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

Optimal. Leaf size=238 \[ \frac{b \left (44 a^2-28 a b+5 b^2\right ) \sinh (c+d x) \cosh ^5(c+d x)}{160 d}+\frac{(4 a-b) \left (8 a^2-2 a b+b^2\right ) \sinh (c+d x) \cosh ^3(c+d x)}{128 d}+\frac{3 (4 a-b) \left (8 a^2-2 a b+b^2\right ) \sinh (c+d x) \cosh (c+d x)}{256 d}+\frac{3}{256} x (4 a-b) \left (8 a^2-2 a b+b^2\right )+\frac{b \sinh (c+d x) \cosh ^9(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{10 d}+\frac{b \sinh (c+d x) \cosh ^7(c+d x) \left (a (10 a-b)-5 (a-b) (2 a-b) \tanh ^2(c+d x)\right )}{80 d} \]

[Out]

(3*(4*a - b)*(8*a^2 - 2*a*b + b^2)*x)/256 + (3*(4*a - b)*(8*a^2 - 2*a*b + b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(2
56*d) + ((4*a - b)*(8*a^2 - 2*a*b + b^2)*Cosh[c + d*x]^3*Sinh[c + d*x])/(128*d) + (b*(44*a^2 - 28*a*b + 5*b^2)
*Cosh[c + d*x]^5*Sinh[c + d*x])/(160*d) + (b*Cosh[c + d*x]^9*Sinh[c + d*x]*(a - (a - b)*Tanh[c + d*x]^2)^2)/(1
0*d) + (b*Cosh[c + d*x]^7*Sinh[c + d*x]*(a*(10*a - b) - 5*(a - b)*(2*a - b)*Tanh[c + d*x]^2))/(80*d)

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Rubi [A]  time = 0.331757, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3191, 413, 526, 385, 199, 206} \[ \frac{b \left (44 a^2-28 a b+5 b^2\right ) \sinh (c+d x) \cosh ^5(c+d x)}{160 d}+\frac{(4 a-b) \left (8 a^2-2 a b+b^2\right ) \sinh (c+d x) \cosh ^3(c+d x)}{128 d}+\frac{3 (4 a-b) \left (8 a^2-2 a b+b^2\right ) \sinh (c+d x) \cosh (c+d x)}{256 d}+\frac{3}{256} x (4 a-b) \left (8 a^2-2 a b+b^2\right )+\frac{b \sinh (c+d x) \cosh ^9(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{10 d}+\frac{b \sinh (c+d x) \cosh ^7(c+d x) \left (a (10 a-b)-5 (a-b) (2 a-b) \tanh ^2(c+d x)\right )}{80 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(4*a - b)*(8*a^2 - 2*a*b + b^2)*x)/256 + (3*(4*a - b)*(8*a^2 - 2*a*b + b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(2
56*d) + ((4*a - b)*(8*a^2 - 2*a*b + b^2)*Cosh[c + d*x]^3*Sinh[c + d*x])/(128*d) + (b*(44*a^2 - 28*a*b + 5*b^2)
*Cosh[c + d*x]^5*Sinh[c + d*x])/(160*d) + (b*Cosh[c + d*x]^9*Sinh[c + d*x]*(a - (a - b)*Tanh[c + d*x]^2)^2)/(1
0*d) + (b*Cosh[c + d*x]^7*Sinh[c + d*x]*(a*(10*a - b) - 5*(a - b)*(2*a - b)*Tanh[c + d*x]^2))/(80*d)

Rule 3191

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

Rule 413

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[((a*d - c*b)*x*(a + b*x^n)^
(p + 1)*(c + d*x^n)^(q - 1))/(a*b*n*(p + 1)), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)
^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /;
FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q
, x]

Rule 526

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

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 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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 \cosh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a-(a-b) x^2\right )^3}{\left (1-x^2\right )^6} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{b \cosh ^9(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{10 d}-\frac{\operatorname{Subst}\left (\int \frac{\left (-a (10 a-b)+5 (a-b) (2 a-b) x^2\right ) \left (a+(-a+b) x^2\right )}{\left (1-x^2\right )^5} \, dx,x,\tanh (c+d x)\right )}{10 d}\\ &=\frac{b \cosh ^9(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{10 d}+\frac{b \cosh ^7(c+d x) \sinh (c+d x) \left (a (10 a-b)-5 (a-b) (2 a-b) \tanh ^2(c+d x)\right )}{80 d}-\frac{\operatorname{Subst}\left (\int \frac{-a (8 a-b) (10 a-b)+5 (8 a-3 b) (a-b) (2 a-b) x^2}{\left (1-x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{80 d}\\ &=\frac{b \left (44 a^2-28 a b+5 b^2\right ) \cosh ^5(c+d x) \sinh (c+d x)}{160 d}+\frac{b \cosh ^9(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{10 d}+\frac{b \cosh ^7(c+d x) \sinh (c+d x) \left (a (10 a-b)-5 (a-b) (2 a-b) \tanh ^2(c+d x)\right )}{80 d}+\frac{\left ((4 a-b) \left (8 a^2-2 a b+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{32 d}\\ &=\frac{(4 a-b) \left (8 a^2-2 a b+b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{128 d}+\frac{b \left (44 a^2-28 a b+5 b^2\right ) \cosh ^5(c+d x) \sinh (c+d x)}{160 d}+\frac{b \cosh ^9(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{10 d}+\frac{b \cosh ^7(c+d x) \sinh (c+d x) \left (a (10 a-b)-5 (a-b) (2 a-b) \tanh ^2(c+d x)\right )}{80 d}+\frac{\left (3 (4 a-b) \left (8 a^2-2 a b+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{128 d}\\ &=\frac{3 (4 a-b) \left (8 a^2-2 a b+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{256 d}+\frac{(4 a-b) \left (8 a^2-2 a b+b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{128 d}+\frac{b \left (44 a^2-28 a b+5 b^2\right ) \cosh ^5(c+d x) \sinh (c+d x)}{160 d}+\frac{b \cosh ^9(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{10 d}+\frac{b \cosh ^7(c+d x) \sinh (c+d x) \left (a (10 a-b)-5 (a-b) (2 a-b) \tanh ^2(c+d x)\right )}{80 d}+\frac{\left (3 (4 a-b) \left (8 a^2-2 a b+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{256 d}\\ &=\frac{3}{256} (4 a-b) \left (8 a^2-2 a b+b^2\right ) x+\frac{3 (4 a-b) \left (8 a^2-2 a b+b^2\right ) \cosh (c+d x) \sinh (c+d x)}{256 d}+\frac{(4 a-b) \left (8 a^2-2 a b+b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{128 d}+\frac{b \left (44 a^2-28 a b+5 b^2\right ) \cosh ^5(c+d x) \sinh (c+d x)}{160 d}+\frac{b \cosh ^9(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{10 d}+\frac{b \cosh ^7(c+d x) \sinh (c+d x) \left (a (10 a-b)-5 (a-b) (2 a-b) \tanh ^2(c+d x)\right )}{80 d}\\ \end{align*}

Mathematica [A]  time = 0.520767, size = 144, normalized size = 0.61 \[ \frac{120 (4 a-b) \left (8 a^2-2 a b+b^2\right ) (c+d x)-10 b \left (b^2-16 a^2\right ) \sinh (6 (c+d x))+20 \left (-24 a^2 b+128 a^3+b^3\right ) \sinh (2 (c+d x))+40 \left (12 a^2 b+8 a^3-6 a b^2+b^3\right ) \sinh (4 (c+d x))+5 b^2 (6 a-b) \sinh (8 (c+d x))+2 b^3 \sinh (10 (c+d x))}{10240 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(120*(4*a - b)*(8*a^2 - 2*a*b + b^2)*(c + d*x) + 20*(128*a^3 - 24*a^2*b + b^3)*Sinh[2*(c + d*x)] + 40*(8*a^3 +
12*a^2*b - 6*a*b^2 + b^3)*Sinh[4*(c + d*x)] - 10*b*(-16*a^2 + b^2)*Sinh[6*(c + d*x)] + 5*(6*a - b)*b^2*Sinh[8
*(c + d*x)] + 2*b^3*Sinh[10*(c + d*x)])/(10240*d)

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Maple [A]  time = 0.032, size = 267, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({b}^{3} \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{5} \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}{10}}-{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3} \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}{16}}+{\frac{\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}{32}}-{\frac{\sinh \left ( dx+c \right ) }{32} \left ({\frac{ \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{3\,\cosh \left ( dx+c \right ) }{8}} \right ) }-{\frac{3\,dx}{256}}-{\frac{3\,c}{256}} \right ) +3\,a{b}^{2} \left ( 1/8\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3} \left ( \cosh \left ( dx+c \right ) \right ) ^{5}-1/16\,\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{5}+1/16\, \left ( 1/4\, \left ( \cosh \left ( dx+c \right ) \right ) ^{3}+3/8\,\cosh \left ( dx+c \right ) \right ) \sinh \left ( dx+c \right ) +{\frac{3\,dx}{128}}+{\frac{3\,c}{128}} \right ) +3\,{a}^{2}b \left ( 1/6\,\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{5}-1/6\, \left ( 1/4\, \left ( \cosh \left ( dx+c \right ) \right ) ^{3}+3/8\,\cosh \left ( dx+c \right ) \right ) \sinh \left ( dx+c \right ) -1/16\,dx-c/16 \right ) +{a}^{3} \left ( \left ({\frac{ \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{3\,\cosh \left ( dx+c \right ) }{8}} \right ) \sinh \left ( dx+c \right ) +{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(b^3*(1/10*sinh(d*x+c)^5*cosh(d*x+c)^5-1/16*sinh(d*x+c)^3*cosh(d*x+c)^5+1/32*sinh(d*x+c)*cosh(d*x+c)^5-1/3
2*(1/4*cosh(d*x+c)^3+3/8*cosh(d*x+c))*sinh(d*x+c)-3/256*d*x-3/256*c)+3*a*b^2*(1/8*sinh(d*x+c)^3*cosh(d*x+c)^5-
1/16*sinh(d*x+c)*cosh(d*x+c)^5+1/16*(1/4*cosh(d*x+c)^3+3/8*cosh(d*x+c))*sinh(d*x+c)+3/128*d*x+3/128*c)+3*a^2*b
*(1/6*sinh(d*x+c)*cosh(d*x+c)^5-1/6*(1/4*cosh(d*x+c)^3+3/8*cosh(d*x+c))*sinh(d*x+c)-1/16*d*x-1/16*c)+a^3*((1/4
*cosh(d*x+c)^3+3/8*cosh(d*x+c))*sinh(d*x+c)+3/8*d*x+3/8*c))

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Maxima [A]  time = 1.10372, size = 490, normalized size = 2.06 \begin{align*} \frac{1}{64} \, a^{3}{\left (24 \, x + \frac{e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac{8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac{8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac{e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac{1}{20480} \, b^{3}{\left (\frac{{\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} - 40 \, e^{\left (-6 \, d x - 6 \, c\right )} - 20 \, e^{\left (-8 \, d x - 8 \, c\right )} - 2\right )} e^{\left (10 \, d x + 10 \, c\right )}}{d} + \frac{240 \,{\left (d x + c\right )}}{d} + \frac{20 \, e^{\left (-2 \, d x - 2 \, c\right )} + 40 \, e^{\left (-4 \, d x - 4 \, c\right )} - 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + 2 \, e^{\left (-10 \, d x - 10 \, c\right )}}{d}\right )} - \frac{3}{2048} \, a b^{2}{\left (\frac{{\left (8 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (8 \, d x + 8 \, c\right )}}{d} - \frac{48 \,{\left (d x + c\right )}}{d} - \frac{8 \, e^{\left (-4 \, d x - 4 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )}}{d}\right )} + \frac{1}{128} \, a^{2} b{\left (\frac{{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} - \frac{24 \,{\left (d x + c\right )}}{d} + \frac{3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, 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(cosh(d*x+c)^4*(a+b*sinh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

1/64*a^3*(24*x + e^(4*d*x + 4*c)/d + 8*e^(2*d*x + 2*c)/d - 8*e^(-2*d*x - 2*c)/d - e^(-4*d*x - 4*c)/d) - 1/2048
0*b^3*((5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) - 40*e^(-6*d*x - 6*c) - 20*e^(-8*d*x - 8*c) - 2)*e^(10*d*x +
10*c)/d + 240*(d*x + c)/d + (20*e^(-2*d*x - 2*c) + 40*e^(-4*d*x - 4*c) - 10*e^(-6*d*x - 6*c) - 5*e^(-8*d*x - 8
*c) + 2*e^(-10*d*x - 10*c))/d) - 3/2048*a*b^2*((8*e^(-4*d*x - 4*c) - 1)*e^(8*d*x + 8*c)/d - 48*(d*x + c)/d - (
8*e^(-4*d*x - 4*c) - e^(-8*d*x - 8*c))/d) + 1/128*a^2*b*((3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + 1)*e^(6*d*
x + 6*c)/d - 24*(d*x + c)/d + (3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) - e^(-6*d*x - 6*c))/d)

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Fricas [A]  time = 1.49742, size = 906, normalized size = 3.81 \begin{align*} \frac{5 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{9} + 10 \,{\left (6 \, b^{3} \cosh \left (d x + c\right )^{3} +{\left (6 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{7} +{\left (126 \, b^{3} \cosh \left (d x + c\right )^{5} + 70 \,{\left (6 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )^{3} + 15 \,{\left (16 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 10 \,{\left (6 \, b^{3} \cosh \left (d x + c\right )^{7} + 7 \,{\left (6 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )^{5} + 5 \,{\left (16 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )^{3} + 4 \,{\left (8 \, a^{3} + 12 \, a^{2} b - 6 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 30 \,{\left (32 \, a^{3} - 16 \, a^{2} b + 6 \, a b^{2} - b^{3}\right )} d x + 5 \,{\left (b^{3} \cosh \left (d x + c\right )^{9} + 2 \,{\left (6 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )^{7} + 3 \,{\left (16 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )^{5} + 8 \,{\left (8 \, a^{3} + 12 \, a^{2} b - 6 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{3} + 2 \,{\left (128 \, a^{3} - 24 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2560 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2560*(5*b^3*cosh(d*x + c)*sinh(d*x + c)^9 + 10*(6*b^3*cosh(d*x + c)^3 + (6*a*b^2 - b^3)*cosh(d*x + c))*sinh(
d*x + c)^7 + (126*b^3*cosh(d*x + c)^5 + 70*(6*a*b^2 - b^3)*cosh(d*x + c)^3 + 15*(16*a^2*b - b^3)*cosh(d*x + c)
)*sinh(d*x + c)^5 + 10*(6*b^3*cosh(d*x + c)^7 + 7*(6*a*b^2 - b^3)*cosh(d*x + c)^5 + 5*(16*a^2*b - b^3)*cosh(d*
x + c)^3 + 4*(8*a^3 + 12*a^2*b - 6*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 30*(32*a^3 - 16*a^2*b + 6*a*b
^2 - b^3)*d*x + 5*(b^3*cosh(d*x + c)^9 + 2*(6*a*b^2 - b^3)*cosh(d*x + c)^7 + 3*(16*a^2*b - b^3)*cosh(d*x + c)^
5 + 8*(8*a^3 + 12*a^2*b - 6*a*b^2 + b^3)*cosh(d*x + c)^3 + 2*(128*a^3 - 24*a^2*b + b^3)*cosh(d*x + c))*sinh(d*
x + c))/d

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Sympy [A]  time = 34.7673, size = 774, normalized size = 3.25 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((3*a**3*x*sinh(c + d*x)**4/8 - 3*a**3*x*sinh(c + d*x)**2*cosh(c + d*x)**2/4 + 3*a**3*x*cosh(c + d*x)
**4/8 - 3*a**3*sinh(c + d*x)**3*cosh(c + d*x)/(8*d) + 5*a**3*sinh(c + d*x)*cosh(c + d*x)**3/(8*d) + 3*a**2*b*x
*sinh(c + d*x)**6/16 - 9*a**2*b*x*sinh(c + d*x)**4*cosh(c + d*x)**2/16 + 9*a**2*b*x*sinh(c + d*x)**2*cosh(c +
d*x)**4/16 - 3*a**2*b*x*cosh(c + d*x)**6/16 - 3*a**2*b*sinh(c + d*x)**5*cosh(c + d*x)/(16*d) + a**2*b*sinh(c +
d*x)**3*cosh(c + d*x)**3/(2*d) + 3*a**2*b*sinh(c + d*x)*cosh(c + d*x)**5/(16*d) + 9*a*b**2*x*sinh(c + d*x)**8
/128 - 9*a*b**2*x*sinh(c + d*x)**6*cosh(c + d*x)**2/32 + 27*a*b**2*x*sinh(c + d*x)**4*cosh(c + d*x)**4/64 - 9*
a*b**2*x*sinh(c + d*x)**2*cosh(c + d*x)**6/32 + 9*a*b**2*x*cosh(c + d*x)**8/128 - 9*a*b**2*sinh(c + d*x)**7*co
sh(c + d*x)/(128*d) + 33*a*b**2*sinh(c + d*x)**5*cosh(c + d*x)**3/(128*d) + 33*a*b**2*sinh(c + d*x)**3*cosh(c
+ d*x)**5/(128*d) - 9*a*b**2*sinh(c + d*x)*cosh(c + d*x)**7/(128*d) + 3*b**3*x*sinh(c + d*x)**10/256 - 15*b**3
*x*sinh(c + d*x)**8*cosh(c + d*x)**2/256 + 15*b**3*x*sinh(c + d*x)**6*cosh(c + d*x)**4/128 - 15*b**3*x*sinh(c
+ d*x)**4*cosh(c + d*x)**6/128 + 15*b**3*x*sinh(c + d*x)**2*cosh(c + d*x)**8/256 - 3*b**3*x*cosh(c + d*x)**10/
256 - 3*b**3*sinh(c + d*x)**9*cosh(c + d*x)/(256*d) + 7*b**3*sinh(c + d*x)**7*cosh(c + d*x)**3/(128*d) + b**3*
sinh(c + d*x)**5*cosh(c + d*x)**5/(10*d) - 7*b**3*sinh(c + d*x)**3*cosh(c + d*x)**7/(128*d) + 3*b**3*sinh(c +
d*x)*cosh(c + d*x)**9/(256*d), Ne(d, 0)), (x*(a + b*sinh(c)**2)**3*cosh(c)**4, True))

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Giac [A]  time = 1.26231, size = 598, normalized size = 2.51 \begin{align*} \frac{2 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 30 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 5 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 160 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 10 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 320 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 480 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 240 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 40 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 2560 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 480 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 20 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 240 \,{\left (32 \, a^{3} - 16 \, a^{2} b + 6 \, a b^{2} - b^{3}\right )}{\left (d x + c\right )} -{\left (8768 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} - 4384 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 1644 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} - 274 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 2560 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} - 480 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 20 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 320 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 480 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 240 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 40 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 160 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 10 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 30 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 5 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, b^{3}\right )} e^{\left (-10 \, d x - 10 \, c\right )}}{20480 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20480*(2*b^3*e^(10*d*x + 10*c) + 30*a*b^2*e^(8*d*x + 8*c) - 5*b^3*e^(8*d*x + 8*c) + 160*a^2*b*e^(6*d*x + 6*c
) - 10*b^3*e^(6*d*x + 6*c) + 320*a^3*e^(4*d*x + 4*c) + 480*a^2*b*e^(4*d*x + 4*c) - 240*a*b^2*e^(4*d*x + 4*c) +
40*b^3*e^(4*d*x + 4*c) + 2560*a^3*e^(2*d*x + 2*c) - 480*a^2*b*e^(2*d*x + 2*c) + 20*b^3*e^(2*d*x + 2*c) + 240*
(32*a^3 - 16*a^2*b + 6*a*b^2 - b^3)*(d*x + c) - (8768*a^3*e^(10*d*x + 10*c) - 4384*a^2*b*e^(10*d*x + 10*c) + 1
644*a*b^2*e^(10*d*x + 10*c) - 274*b^3*e^(10*d*x + 10*c) + 2560*a^3*e^(8*d*x + 8*c) - 480*a^2*b*e^(8*d*x + 8*c)
+ 20*b^3*e^(8*d*x + 8*c) + 320*a^3*e^(6*d*x + 6*c) + 480*a^2*b*e^(6*d*x + 6*c) - 240*a*b^2*e^(6*d*x + 6*c) +
40*b^3*e^(6*d*x + 6*c) + 160*a^2*b*e^(4*d*x + 4*c) - 10*b^3*e^(4*d*x + 4*c) + 30*a*b^2*e^(2*d*x + 2*c) - 5*b^3
*e^(2*d*x + 2*c) + 2*b^3)*e^(-10*d*x - 10*c))/d