3.216 \(\int \text{csch}^{13}(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\)

Optimal. Leaf size=220 \[ -\frac{\left (840 a^2 b+231 a^3+1152 a b^2+1024 b^3\right ) \tanh ^{-1}(\cosh (c+d x))}{1024 d}-\frac{a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text{csch}^3(c+d x)}{512 d}+\frac{3 a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text{csch}(c+d x)}{1024 d}-\frac{3 a^2 (11 a+40 b) \coth (c+d x) \text{csch}^7(c+d x)}{320 d}+\frac{7 a^2 (11 a+40 b) \coth (c+d x) \text{csch}^5(c+d x)}{640 d}-\frac{a^3 \coth (c+d x) \text{csch}^{11}(c+d x)}{12 d}+\frac{11 a^3 \coth (c+d x) \text{csch}^9(c+d x)}{120 d} \]

[Out]

-((231*a^3 + 840*a^2*b + 1152*a*b^2 + 1024*b^3)*ArcTanh[Cosh[c + d*x]])/(1024*d) + (3*a*(77*a^2 + 280*a*b + 38
4*b^2)*Coth[c + d*x]*Csch[c + d*x])/(1024*d) - (a*(77*a^2 + 280*a*b + 384*b^2)*Coth[c + d*x]*Csch[c + d*x]^3)/
(512*d) + (7*a^2*(11*a + 40*b)*Coth[c + d*x]*Csch[c + d*x]^5)/(640*d) - (3*a^2*(11*a + 40*b)*Coth[c + d*x]*Csc
h[c + d*x]^7)/(320*d) + (11*a^3*Coth[c + d*x]*Csch[c + d*x]^9)/(120*d) - (a^3*Coth[c + d*x]*Csch[c + d*x]^11)/
(12*d)

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Rubi [A]  time = 0.397589, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3215, 1157, 1814, 385, 206} \[ -\frac{\left (840 a^2 b+231 a^3+1152 a b^2+1024 b^3\right ) \tanh ^{-1}(\cosh (c+d x))}{1024 d}-\frac{a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text{csch}^3(c+d x)}{512 d}+\frac{3 a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text{csch}(c+d x)}{1024 d}-\frac{3 a^2 (11 a+40 b) \coth (c+d x) \text{csch}^7(c+d x)}{320 d}+\frac{7 a^2 (11 a+40 b) \coth (c+d x) \text{csch}^5(c+d x)}{640 d}-\frac{a^3 \coth (c+d x) \text{csch}^{11}(c+d x)}{12 d}+\frac{11 a^3 \coth (c+d x) \text{csch}^9(c+d x)}{120 d} \]

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^13*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

-((231*a^3 + 840*a^2*b + 1152*a*b^2 + 1024*b^3)*ArcTanh[Cosh[c + d*x]])/(1024*d) + (3*a*(77*a^2 + 280*a*b + 38
4*b^2)*Coth[c + d*x]*Csch[c + d*x])/(1024*d) - (a*(77*a^2 + 280*a*b + 384*b^2)*Coth[c + d*x]*Csch[c + d*x]^3)/
(512*d) + (7*a^2*(11*a + 40*b)*Coth[c + d*x]*Csch[c + d*x]^5)/(640*d) - (3*a^2*(11*a + 40*b)*Coth[c + d*x]*Csc
h[c + d*x]^7)/(320*d) + (11*a^3*Coth[c + d*x]*Csch[c + d*x]^9)/(120*d) - (a^3*Coth[c + d*x]*Csch[c + d*x]^11)/
(12*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 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 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}^{13}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (a+b-2 b x^2+b x^4\right )^3}{\left (1-x^2\right )^7} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{a^3 \coth (c+d x) \text{csch}^{11}(c+d x)}{12 d}+\frac{\operatorname{Subst}\left (\int \frac{-11 a^3-36 a^2 b-36 a b^2-12 b^3+12 b \left (3 a^2+9 a b+5 b^2\right ) x^2-12 b^2 (9 a+10 b) x^4+12 b^2 (3 a+10 b) x^6-60 b^3 x^8+12 b^3 x^{10}}{\left (1-x^2\right )^6} \, dx,x,\cosh (c+d x)\right )}{12 d}\\ &=\frac{11 a^3 \coth (c+d x) \text{csch}^9(c+d x)}{120 d}-\frac{a^3 \coth (c+d x) \text{csch}^{11}(c+d x)}{12 d}-\frac{\operatorname{Subst}\left (\int \frac{3 \left (33 a^3+120 a^2 b+120 a b^2+40 b^3\right )-240 b^2 (3 a+2 b) x^2+360 b^2 (a+2 b) x^4-480 b^3 x^6+120 b^3 x^8}{\left (1-x^2\right )^5} \, dx,x,\cosh (c+d x)\right )}{120 d}\\ &=-\frac{3 a^2 (11 a+40 b) \coth (c+d x) \text{csch}^7(c+d x)}{320 d}+\frac{11 a^3 \coth (c+d x) \text{csch}^9(c+d x)}{120 d}-\frac{a^3 \coth (c+d x) \text{csch}^{11}(c+d x)}{12 d}+\frac{\operatorname{Subst}\left (\int \frac{-3 \left (231 a^3+840 a^2 b+960 a b^2+320 b^3\right )+2880 b^2 (a+b) x^2-2880 b^3 x^4+960 b^3 x^6}{\left (1-x^2\right )^4} \, dx,x,\cosh (c+d x)\right )}{960 d}\\ &=\frac{7 a^2 (11 a+40 b) \coth (c+d x) \text{csch}^5(c+d x)}{640 d}-\frac{3 a^2 (11 a+40 b) \coth (c+d x) \text{csch}^7(c+d x)}{320 d}+\frac{11 a^3 \coth (c+d x) \text{csch}^9(c+d x)}{120 d}-\frac{a^3 \coth (c+d x) \text{csch}^{11}(c+d x)}{12 d}-\frac{\operatorname{Subst}\left (\int \frac{45 \left (77 a^3+280 a^2 b+384 a b^2+128 b^3\right )-11520 b^3 x^2+5760 b^3 x^4}{\left (1-x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{5760 d}\\ &=-\frac{a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text{csch}^3(c+d x)}{512 d}+\frac{7 a^2 (11 a+40 b) \coth (c+d x) \text{csch}^5(c+d x)}{640 d}-\frac{3 a^2 (11 a+40 b) \coth (c+d x) \text{csch}^7(c+d x)}{320 d}+\frac{11 a^3 \coth (c+d x) \text{csch}^9(c+d x)}{120 d}-\frac{a^3 \coth (c+d x) \text{csch}^{11}(c+d x)}{12 d}+\frac{\operatorname{Subst}\left (\int \frac{-45 \left (231 a^3+840 a^2 b+1152 a b^2+512 b^3\right )+23040 b^3 x^2}{\left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{23040 d}\\ &=\frac{3 a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text{csch}(c+d x)}{1024 d}-\frac{a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text{csch}^3(c+d x)}{512 d}+\frac{7 a^2 (11 a+40 b) \coth (c+d x) \text{csch}^5(c+d x)}{640 d}-\frac{3 a^2 (11 a+40 b) \coth (c+d x) \text{csch}^7(c+d x)}{320 d}+\frac{11 a^3 \coth (c+d x) \text{csch}^9(c+d x)}{120 d}-\frac{a^3 \coth (c+d x) \text{csch}^{11}(c+d x)}{12 d}-\frac{\left (231 a^3+840 a^2 b+1152 a b^2+1024 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{1024 d}\\ &=-\frac{\left (231 a^3+840 a^2 b+1152 a b^2+1024 b^3\right ) \tanh ^{-1}(\cosh (c+d x))}{1024 d}+\frac{3 a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text{csch}(c+d x)}{1024 d}-\frac{a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text{csch}^3(c+d x)}{512 d}+\frac{7 a^2 (11 a+40 b) \coth (c+d x) \text{csch}^5(c+d x)}{640 d}-\frac{3 a^2 (11 a+40 b) \coth (c+d x) \text{csch}^7(c+d x)}{320 d}+\frac{11 a^3 \coth (c+d x) \text{csch}^9(c+d x)}{120 d}-\frac{a^3 \coth (c+d x) \text{csch}^{11}(c+d x)}{12 d}\\ \end{align*}

Mathematica [A]  time = 2.06945, size = 246, normalized size = 1.12 \[ \frac{15360 \left (840 a^2 b+231 a^3+1152 a b^2+1024 b^3\right ) \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )+2 a \left (750629 a^2+2074200 a b+1422720 b^2\right ) \cosh (3 (c+d x)) \text{csch}^{12}(c+d x)-9 a \left (77099 a^2+280360 a b+246400 b^2\right ) \cosh (5 (c+d x)) \text{csch}^{12}(c+d x)+63 a \left (3421 a^2+12440 a b+14720 b^2\right ) \cosh (7 (c+d x)) \text{csch}^{12}(c+d x)-525 a \left (77 a^2+280 a b+384 b^2\right ) \cosh (9 (c+d x)) \text{csch}^{12}(c+d x)+45 a \left (77 a^2+280 a b+384 b^2\right ) \cosh (11 (c+d x)) \text{csch}^{12}(c+d x)-30 a \left (76555 a^2+75816 a b+45696 b^2\right ) \coth (c+d x) \text{csch}^{11}(c+d x)}{15728640 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^13*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

(-30*a*(76555*a^2 + 75816*a*b + 45696*b^2)*Coth[c + d*x]*Csch[c + d*x]^11 + 2*a*(750629*a^2 + 2074200*a*b + 14
22720*b^2)*Cosh[3*(c + d*x)]*Csch[c + d*x]^12 - 9*a*(77099*a^2 + 280360*a*b + 246400*b^2)*Cosh[5*(c + d*x)]*Cs
ch[c + d*x]^12 + 63*a*(3421*a^2 + 12440*a*b + 14720*b^2)*Cosh[7*(c + d*x)]*Csch[c + d*x]^12 - 525*a*(77*a^2 +
280*a*b + 384*b^2)*Cosh[9*(c + d*x)]*Csch[c + d*x]^12 + 45*a*(77*a^2 + 280*a*b + 384*b^2)*Cosh[11*(c + d*x)]*C
sch[c + d*x]^12 + 15360*(231*a^3 + 840*a^2*b + 1152*a*b^2 + 1024*b^3)*Log[Tanh[(c + d*x)/2]])/(15728640*d)

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Maple [A]  time = 0.085, size = 202, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( \left ( -{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{11}}{12}}+{\frac{11\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{9}}{120}}-{\frac{33\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{7}}{320}}+{\frac{77\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{5}}{640}}-{\frac{77\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{3}}{512}}+{\frac{231\,{\rm csch} \left (dx+c\right )}{1024}} \right ){\rm coth} \left (dx+c\right )-{\frac{231\,{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) }{512}} \right ) +3\,{a}^{2}b \left ( \left ( -1/8\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{7}+{\frac{7\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{5}}{48}}-{\frac{35\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{3}}{192}}+{\frac{35\,{\rm csch} \left (dx+c\right )}{128}} \right ){\rm coth} \left (dx+c\right )-{\frac{35\,{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) }{64}} \right ) +3\,a{b}^{2} \left ( \left ( -1/4\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{3}+3/8\,{\rm csch} \left (dx+c\right ) \right ){\rm coth} \left (dx+c\right )-3/4\,{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \right ) -2\,{b}^{3}{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^13*(a+b*sinh(d*x+c)^4)^3,x)

[Out]

1/d*(a^3*((-1/12*csch(d*x+c)^11+11/120*csch(d*x+c)^9-33/320*csch(d*x+c)^7+77/640*csch(d*x+c)^5-77/512*csch(d*x
+c)^3+231/1024*csch(d*x+c))*coth(d*x+c)-231/512*arctanh(exp(d*x+c)))+3*a^2*b*((-1/8*csch(d*x+c)^7+7/48*csch(d*
x+c)^5-35/192*csch(d*x+c)^3+35/128*csch(d*x+c))*coth(d*x+c)-35/64*arctanh(exp(d*x+c)))+3*a*b^2*((-1/4*csch(d*x
+c)^3+3/8*csch(d*x+c))*coth(d*x+c)-3/4*arctanh(exp(d*x+c)))-2*b^3*arctanh(exp(d*x+c)))

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Maxima [B]  time = 1.15229, size = 972, normalized size = 4.42 \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)^13*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

-1/15360*a^3*(3465*log(e^(-d*x - c) + 1)/d - 3465*log(e^(-d*x - c) - 1)/d + 2*(3465*e^(-d*x - c) - 40425*e^(-3
*d*x - 3*c) + 215523*e^(-5*d*x - 5*c) - 693891*e^(-7*d*x - 7*c) + 1501258*e^(-9*d*x - 9*c) - 2296650*e^(-11*d*
x - 11*c) - 2296650*e^(-13*d*x - 13*c) + 1501258*e^(-15*d*x - 15*c) - 693891*e^(-17*d*x - 17*c) + 215523*e^(-1
9*d*x - 19*c) - 40425*e^(-21*d*x - 21*c) + 3465*e^(-23*d*x - 23*c))/(d*(12*e^(-2*d*x - 2*c) - 66*e^(-4*d*x - 4
*c) + 220*e^(-6*d*x - 6*c) - 495*e^(-8*d*x - 8*c) + 792*e^(-10*d*x - 10*c) - 924*e^(-12*d*x - 12*c) + 792*e^(-
14*d*x - 14*c) - 495*e^(-16*d*x - 16*c) + 220*e^(-18*d*x - 18*c) - 66*e^(-20*d*x - 20*c) + 12*e^(-22*d*x - 22*
c) - e^(-24*d*x - 24*c) - 1))) - 1/128*a^2*b*(105*log(e^(-d*x - c) + 1)/d - 105*log(e^(-d*x - c) - 1)/d + 2*(1
05*e^(-d*x - c) - 805*e^(-3*d*x - 3*c) + 2681*e^(-5*d*x - 5*c) - 5053*e^(-7*d*x - 7*c) - 5053*e^(-9*d*x - 9*c)
+ 2681*e^(-11*d*x - 11*c) - 805*e^(-13*d*x - 13*c) + 105*e^(-15*d*x - 15*c))/(d*(8*e^(-2*d*x - 2*c) - 28*e^(-
4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) - 70*e^(-8*d*x - 8*c) + 56*e^(-10*d*x - 10*c) - 28*e^(-12*d*x - 12*c) + 8*e
^(-14*d*x - 14*c) - e^(-16*d*x - 16*c) - 1))) - 3/8*a*b^2*(3*log(e^(-d*x - c) + 1)/d - 3*log(e^(-d*x - c) - 1)
/d + 2*(3*e^(-d*x - c) - 11*e^(-3*d*x - 3*c) - 11*e^(-5*d*x - 5*c) + 3*e^(-7*d*x - 7*c))/(d*(4*e^(-2*d*x - 2*c
) - 6*e^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) - e^(-8*d*x - 8*c) - 1))) - b^3*(log(e^(-d*x - c) + 1)/d - log(e^(
-d*x - c) - 1)/d)

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Fricas [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)^13*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

Timed out

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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)**13*(a+b*sinh(d*x+c)**4)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.65125, size = 730, normalized size = 3.32 \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)^13*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")

[Out]

-1/2048*(231*a^3 + 840*a^2*b + 1152*a*b^2 + 1024*b^3)*log(e^(d*x + c) + e^(-d*x - c) + 2)/d + 1/2048*(231*a^3
+ 840*a^2*b + 1152*a*b^2 + 1024*b^3)*log(e^(d*x + c) + e^(-d*x - c) - 2)/d + 1/7680*(3465*a^3*(e^(d*x + c) + e
^(-d*x - c))^11 + 12600*a^2*b*(e^(d*x + c) + e^(-d*x - c))^11 + 17280*a*b^2*(e^(d*x + c) + e^(-d*x - c))^11 -
78540*a^3*(e^(d*x + c) + e^(-d*x - c))^9 - 285600*a^2*b*(e^(d*x + c) + e^(-d*x - c))^9 - 391680*a*b^2*(e^(d*x
+ c) + e^(-d*x - c))^9 + 731808*a^3*(e^(d*x + c) + e^(-d*x - c))^7 + 2661120*a^2*b*(e^(d*x + c) + e^(-d*x - c)
)^7 + 3502080*a*b^2*(e^(d*x + c) + e^(-d*x - c))^7 - 3560832*a^3*(e^(d*x + c) + e^(-d*x - c))^5 - 12948480*a^2
*b*(e^(d*x + c) + e^(-d*x - c))^5 - 15482880*a*b^2*(e^(d*x + c) + e^(-d*x - c))^5 + 9391360*a^3*(e^(d*x + c) +
e^(-d*x - c))^3 + 32839680*a^2*b*(e^(d*x + c) + e^(-d*x - c))^3 + 33914880*a*b^2*(e^(d*x + c) + e^(-d*x - c))
^3 - 12180480*a^3*(e^(d*x + c) + e^(-d*x - c)) - 34283520*a^2*b*(e^(d*x + c) + e^(-d*x - c)) - 29491200*a*b^2*
(e^(d*x + c) + e^(-d*x - c)))/(((e^(d*x + c) + e^(-d*x - c))^2 - 4)^6*d)