3.217 \(\int \sinh ^2(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\)

Optimal. Leaf size=255 \[ \frac{b \left (1920 a^2+12312 a b+10579 b^2\right ) \sinh (c+d x) \cosh ^5(c+d x)}{3840 d}-\frac{b \left (4992 a^2+10728 a b+5549 b^2\right ) \sinh (c+d x) \cosh ^3(c+d x)}{3072 d}+\frac{\left (4224 a^2 b+1024 a^3+4632 a b^2+1619 b^3\right ) \sinh (c+d x) \cosh (c+d x)}{2048 d}-\frac{x \left (1920 a^2 b+1024 a^3+1512 a b^2+429 b^3\right )}{2048}+\frac{b^2 (504 a+2593 b) \sinh (c+d x) \cosh ^9(c+d x)}{1680 d}-\frac{b^2 (6888 a+11821 b) \sinh (c+d x) \cosh ^7(c+d x)}{4480 d}+\frac{b^3 \sinh (c+d x) \cosh ^{13}(c+d x)}{14 d}-\frac{85 b^3 \sinh (c+d x) \cosh ^{11}(c+d x)}{168 d} \]

[Out]

-((1024*a^3 + 1920*a^2*b + 1512*a*b^2 + 429*b^3)*x)/2048 + ((1024*a^3 + 4224*a^2*b + 4632*a*b^2 + 1619*b^3)*Co
sh[c + d*x]*Sinh[c + d*x])/(2048*d) - (b*(4992*a^2 + 10728*a*b + 5549*b^2)*Cosh[c + d*x]^3*Sinh[c + d*x])/(307
2*d) + (b*(1920*a^2 + 12312*a*b + 10579*b^2)*Cosh[c + d*x]^5*Sinh[c + d*x])/(3840*d) - (b^2*(6888*a + 11821*b)
*Cosh[c + d*x]^7*Sinh[c + d*x])/(4480*d) + (b^2*(504*a + 2593*b)*Cosh[c + d*x]^9*Sinh[c + d*x])/(1680*d) - (85
*b^3*Cosh[c + d*x]^11*Sinh[c + d*x])/(168*d) + (b^3*Cosh[c + d*x]^13*Sinh[c + d*x])/(14*d)

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Rubi [A]  time = 0.557087, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3217, 1257, 1814, 1157, 385, 206} \[ \frac{b \left (1920 a^2+12312 a b+10579 b^2\right ) \sinh (c+d x) \cosh ^5(c+d x)}{3840 d}-\frac{b \left (4992 a^2+10728 a b+5549 b^2\right ) \sinh (c+d x) \cosh ^3(c+d x)}{3072 d}+\frac{\left (4224 a^2 b+1024 a^3+4632 a b^2+1619 b^3\right ) \sinh (c+d x) \cosh (c+d x)}{2048 d}-\frac{x \left (1920 a^2 b+1024 a^3+1512 a b^2+429 b^3\right )}{2048}+\frac{b^2 (504 a+2593 b) \sinh (c+d x) \cosh ^9(c+d x)}{1680 d}-\frac{b^2 (6888 a+11821 b) \sinh (c+d x) \cosh ^7(c+d x)}{4480 d}+\frac{b^3 \sinh (c+d x) \cosh ^{13}(c+d x)}{14 d}-\frac{85 b^3 \sinh (c+d x) \cosh ^{11}(c+d x)}{168 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1024*a^3 + 1920*a^2*b + 1512*a*b^2 + 429*b^3)*x)/2048 + ((1024*a^3 + 4224*a^2*b + 4632*a*b^2 + 1619*b^3)*Co
sh[c + d*x]*Sinh[c + d*x])/(2048*d) - (b*(4992*a^2 + 10728*a*b + 5549*b^2)*Cosh[c + d*x]^3*Sinh[c + d*x])/(307
2*d) + (b*(1920*a^2 + 12312*a*b + 10579*b^2)*Cosh[c + d*x]^5*Sinh[c + d*x])/(3840*d) - (b^2*(6888*a + 11821*b)
*Cosh[c + d*x]^7*Sinh[c + d*x])/(4480*d) + (b^2*(504*a + 2593*b)*Cosh[c + d*x]^9*Sinh[c + d*x])/(1680*d) - (85
*b^3*Cosh[c + d*x]^11*Sinh[c + d*x])/(168*d) + (b^3*Cosh[c + d*x]^13*Sinh[c + d*x])/(14*d)

Rule 3217

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p)/(1 + ff^2
*x^2)^(m/2 + 2*p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rule 1257

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[((-d)^
(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*(d + e*x^2)^(q + 1))/(2*e^(2*p + m/2)*(q + 1)), x] + Dist[1/(2*e^(2*p +
m/2)*(q + 1)), Int[(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1*(2*e^(2*p + m/2)*(q + 1)*x^m*(a + b*x^2 + c*x^4
)^p - (-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*(d + e*(2*q + 3)*x^2)))/(d + e*x^2)], x], x], x] /; FreeQ[{a, b
, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && IGtQ[m/2, 0]

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 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 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 \sinh ^2(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (a-2 a x^2+(a+b) x^4\right )^3}{\left (1-x^2\right )^8} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d}+\frac{\operatorname{Subst}\left (\int \frac{-b^3+14 \left (a^3-b^3\right ) x^2-14 \left (5 a^3+b^3\right ) x^4+14 \left (10 a^3+3 a^2 b-b^3\right ) x^6-14 \left (10 a^3+9 a^2 b+b^3\right ) x^8+14 (5 a-b) (a+b)^2 x^{10}-14 (a+b)^3 x^{12}}{\left (1-x^2\right )^7} \, dx,x,\tanh (c+d x)\right )}{14 d}\\ &=-\frac{85 b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{168 d}+\frac{b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d}-\frac{\operatorname{Subst}\left (\int \frac{-73 b^3-168 \left (a^3+5 b^3\right ) x^2+672 \left (a^3-b^3\right ) x^4-504 \left (2 a^3+a^2 b+b^3\right ) x^6+336 (2 a-b) (a+b)^2 x^8-168 (a+b)^3 x^{10}}{\left (1-x^2\right )^6} \, dx,x,\tanh (c+d x)\right )}{168 d}\\ &=\frac{b^2 (504 a+2593 b) \cosh ^9(c+d x) \sinh (c+d x)}{1680 d}-\frac{85 b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{168 d}+\frac{b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d}+\frac{\operatorname{Subst}\left (\int \frac{-9 b^2 (56 a+207 b)+1680 \left (a^3-3 a b^2-10 b^3\right ) x^2-5040 \left (a^3+a b^2+2 b^3\right ) x^4+5040 (a-b) (a+b)^2 x^6-1680 (a+b)^3 x^8}{\left (1-x^2\right )^5} \, dx,x,\tanh (c+d x)\right )}{1680 d}\\ &=-\frac{b^2 (6888 a+11821 b) \cosh ^7(c+d x) \sinh (c+d x)}{4480 d}+\frac{b^2 (504 a+2593 b) \cosh ^9(c+d x) \sinh (c+d x)}{1680 d}-\frac{85 b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{168 d}+\frac{b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d}-\frac{\operatorname{Subst}\left (\int \frac{-231 b^2 (72 a+89 b)-13440 \left (a^3+9 a b^2+10 b^3\right ) x^2+26880 (a-2 b) (a+b)^2 x^4-13440 (a+b)^3 x^6}{\left (1-x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{13440 d}\\ &=\frac{b \left (1920 a^2+12312 a b+10579 b^2\right ) \cosh ^5(c+d x) \sinh (c+d x)}{3840 d}-\frac{b^2 (6888 a+11821 b) \cosh ^7(c+d x) \sinh (c+d x)}{4480 d}+\frac{b^2 (504 a+2593 b) \cosh ^9(c+d x) \sinh (c+d x)}{1680 d}-\frac{85 b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{168 d}+\frac{b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d}+\frac{\operatorname{Subst}\left (\int \frac{-105 b \left (384 a^2+1512 a b+941 b^2\right )+80640 (a-5 b) (a+b)^2 x^2-80640 (a+b)^3 x^4}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{80640 d}\\ &=-\frac{b \left (4992 a^2+10728 a b+5549 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{3072 d}+\frac{b \left (1920 a^2+12312 a b+10579 b^2\right ) \cosh ^5(c+d x) \sinh (c+d x)}{3840 d}-\frac{b^2 (6888 a+11821 b) \cosh ^7(c+d x) \sinh (c+d x)}{4480 d}+\frac{b^2 (504 a+2593 b) \cosh ^9(c+d x) \sinh (c+d x)}{1680 d}-\frac{85 b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{168 d}+\frac{b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d}-\frac{\operatorname{Subst}\left (\int \frac{-315 b \left (1152 a^2+1560 a b+595 b^2\right )-322560 (a+b)^3 x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{322560 d}\\ &=\frac{\left (1024 a^3+4224 a^2 b+4632 a b^2+1619 b^3\right ) \cosh (c+d x) \sinh (c+d x)}{2048 d}-\frac{b \left (4992 a^2+10728 a b+5549 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{3072 d}+\frac{b \left (1920 a^2+12312 a b+10579 b^2\right ) \cosh ^5(c+d x) \sinh (c+d x)}{3840 d}-\frac{b^2 (6888 a+11821 b) \cosh ^7(c+d x) \sinh (c+d x)}{4480 d}+\frac{b^2 (504 a+2593 b) \cosh ^9(c+d x) \sinh (c+d x)}{1680 d}-\frac{85 b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{168 d}+\frac{b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d}-\frac{\left (1024 a^3+1920 a^2 b+1512 a b^2+429 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2048 d}\\ &=-\frac{\left (1024 a^3+1920 a^2 b+1512 a b^2+429 b^3\right ) x}{2048}+\frac{\left (1024 a^3+4224 a^2 b+4632 a b^2+1619 b^3\right ) \cosh (c+d x) \sinh (c+d x)}{2048 d}-\frac{b \left (4992 a^2+10728 a b+5549 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{3072 d}+\frac{b \left (1920 a^2+12312 a b+10579 b^2\right ) \cosh ^5(c+d x) \sinh (c+d x)}{3840 d}-\frac{b^2 (6888 a+11821 b) \cosh ^7(c+d x) \sinh (c+d x)}{4480 d}+\frac{b^2 (504 a+2593 b) \cosh ^9(c+d x) \sinh (c+d x)}{1680 d}-\frac{85 b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{168 d}+\frac{b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d}\\ \end{align*}

Mathematica [A]  time = 0.680134, size = 189, normalized size = 0.74 \[ \frac{-840 \left (1920 a^2 b+1024 a^3+1512 a b^2+429 b^3\right ) (c+d x)-105 b \left (2304 a^2+2880 a b+1001 b^2\right ) \sinh (4 (c+d x))+35 b \left (768 a^2+2160 a b+1001 b^2\right ) \sinh (6 (c+d x))+105 \left (11520 a^2 b+4096 a^3+10080 a b^2+3003 b^3\right ) \sinh (2 (c+d x))-105 b^2 (120 a+91 b) \sinh (8 (c+d x))+21 b^2 (48 a+91 b) \sinh (10 (c+d x))-245 b^3 \sinh (12 (c+d x))+15 b^3 \sinh (14 (c+d x))}{1720320 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-840*(1024*a^3 + 1920*a^2*b + 1512*a*b^2 + 429*b^3)*(c + d*x) + 105*(4096*a^3 + 11520*a^2*b + 10080*a*b^2 + 3
003*b^3)*Sinh[2*(c + d*x)] - 105*b*(2304*a^2 + 2880*a*b + 1001*b^2)*Sinh[4*(c + d*x)] + 35*b*(768*a^2 + 2160*a
*b + 1001*b^2)*Sinh[6*(c + d*x)] - 105*b^2*(120*a + 91*b)*Sinh[8*(c + d*x)] + 21*b^2*(48*a + 91*b)*Sinh[10*(c
+ d*x)] - 245*b^3*Sinh[12*(c + d*x)] + 15*b^3*Sinh[14*(c + d*x)])/(1720320*d)

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Maple [A]  time = 0.054, size = 240, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({b}^{3} \left ( \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{13}}{14}}-{\frac{13\, \left ( \sinh \left ( dx+c \right ) \right ) ^{11}}{168}}+{\frac{143\, \left ( \sinh \left ( dx+c \right ) \right ) ^{9}}{1680}}-{\frac{429\, \left ( \sinh \left ( dx+c \right ) \right ) ^{7}}{4480}}+{\frac{143\, \left ( \sinh \left ( dx+c \right ) \right ) ^{5}}{1280}}-{\frac{143\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{1024}}+{\frac{429\,\sinh \left ( dx+c \right ) }{2048}} \right ) \cosh \left ( dx+c \right ) -{\frac{429\,dx}{2048}}-{\frac{429\,c}{2048}} \right ) +3\,a{b}^{2} \left ( \left ( 1/10\, \left ( \sinh \left ( dx+c \right ) \right ) ^{9}-{\frac{9\, \left ( \sinh \left ( dx+c \right ) \right ) ^{7}}{80}}+{\frac{21\, \left ( \sinh \left ( dx+c \right ) \right ) ^{5}}{160}}-{\frac{21\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{128}}+{\frac{63\,\sinh \left ( dx+c \right ) }{256}} \right ) \cosh \left ( dx+c \right ) -{\frac{63\,dx}{256}}-{\frac{63\,c}{256}} \right ) +3\,{a}^{2}b \left ( \left ( 1/6\, \left ( \sinh \left ( dx+c \right ) \right ) ^{5}-{\frac{5\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{24}}+{\frac{5\,\sinh \left ( dx+c \right ) }{16}} \right ) \cosh \left ( dx+c \right ) -{\frac{5\,dx}{16}}-{\frac{5\,c}{16}} \right ) +{a}^{3} \left ({\frac{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{2}}-{\frac{dx}{2}}-{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^2*(a+b*sinh(d*x+c)^4)^3,x)

[Out]

1/d*(b^3*((1/14*sinh(d*x+c)^13-13/168*sinh(d*x+c)^11+143/1680*sinh(d*x+c)^9-429/4480*sinh(d*x+c)^7+143/1280*si
nh(d*x+c)^5-143/1024*sinh(d*x+c)^3+429/2048*sinh(d*x+c))*cosh(d*x+c)-429/2048*d*x-429/2048*c)+3*a*b^2*((1/10*s
inh(d*x+c)^9-9/80*sinh(d*x+c)^7+21/160*sinh(d*x+c)^5-21/128*sinh(d*x+c)^3+63/256*sinh(d*x+c))*cosh(d*x+c)-63/2
56*d*x-63/256*c)+3*a^2*b*((1/6*sinh(d*x+c)^5-5/24*sinh(d*x+c)^3+5/16*sinh(d*x+c))*cosh(d*x+c)-5/16*d*x-5/16*c)
+a^3*(1/2*cosh(d*x+c)*sinh(d*x+c)-1/2*d*x-1/2*c))

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Maxima [A]  time = 1.21768, size = 597, normalized size = 2.34 \begin{align*} -\frac{1}{8} \, a^{3}{\left (4 \, x - \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac{1}{3440640} \, b^{3}{\left (\frac{{\left (245 \, e^{\left (-2 \, d x - 2 \, c\right )} - 1911 \, e^{\left (-4 \, d x - 4 \, c\right )} + 9555 \, e^{\left (-6 \, d x - 6 \, c\right )} - 35035 \, e^{\left (-8 \, d x - 8 \, c\right )} + 105105 \, e^{\left (-10 \, d x - 10 \, c\right )} - 315315 \, e^{\left (-12 \, d x - 12 \, c\right )} - 15\right )} e^{\left (14 \, d x + 14 \, c\right )}}{d} + \frac{720720 \,{\left (d x + c\right )}}{d} + \frac{315315 \, e^{\left (-2 \, d x - 2 \, c\right )} - 105105 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35035 \, e^{\left (-6 \, d x - 6 \, c\right )} - 9555 \, e^{\left (-8 \, d x - 8 \, c\right )} + 1911 \, e^{\left (-10 \, d x - 10 \, c\right )} - 245 \, e^{\left (-12 \, d x - 12 \, c\right )} + 15 \, e^{\left (-14 \, d x - 14 \, c\right )}}{d}\right )} - \frac{3}{20480} \, a b^{2}{\left (\frac{{\left (25 \, e^{\left (-2 \, d x - 2 \, c\right )} - 150 \, e^{\left (-4 \, d x - 4 \, c\right )} + 600 \, e^{\left (-6 \, d x - 6 \, c\right )} - 2100 \, e^{\left (-8 \, d x - 8 \, c\right )} - 2\right )} e^{\left (10 \, d x + 10 \, c\right )}}{d} + \frac{5040 \,{\left (d x + c\right )}}{d} + \frac{2100 \, e^{\left (-2 \, d x - 2 \, c\right )} - 600 \, e^{\left (-4 \, d x - 4 \, c\right )} + 150 \, e^{\left (-6 \, d x - 6 \, c\right )} - 25 \, e^{\left (-8 \, d x - 8 \, c\right )} + 2 \, e^{\left (-10 \, d x - 10 \, c\right )}}{d}\right )} - \frac{1}{128} \, a^{2} b{\left (\frac{{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac{120 \,{\left (d x + c\right )}}{d} + \frac{45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*a^3*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) - 1/3440640*b^3*((245*e^(-2*d*x - 2*c) - 1911*e^(-4*d*
x - 4*c) + 9555*e^(-6*d*x - 6*c) - 35035*e^(-8*d*x - 8*c) + 105105*e^(-10*d*x - 10*c) - 315315*e^(-12*d*x - 12
*c) - 15)*e^(14*d*x + 14*c)/d + 720720*(d*x + c)/d + (315315*e^(-2*d*x - 2*c) - 105105*e^(-4*d*x - 4*c) + 3503
5*e^(-6*d*x - 6*c) - 9555*e^(-8*d*x - 8*c) + 1911*e^(-10*d*x - 10*c) - 245*e^(-12*d*x - 12*c) + 15*e^(-14*d*x
- 14*c))/d) - 3/20480*a*b^2*((25*e^(-2*d*x - 2*c) - 150*e^(-4*d*x - 4*c) + 600*e^(-6*d*x - 6*c) - 2100*e^(-8*d
*x - 8*c) - 2)*e^(10*d*x + 10*c)/d + 5040*(d*x + c)/d + (2100*e^(-2*d*x - 2*c) - 600*e^(-4*d*x - 4*c) + 150*e^
(-6*d*x - 6*c) - 25*e^(-8*d*x - 8*c) + 2*e^(-10*d*x - 10*c))/d) - 1/128*a^2*b*((9*e^(-2*d*x - 2*c) - 45*e^(-4*
d*x - 4*c) - 1)*e^(6*d*x + 6*c)/d + 120*(d*x + c)/d + (45*e^(-2*d*x - 2*c) - 9*e^(-4*d*x - 4*c) + e^(-6*d*x -
6*c))/d)

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Fricas [B]  time = 1.40475, size = 1688, normalized size = 6.62 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/860160*(105*b^3*cosh(d*x + c)*sinh(d*x + c)^13 + 210*(13*b^3*cosh(d*x + c)^3 - 7*b^3*cosh(d*x + c))*sinh(d*x
+ c)^11 + 35*(429*b^3*cosh(d*x + c)^5 - 770*b^3*cosh(d*x + c)^3 + 3*(48*a*b^2 + 91*b^3)*cosh(d*x + c))*sinh(d
*x + c)^9 + 60*(429*b^3*cosh(d*x + c)^7 - 1617*b^3*cosh(d*x + c)^5 + 21*(48*a*b^2 + 91*b^3)*cosh(d*x + c)^3 -
7*(120*a*b^2 + 91*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 + 21*(715*b^3*cosh(d*x + c)^9 - 4620*b^3*cosh(d*x + c)^7
+ 126*(48*a*b^2 + 91*b^3)*cosh(d*x + c)^5 - 140*(120*a*b^2 + 91*b^3)*cosh(d*x + c)^3 + 5*(768*a^2*b + 2160*a*
b^2 + 1001*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 70*(39*b^3*cosh(d*x + c)^11 - 385*b^3*cosh(d*x + c)^9 + 18*(4
8*a*b^2 + 91*b^3)*cosh(d*x + c)^7 - 42*(120*a*b^2 + 91*b^3)*cosh(d*x + c)^5 + 5*(768*a^2*b + 2160*a*b^2 + 1001
*b^3)*cosh(d*x + c)^3 - 3*(2304*a^2*b + 2880*a*b^2 + 1001*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - 420*(1024*a^3
+ 1920*a^2*b + 1512*a*b^2 + 429*b^3)*d*x + 105*(b^3*cosh(d*x + c)^13 - 14*b^3*cosh(d*x + c)^11 + (48*a*b^2 + 9
1*b^3)*cosh(d*x + c)^9 - 4*(120*a*b^2 + 91*b^3)*cosh(d*x + c)^7 + (768*a^2*b + 2160*a*b^2 + 1001*b^3)*cosh(d*x
+ c)^5 - 2*(2304*a^2*b + 2880*a*b^2 + 1001*b^3)*cosh(d*x + c)^3 + (4096*a^3 + 11520*a^2*b + 10080*a*b^2 + 300
3*b^3)*cosh(d*x + c))*sinh(d*x + c))/d

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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(sinh(d*x+c)**2*(a+b*sinh(d*x+c)**4)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.70269, size = 756, normalized size = 2.96 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3440640*(15*b^3*e^(14*d*x + 14*c) - 245*b^3*e^(12*d*x + 12*c) + 1008*a*b^2*e^(10*d*x + 10*c) + 1911*b^3*e^(1
0*d*x + 10*c) - 12600*a*b^2*e^(8*d*x + 8*c) - 9555*b^3*e^(8*d*x + 8*c) + 26880*a^2*b*e^(6*d*x + 6*c) + 75600*a
*b^2*e^(6*d*x + 6*c) + 35035*b^3*e^(6*d*x + 6*c) - 241920*a^2*b*e^(4*d*x + 4*c) - 302400*a*b^2*e^(4*d*x + 4*c)
- 105105*b^3*e^(4*d*x + 4*c) + 430080*a^3*e^(2*d*x + 2*c) + 1209600*a^2*b*e^(2*d*x + 2*c) + 1058400*a*b^2*e^(
2*d*x + 2*c) + 315315*b^3*e^(2*d*x + 2*c) - 1680*(1024*a^3 + 1920*a^2*b + 1512*a*b^2 + 429*b^3)*(d*x + c) + (2
230272*a^3*e^(14*d*x + 14*c) + 4181760*a^2*b*e^(14*d*x + 14*c) + 3293136*a*b^2*e^(14*d*x + 14*c) + 934362*b^3*
e^(14*d*x + 14*c) - 430080*a^3*e^(12*d*x + 12*c) - 1209600*a^2*b*e^(12*d*x + 12*c) - 1058400*a*b^2*e^(12*d*x +
12*c) - 315315*b^3*e^(12*d*x + 12*c) + 241920*a^2*b*e^(10*d*x + 10*c) + 302400*a*b^2*e^(10*d*x + 10*c) + 1051
05*b^3*e^(10*d*x + 10*c) - 26880*a^2*b*e^(8*d*x + 8*c) - 75600*a*b^2*e^(8*d*x + 8*c) - 35035*b^3*e^(8*d*x + 8*
c) + 12600*a*b^2*e^(6*d*x + 6*c) + 9555*b^3*e^(6*d*x + 6*c) - 1008*a*b^2*e^(4*d*x + 4*c) - 1911*b^3*e^(4*d*x +
4*c) + 245*b^3*e^(2*d*x + 2*c) - 15*b^3)*e^(-14*d*x - 14*c))/d