (a^2*Csch[(
c + d*x)/2]^4)/(64*d) - (a^2*Csch[(c + d*x)/2]^6)/(384*d) - (5*a^2*Log[Tanh[(c + d*x)/2]])/(16*d) - (a*b*Log[T
anh[(c + d*x)/2]])/d - (5*a^2*Sech[(c + d*x)/2]^2)/(64*d) - (a*b*Sech[(c + d*x)/2]^2)/(4*d) - (a^2*Sech[(c + d
*x)/2]^4)/(64*d) - (a^2*Sech[(c + d*x)/2]^6)/(384*d) + (b^2*Sinh[c]*Sinh[d*x])/d

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Maple [A]  time = 0.048, size = 92, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( \left ( -{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{5}}{6}}+{\frac{5\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{3}}{24}}-{\frac{5\,{\rm csch} \left (dx+c\right )}{16}} \right ){\rm coth} \left (dx+c\right )+{\frac{5\,{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) }{8}} \right ) +2\,ab \left ( -1/2\,{\rm csch} \left (dx+c\right ){\rm coth} \left (dx+c\right )+{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \right ) +{b}^{2}\cosh \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^7*(a+b*sinh(d*x+c)^4)^2,x)

[Out]

1/d*(a^2*((-1/6*csch(d*x+c)^5+5/24*csch(d*x+c)^3-5/16*csch(d*x+c))*coth(d*x+c)+5/8*arctanh(exp(d*x+c)))+2*a*b*
(-1/2*csch(d*x+c)*coth(d*x+c)+arctanh(exp(d*x+c)))+b^2*cosh(d*x+c))

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Maxima [B]  time = 1.04059, size = 404, normalized size = 3.64 \begin{align*} \frac{1}{2} \, b^{2}{\left (\frac{e^{\left (d x + c\right )}}{d} + \frac{e^{\left (-d x - c\right )}}{d}\right )} + \frac{1}{48} \, a^{2}{\left (\frac{15 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac{15 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac{2 \,{\left (15 \, e^{\left (-d x - c\right )} - 85 \, e^{\left (-3 \, d x - 3 \, c\right )} + 198 \, e^{\left (-5 \, d x - 5 \, c\right )} + 198 \, e^{\left (-7 \, d x - 7 \, c\right )} - 85 \, e^{\left (-9 \, d x - 9 \, c\right )} + 15 \, e^{\left (-11 \, d x - 11 \, c\right )}\right )}}{d{\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}}\right )} + a b{\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)^7*(a+b*sinh(d*x+c)^4)^2,x, algorithm="maxima")

[Out]

1/2*b^2*(e^(d*x + c)/d + e^(-d*x - c)/d) + 1/48*a^2*(15*log(e^(-d*x - c) + 1)/d - 15*log(e^(-d*x - c) - 1)/d +
2*(15*e^(-d*x - c) - 85*e^(-3*d*x - 3*c) + 198*e^(-5*d*x - 5*c) + 198*e^(-7*d*x - 7*c) - 85*e^(-9*d*x - 9*c)
+ 15*e^(-11*d*x - 11*c))/(d*(6*e^(-2*d*x - 2*c) - 15*e^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) - 15*e^(-8*d*x - 8
*c) + 6*e^(-10*d*x - 10*c) - e^(-12*d*x - 12*c) - 1))) + a*b*(log(e^(-d*x - c) + 1)/d - log(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 = 2.11855, size = 11957, normalized size = 107.72 \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)^7*(a+b*sinh(d*x+c)^4)^2,x, algorithm="fricas")

[Out]

1/48*(24*b^2*cosh(d*x + c)^14 + 336*b^2*cosh(d*x + c)*sinh(d*x + c)^13 + 24*b^2*sinh(d*x + c)^14 - 6*(5*a^2 +
16*a*b + 20*b^2)*cosh(d*x + c)^12 + 6*(364*b^2*cosh(d*x + c)^2 - 5*a^2 - 16*a*b - 20*b^2)*sinh(d*x + c)^12 + 2
4*(364*b^2*cosh(d*x + c)^3 - 3*(5*a^2 + 16*a*b + 20*b^2)*cosh(d*x + c))*sinh(d*x + c)^11 + 2*(85*a^2 + 144*a*b
+ 108*b^2)*cosh(d*x + c)^10 + 2*(12012*b^2*cosh(d*x + c)^4 - 198*(5*a^2 + 16*a*b + 20*b^2)*cosh(d*x + c)^2 +
85*a^2 + 144*a*b + 108*b^2)*sinh(d*x + c)^10 + 4*(12012*b^2*cosh(d*x + c)^5 - 330*(5*a^2 + 16*a*b + 20*b^2)*co
sh(d*x + c)^3 + 5*(85*a^2 + 144*a*b + 108*b^2)*cosh(d*x + c))*sinh(d*x + c)^9 - 12*(33*a^2 + 16*a*b + 10*b^2)*
cosh(d*x + c)^8 + 6*(12012*b^2*cosh(d*x + c)^6 - 495*(5*a^2 + 16*a*b + 20*b^2)*cosh(d*x + c)^4 + 15*(85*a^2 +
144*a*b + 108*b^2)*cosh(d*x + c)^2 - 66*a^2 - 32*a*b - 20*b^2)*sinh(d*x + c)^8 + 48*(1716*b^2*cosh(d*x + c)^7
- 99*(5*a^2 + 16*a*b + 20*b^2)*cosh(d*x + c)^5 + 5*(85*a^2 + 144*a*b + 108*b^2)*cosh(d*x + c)^3 - 2*(33*a^2 +
16*a*b + 10*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 12*(33*a^2 + 16*a*b + 10*b^2)*cosh(d*x + c)^6 + 12*(6006*b^2
*cosh(d*x + c)^8 - 462*(5*a^2 + 16*a*b + 20*b^2)*cosh(d*x + c)^6 + 35*(85*a^2 + 144*a*b + 108*b^2)*cosh(d*x +
c)^4 - 28*(33*a^2 + 16*a*b + 10*b^2)*cosh(d*x + c)^2 - 33*a^2 - 16*a*b - 10*b^2)*sinh(d*x + c)^6 + 24*(2002*b^
2*cosh(d*x + c)^9 - 198*(5*a^2 + 16*a*b + 20*b^2)*cosh(d*x + c)^7 + 21*(85*a^2 + 144*a*b + 108*b^2)*cosh(d*x +
c)^5 - 28*(33*a^2 + 16*a*b + 10*b^2)*cosh(d*x + c)^3 - 3*(33*a^2 + 16*a*b + 10*b^2)*cosh(d*x + c))*sinh(d*x +
c)^5 + 2*(85*a^2 + 144*a*b + 108*b^2)*cosh(d*x + c)^4 + 2*(12012*b^2*cosh(d*x + c)^10 - 1485*(5*a^2 + 16*a*b
+ 20*b^2)*cosh(d*x + c)^8 + 210*(85*a^2 + 144*a*b + 108*b^2)*cosh(d*x + c)^6 - 420*(33*a^2 + 16*a*b + 10*b^2)*
cosh(d*x + c)^4 - 90*(33*a^2 + 16*a*b + 10*b^2)*cosh(d*x + c)^2 + 85*a^2 + 144*a*b + 108*b^2)*sinh(d*x + c)^4
+ 8*(1092*b^2*cosh(d*x + c)^11 - 165*(5*a^2 + 16*a*b + 20*b^2)*cosh(d*x + c)^9 + 30*(85*a^2 + 144*a*b + 108*b^
2)*cosh(d*x + c)^7 - 84*(33*a^2 + 16*a*b + 10*b^2)*cosh(d*x + c)^5 - 30*(33*a^2 + 16*a*b + 10*b^2)*cosh(d*x +
c)^3 + (85*a^2 + 144*a*b + 108*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 6*(5*a^2 + 16*a*b + 20*b^2)*cosh(d*x + c)
^2 + 6*(364*b^2*cosh(d*x + c)^12 - 66*(5*a^2 + 16*a*b + 20*b^2)*cosh(d*x + c)^10 + 15*(85*a^2 + 144*a*b + 108*
b^2)*cosh(d*x + c)^8 - 56*(33*a^2 + 16*a*b + 10*b^2)*cosh(d*x + c)^6 - 30*(33*a^2 + 16*a*b + 10*b^2)*cosh(d*x
+ c)^4 + 2*(85*a^2 + 144*a*b + 108*b^2)*cosh(d*x + c)^2 - 5*a^2 - 16*a*b - 20*b^2)*sinh(d*x + c)^2 + 24*b^2 +
3*((5*a^2 + 16*a*b)*cosh(d*x + c)^13 + 13*(5*a^2 + 16*a*b)*cosh(d*x + c)*sinh(d*x + c)^12 + (5*a^2 + 16*a*b)*s
inh(d*x + c)^13 - 6*(5*a^2 + 16*a*b)*cosh(d*x + c)^11 + 6*(13*(5*a^2 + 16*a*b)*cosh(d*x + c)^2 - 5*a^2 - 16*a*
b)*sinh(d*x + c)^11 + 22*(13*(5*a^2 + 16*a*b)*cosh(d*x + c)^3 - 3*(5*a^2 + 16*a*b)*cosh(d*x + c))*sinh(d*x + c
)^10 + 15*(5*a^2 + 16*a*b)*cosh(d*x + c)^9 + 5*(143*(5*a^2 + 16*a*b)*cosh(d*x + c)^4 - 66*(5*a^2 + 16*a*b)*cos
h(d*x + c)^2 + 15*a^2 + 48*a*b)*sinh(d*x + c)^9 + 9*(143*(5*a^2 + 16*a*b)*cosh(d*x + c)^5 - 110*(5*a^2 + 16*a*
b)*cosh(d*x + c)^3 + 15*(5*a^2 + 16*a*b)*cosh(d*x + c))*sinh(d*x + c)^8 - 20*(5*a^2 + 16*a*b)*cosh(d*x + c)^7
+ 4*(429*(5*a^2 + 16*a*b)*cosh(d*x + c)^6 - 495*(5*a^2 + 16*a*b)*cosh(d*x + c)^4 + 135*(5*a^2 + 16*a*b)*cosh(d
*x + c)^2 - 25*a^2 - 80*a*b)*sinh(d*x + c)^7 + 4*(429*(5*a^2 + 16*a*b)*cosh(d*x + c)^7 - 693*(5*a^2 + 16*a*b)*
cosh(d*x + c)^5 + 315*(5*a^2 + 16*a*b)*cosh(d*x + c)^3 - 35*(5*a^2 + 16*a*b)*cosh(d*x + c))*sinh(d*x + c)^6 +
15*(5*a^2 + 16*a*b)*cosh(d*x + c)^5 + 3*(429*(5*a^2 + 16*a*b)*cosh(d*x + c)^8 - 924*(5*a^2 + 16*a*b)*cosh(d*x
+ c)^6 + 630*(5*a^2 + 16*a*b)*cosh(d*x + c)^4 - 140*(5*a^2 + 16*a*b)*cosh(d*x + c)^2 + 25*a^2 + 80*a*b)*sinh(d
*x + c)^5 + 5*(143*(5*a^2 + 16*a*b)*cosh(d*x + c)^9 - 396*(5*a^2 + 16*a*b)*cosh(d*x + c)^7 + 378*(5*a^2 + 16*a
*b)*cosh(d*x + c)^5 - 140*(5*a^2 + 16*a*b)*cosh(d*x + c)^3 + 15*(5*a^2 + 16*a*b)*cosh(d*x + c))*sinh(d*x + c)^
4 - 6*(5*a^2 + 16*a*b)*cosh(d*x + c)^3 + 2*(143*(5*a^2 + 16*a*b)*cosh(d*x + c)^10 - 495*(5*a^2 + 16*a*b)*cosh(
d*x + c)^8 + 630*(5*a^2 + 16*a*b)*cosh(d*x + c)^6 - 350*(5*a^2 + 16*a*b)*cosh(d*x + c)^4 + 75*(5*a^2 + 16*a*b)
*cosh(d*x + c)^2 - 15*a^2 - 48*a*b)*sinh(d*x + c)^3 + 6*(13*(5*a^2 + 16*a*b)*cosh(d*x + c)^11 - 55*(5*a^2 + 16
*a*b)*cosh(d*x + c)^9 + 90*(5*a^2 + 16*a*b)*cosh(d*x + c)^7 - 70*(5*a^2 + 16*a*b)*cosh(d*x + c)^5 + 25*(5*a^2
+ 16*a*b)*cosh(d*x + c)^3 - 3*(5*a^2 + 16*a*b)*cosh(d*x + c))*sinh(d*x + c)^2 + (5*a^2 + 16*a*b)*cosh(d*x + c)
+ (13*(5*a^2 + 16*a*b)*cosh(d*x + c)^12 - 66*(5*a^2 + 16*a*b)*cosh(d*x + c)^10 + 135*(5*a^2 + 16*a*b)*cosh(d*
x + c)^8 - 140*(5*a^2 + 16*a*b)*cosh(d*x + c)^6 + 75*(5*a^2 + 16*a*b)*cosh(d*x + c)^4 - 18*(5*a^2 + 16*a*b)*co
sh(d*x + c)^2 + 5*a^2 + 16*a*b)*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) - 3*((5*a^2 + 16*a*b)*co
sh(d*x + c)^13 + 13*(5*a^2 + 16*a*b)*cosh(d*x + c)*sinh(d*x + c)^12 + (5*a^2 + 16*a*b)*sinh(d*x + c)^13 - 6*(5
*a^2 + 16*a*b)*cosh(d*x + c)^11 + 6*(13*(5*a^2 + 16*a*b)*cosh(d*x + c)^2 - 5*a^2 - 16*a*b)*sinh(d*x + c)^11 +
22*(13*(5*a^2 + 16*a*b)*cosh(d*x + c)^3 - 3*(5*a^2 + 16*a*b)*cosh(d*x + c))*sinh(d*x + c)^10 + 15*(5*a^2 + 16*
a*b)*cosh(d*x + c)^9 + 5*(143*(5*a^2 + 16*a*b)*cosh(d*x + c)^4 - 66*(5*a^2 + 16*a*b)*cosh(d*x + c)^2 + 15*a^2
+ 48*a*b)*sinh(d*x + c)^9 + 9*(143*(5*a^2 + 16*a*b)*cosh(d*x + c)^5 - 110*(5*a^2 + 16*a*b)*cosh(d*x + c)^3 + 1
5*(5*a^2 + 16*a*b)*cosh(d*x + c))*sinh(d*x + c)^8 - 20*(5*a^2 + 16*a*b)*cosh(d*x + c)^7 + 4*(429*(5*a^2 + 16*a
*b)*cosh(d*x + c)^6 - 495*(5*a^2 + 16*a*b)*cosh(d*x + c)^4 + 135*(5*a^2 + 16*a*b)*cosh(d*x + c)^2 - 25*a^2 - 8
0*a*b)*sinh(d*x + c)^7 + 4*(429*(5*a^2 + 16*a*b)*cosh(d*x + c)^7 - 693*(5*a^2 + 16*a*b)*cosh(d*x + c)^5 + 315*
(5*a^2 + 16*a*b)*cosh(d*x + c)^3 - 35*(5*a^2 + 16*a*b)*cosh(d*x + c))*sinh(d*x + c)^6 + 15*(5*a^2 + 16*a*b)*co
sh(d*x + c)^5 + 3*(429*(5*a^2 + 16*a*b)*cosh(d*x + c)^8 - 924*(5*a^2 + 16*a*b)*cosh(d*x + c)^6 + 630*(5*a^2 +
16*a*b)*cosh(d*x + c)^4 - 140*(5*a^2 + 16*a*b)*cosh(d*x + c)^2 + 25*a^2 + 80*a*b)*sinh(d*x + c)^5 + 5*(143*(5*
a^2 + 16*a*b)*cosh(d*x + c)^9 - 396*(5*a^2 + 16*a*b)*cosh(d*x + c)^7 + 378*(5*a^2 + 16*a*b)*cosh(d*x + c)^5 -
140*(5*a^2 + 16*a*b)*cosh(d*x + c)^3 + 15*(5*a^2 + 16*a*b)*cosh(d*x + c))*sinh(d*x + c)^4 - 6*(5*a^2 + 16*a*b)
*cosh(d*x + c)^3 + 2*(143*(5*a^2 + 16*a*b)*cosh(d*x + c)^10 - 495*(5*a^2 + 16*a*b)*cosh(d*x + c)^8 + 630*(5*a^
2 + 16*a*b)*cosh(d*x + c)^6 - 350*(5*a^2 + 16*a*b)*cosh(d*x + c)^4 + 75*(5*a^2 + 16*a*b)*cosh(d*x + c)^2 - 15*
a^2 - 48*a*b)*sinh(d*x + c)^3 + 6*(13*(5*a^2 + 16*a*b)*cosh(d*x + c)^11 - 55*(5*a^2 + 16*a*b)*cosh(d*x + c)^9
+ 90*(5*a^2 + 16*a*b)*cosh(d*x + c)^7 - 70*(5*a^2 + 16*a*b)*cosh(d*x + c)^5 + 25*(5*a^2 + 16*a*b)*cosh(d*x + c
)^3 - 3*(5*a^2 + 16*a*b)*cosh(d*x + c))*sinh(d*x + c)^2 + (5*a^2 + 16*a*b)*cosh(d*x + c) + (13*(5*a^2 + 16*a*b
)*cosh(d*x + c)^12 - 66*(5*a^2 + 16*a*b)*cosh(d*x + c)^10 + 135*(5*a^2 + 16*a*b)*cosh(d*x + c)^8 - 140*(5*a^2
+ 16*a*b)*cosh(d*x + c)^6 + 75*(5*a^2 + 16*a*b)*cosh(d*x + c)^4 - 18*(5*a^2 + 16*a*b)*cosh(d*x + c)^2 + 5*a^2
+ 16*a*b)*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 4*(84*b^2*cosh(d*x + c)^13 - 18*(5*a^2 + 16*
a*b + 20*b^2)*cosh(d*x + c)^11 + 5*(85*a^2 + 144*a*b + 108*b^2)*cosh(d*x + c)^9 - 24*(33*a^2 + 16*a*b + 10*b^2
)*cosh(d*x + c)^7 - 18*(33*a^2 + 16*a*b + 10*b^2)*cosh(d*x + c)^5 + 2*(85*a^2 + 144*a*b + 108*b^2)*cosh(d*x +
c)^3 - 3*(5*a^2 + 16*a*b + 20*b^2)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^13 + 13*d*cosh(d*x + c)*sinh
(d*x + c)^12 + d*sinh(d*x + c)^13 - 6*d*cosh(d*x + c)^11 + 6*(13*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^11 + 22*
(13*d*cosh(d*x + c)^3 - 3*d*cosh(d*x + c))*sinh(d*x + c)^10 + 15*d*cosh(d*x + c)^9 + 5*(143*d*cosh(d*x + c)^4
- 66*d*cosh(d*x + c)^2 + 3*d)*sinh(d*x + c)^9 + 9*(143*d*cosh(d*x + c)^5 - 110*d*cosh(d*x + c)^3 + 15*d*cosh(d
*x + c))*sinh(d*x + c)^8 - 20*d*cosh(d*x + c)^7 + 4*(429*d*cosh(d*x + c)^6 - 495*d*cosh(d*x + c)^4 + 135*d*cos
h(d*x + c)^2 - 5*d)*sinh(d*x + c)^7 + 4*(429*d*cosh(d*x + c)^7 - 693*d*cosh(d*x + c)^5 + 315*d*cosh(d*x + c)^3
- 35*d*cosh(d*x + c))*sinh(d*x + c)^6 + 15*d*cosh(d*x + c)^5 + 3*(429*d*cosh(d*x + c)^8 - 924*d*cosh(d*x + c)
^6 + 630*d*cosh(d*x + c)^4 - 140*d*cosh(d*x + c)^2 + 5*d)*sinh(d*x + c)^5 + 5*(143*d*cosh(d*x + c)^9 - 396*d*c
osh(d*x + c)^7 + 378*d*cosh(d*x + c)^5 - 140*d*cosh(d*x + c)^3 + 15*d*cosh(d*x + c))*sinh(d*x + c)^4 - 6*d*cos
h(d*x + c)^3 + 2*(143*d*cosh(d*x + c)^10 - 495*d*cosh(d*x + c)^8 + 630*d*cosh(d*x + c)^6 - 350*d*cosh(d*x + c)
^4 + 75*d*cosh(d*x + c)^2 - 3*d)*sinh(d*x + c)^3 + 6*(13*d*cosh(d*x + c)^11 - 55*d*cosh(d*x + c)^9 + 90*d*cosh
(d*x + c)^7 - 70*d*cosh(d*x + c)^5 + 25*d*cosh(d*x + c)^3 - 3*d*cosh(d*x + c))*sinh(d*x + c)^2 + d*cosh(d*x +
c) + (13*d*cosh(d*x + c)^12 - 66*d*cosh(d*x + c)^10 + 135*d*cosh(d*x + c)^8 - 140*d*cosh(d*x + c)^6 + 75*d*cos
h(d*x + c)^4 - 18*d*cosh(d*x + c)^2 + d)*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)**7*(a+b*sinh(d*x+c)**4)**2,x)

[Out]

Timed out

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Giac [B]  time = 1.27771, size = 338, normalized size = 3.05 \begin{align*} \frac{b^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{2 \, d} + \frac{{\left (5 \, a^{2} + 16 \, a b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right )}{32 \, d} - \frac{{\left (5 \, a^{2} + 16 \, a b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right )}{32 \, d} - \frac{15 \, a^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 48 \, a b{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 160 \, a^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 384 \, a b{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 528 \, a^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 768 \, a b{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{24 \,{\left ({\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4\right )}^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^7*(a+b*sinh(d*x+c)^4)^2,x, algorithm="giac")

[Out]

1/2*b^2*(e^(d*x + c) + e^(-d*x - c))/d + 1/32*(5*a^2 + 16*a*b)*log(e^(d*x + c) + e^(-d*x - c) + 2)/d - 1/32*(5
*a^2 + 16*a*b)*log(e^(d*x + c) + e^(-d*x - c) - 2)/d - 1/24*(15*a^2*(e^(d*x + c) + e^(-d*x - c))^5 + 48*a*b*(e
^(d*x + c) + e^(-d*x - c))^5 - 160*a^2*(e^(d*x + c) + e^(-d*x - c))^3 - 384*a*b*(e^(d*x + c) + e^(-d*x - c))^3
+ 528*a^2*(e^(d*x + c) + e^(-d*x - c)) + 768*a*b*(e^(d*x + c) + e^(-d*x - c)))/(((e^(d*x + c) + e^(-d*x - c))
^2 - 4)^3*d)