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

Optimal. Leaf size=261 \[ \frac{\left (-272 a^2 b+48 a^3+314 a b^2-105 b^3\right ) \sinh (c+d x) \cosh ^3(c+d x)}{640 d}-\frac{\left (-1744 a^2 b+576 a^3+1678 a b^2-525 b^3\right ) \sinh (c+d x) \cosh (c+d x)}{1280 d}+\frac{3}{256} x (4 a-3 b) \left (8 a^2-14 a b+7 b^2\right )+\frac{\sinh ^3(c+d x) \cosh ^7(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^3}{10 d}+\frac{3 (2 a-3 b) \sinh ^3(c+d x) \cosh ^5(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{80 d}-\frac{b \sinh ^3(c+d x) \cosh ^3(c+d x) \left (a (14 a-9 b)-(22 a-21 b) (a-b) \tanh ^2(c+d x)\right )}{160 d} \]

[Out]

(3*(4*a - 3*b)*(8*a^2 - 14*a*b + 7*b^2)*x)/256 - ((576*a^3 - 1744*a^2*b + 1678*a*b^2 - 525*b^3)*Cosh[c + d*x]*
Sinh[c + d*x])/(1280*d) + ((48*a^3 - 272*a^2*b + 314*a*b^2 - 105*b^3)*Cosh[c + d*x]^3*Sinh[c + d*x])/(640*d) +
(3*(2*a - 3*b)*Cosh[c + d*x]^5*Sinh[c + d*x]^3*(a - (a - b)*Tanh[c + d*x]^2)^2)/(80*d) + (Cosh[c + d*x]^7*Sin
h[c + d*x]^3*(a - (a - b)*Tanh[c + d*x]^2)^3)/(10*d) - (b*Cosh[c + d*x]^3*Sinh[c + d*x]^3*(a*(14*a - 9*b) - (2
2*a - 21*b)*(a - b)*Tanh[c + d*x]^2))/(160*d)

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Rubi [A]  time = 0.429778, antiderivative size = 261, 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 = {3187, 467, 577, 455, 385, 206} \[ \frac{\left (-272 a^2 b+48 a^3+314 a b^2-105 b^3\right ) \sinh (c+d x) \cosh ^3(c+d x)}{640 d}-\frac{\left (-1744 a^2 b+576 a^3+1678 a b^2-525 b^3\right ) \sinh (c+d x) \cosh (c+d x)}{1280 d}+\frac{3}{256} x (4 a-3 b) \left (8 a^2-14 a b+7 b^2\right )+\frac{\sinh ^3(c+d x) \cosh ^7(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^3}{10 d}+\frac{3 (2 a-3 b) \sinh ^3(c+d x) \cosh ^5(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{80 d}-\frac{b \sinh ^3(c+d x) \cosh ^3(c+d x) \left (a (14 a-9 b)-(22 a-21 b) (a-b) \tanh ^2(c+d x)\right )}{160 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(4*a - 3*b)*(8*a^2 - 14*a*b + 7*b^2)*x)/256 - ((576*a^3 - 1744*a^2*b + 1678*a*b^2 - 525*b^3)*Cosh[c + d*x]*
Sinh[c + d*x])/(1280*d) + ((48*a^3 - 272*a^2*b + 314*a*b^2 - 105*b^3)*Cosh[c + d*x]^3*Sinh[c + d*x])/(640*d) +
(3*(2*a - 3*b)*Cosh[c + d*x]^5*Sinh[c + d*x]^3*(a - (a - b)*Tanh[c + d*x]^2)^2)/(80*d) + (Cosh[c + d*x]^7*Sin
h[c + d*x]^3*(a - (a - b)*Tanh[c + d*x]^2)^3)/(10*d) - (b*Cosh[c + d*x]^3*Sinh[c + d*x]^3*(a*(14*a - 9*b) - (2
2*a - 21*b)*(a - b)*Tanh[c + d*x]^2))/(160*d)

Rule 3187

Int[sin[(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^(m + 1)/f, Subst[Int[(x^m*(a + (a + b)*ff^2*x^2)^p)/(1 + ff^2*x^2)^(m/2 + p
+ 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rule 467

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

Rule 577

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

Rule 455

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[(a + b*x^2)^(p + 1)*E
xpandToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)]
- (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[
m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 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 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 ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4 \left (a-(a-b) x^2\right )^3}{\left (1-x^2\right )^6} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\cosh ^7(c+d x) \sinh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^3}{10 d}-\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (3 a-9 (a-b) x^2\right ) \left (a+(-a+b) x^2\right )^2}{\left (1-x^2\right )^5} \, dx,x,\tanh (c+d x)\right )}{10 d}\\ &=\frac{3 (2 a-3 b) \cosh ^5(c+d x) \sinh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{80 d}+\frac{\cosh ^7(c+d x) \sinh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^3}{10 d}-\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (3 a (14 a-9 b)-3 (22 a-21 b) (a-b) x^2\right ) \left (a+(-a+b) x^2\right )}{\left (1-x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{80 d}\\ &=\frac{3 (2 a-3 b) \cosh ^5(c+d x) \sinh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{80 d}+\frac{\cosh ^7(c+d x) \sinh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^3}{10 d}-\frac{b \cosh ^3(c+d x) \sinh ^3(c+d x) \left (a (14 a-9 b)-(22 a-21 b) (a-b) \tanh ^2(c+d x)\right )}{160 d}-\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (9 a (14 a-9 b) (2 a-b)-3 (22 a-21 b) (6 a-5 b) (a-b) x^2\right )}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{480 d}\\ &=\frac{\left (48 a^3-272 a^2 b+314 a b^2-105 b^3\right ) \cosh ^3(c+d x) \sinh (c+d x)}{640 d}+\frac{3 (2 a-3 b) \cosh ^5(c+d x) \sinh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{80 d}+\frac{\cosh ^7(c+d x) \sinh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^3}{10 d}-\frac{b \cosh ^3(c+d x) \sinh ^3(c+d x) \left (a (14 a-9 b)-(22 a-21 b) (a-b) \tanh ^2(c+d x)\right )}{160 d}+\frac{\operatorname{Subst}\left (\int \frac{-3 \left (48 a^3-272 a^2 b+314 a b^2-105 b^3\right )-12 (22 a-21 b) (6 a-5 b) (a-b) x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{1920 d}\\ &=-\frac{\left (576 a^3-1744 a^2 b+1678 a b^2-525 b^3\right ) \cosh (c+d x) \sinh (c+d x)}{1280 d}+\frac{\left (48 a^3-272 a^2 b+314 a b^2-105 b^3\right ) \cosh ^3(c+d x) \sinh (c+d x)}{640 d}+\frac{3 (2 a-3 b) \cosh ^5(c+d x) \sinh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{80 d}+\frac{\cosh ^7(c+d x) \sinh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^3}{10 d}-\frac{b \cosh ^3(c+d x) \sinh ^3(c+d x) \left (a (14 a-9 b)-(22 a-21 b) (a-b) \tanh ^2(c+d x)\right )}{160 d}+\frac{\left (3 (4 a-3 b) \left (8 a^2-14 a b+7 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-3 b) \left (8 a^2-14 a b+7 b^2\right ) x-\frac{\left (576 a^3-1744 a^2 b+1678 a b^2-525 b^3\right ) \cosh (c+d x) \sinh (c+d x)}{1280 d}+\frac{\left (48 a^3-272 a^2 b+314 a b^2-105 b^3\right ) \cosh ^3(c+d x) \sinh (c+d x)}{640 d}+\frac{3 (2 a-3 b) \cosh ^5(c+d x) \sinh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{80 d}+\frac{\cosh ^7(c+d x) \sinh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^3}{10 d}-\frac{b \cosh ^3(c+d x) \sinh ^3(c+d x) \left (a (14 a-9 b)-(22 a-21 b) (a-b) \tanh ^2(c+d x)\right )}{160 d}\\ \end{align*}

Mathematica [A]  time = 0.42218, size = 162, normalized size = 0.62 \[ \frac{120 (4 a-3 b) \left (8 a^2-14 a b+7 b^2\right ) (c+d x)+10 b \left (16 a^2-32 a b+15 b^2\right ) \sinh (6 (c+d x))-20 \left (-360 a^2 b+128 a^3+336 a b^2-105 b^3\right ) \sinh (2 (c+d x))+40 \left (-36 a^2 b+8 a^3+42 a b^2-15 b^3\right ) \sinh (4 (c+d x))+5 b^2 (6 a-5 b) \sinh (8 (c+d x))+2 b^3 \sinh (10 (c+d x))}{10240 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(120*(4*a - 3*b)*(8*a^2 - 14*a*b + 7*b^2)*(c + d*x) - 20*(128*a^3 - 360*a^2*b + 336*a*b^2 - 105*b^3)*Sinh[2*(c
+ d*x)] + 40*(8*a^3 - 36*a^2*b + 42*a*b^2 - 15*b^3)*Sinh[4*(c + d*x)] + 10*b*(16*a^2 - 32*a*b + 15*b^2)*Sinh[
6*(c + d*x)] + 5*(6*a - 5*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 = 0.052, size = 222, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({b}^{3} \left ( \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{9}}{10}}-{\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{b}^{2} \left ( \left ( 1/8\, \left ( \sinh \left ( dx+c \right ) \right ) ^{7}-{\frac{7\, \left ( \sinh \left ( dx+c \right ) \right ) ^{5}}{48}}+{\frac{35\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{192}}-{\frac{35\,\sinh \left ( dx+c \right ) }{128}} \right ) \cosh \left ( dx+c \right ) +{\frac{35\,dx}{128}}+{\frac{35\,c}{128}} \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 ( \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{4}}-{\frac{3\,\sinh \left ( dx+c \right ) }{8}} \right ) \cosh \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(sinh(d*x+c)^4*(a+b*sinh(d*x+c)^2)^3,x)

[Out]

1/d*(b^3*((1/10*sinh(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)-63/256*d*x-63/256*c)+3*a*b^2*((1/8*sinh(d*x+c)^7-7/48*sinh(d*x+c)^5+35/192*sinh(d*x+c)^3-35/128*s
inh(d*x+c))*cosh(d*x+c)+35/128*d*x+35/128*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*((1/4*sinh(d*x+c)^3-3/8*sinh(d*x+c))*cosh(d*x+c)+3/8*d*x+3/8*c))

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Maxima [A]  time = 1.0771, size = 547, normalized size = 2.1 \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 (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}{2048} \, a b^{2}{\left (\frac{{\left (32 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 672 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3\right )} e^{\left (8 \, d x + 8 \, c\right )}}{d} - \frac{1680 \,{\left (d x + c\right )}}{d} - \frac{672 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 32 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3 \, e^{\left (-8 \, d x - 8 \, 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)^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*((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) - 1/2048*a*b^2*((32*e^(-2*d*x - 2*c) - 168*e^(-4*d*x - 4*c) + 672*e^
(-6*d*x - 6*c) - 3)*e^(8*d*x + 8*c)/d - 1680*(d*x + c)/d - (672*e^(-2*d*x - 2*c) - 168*e^(-4*d*x - 4*c) + 32*e
^(-6*d*x - 6*c) - 3*e^(-8*d*x - 8*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 [A]  time = 1.80972, size = 1013, normalized size = 3.88 \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} - 5 \, 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} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 15 \,{\left (16 \, a^{2} b - 32 \, a b^{2} + 15 \, 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} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 5 \,{\left (16 \, a^{2} b - 32 \, a b^{2} + 15 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 4 \,{\left (8 \, a^{3} - 36 \, a^{2} b + 42 \, a b^{2} - 15 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 30 \,{\left (32 \, a^{3} - 80 \, a^{2} b + 70 \, a b^{2} - 21 \, b^{3}\right )} d x + 5 \,{\left (b^{3} \cosh \left (d x + c\right )^{9} + 2 \,{\left (6 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} + 3 \,{\left (16 \, a^{2} b - 32 \, a b^{2} + 15 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 8 \,{\left (8 \, a^{3} - 36 \, a^{2} b + 42 \, a b^{2} - 15 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 2 \,{\left (128 \, a^{3} - 360 \, a^{2} b + 336 \, a b^{2} - 105 \, 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(sinh(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 - 5*b^3)*cosh(d*x + c))*sin
h(d*x + c)^7 + (126*b^3*cosh(d*x + c)^5 + 70*(6*a*b^2 - 5*b^3)*cosh(d*x + c)^3 + 15*(16*a^2*b - 32*a*b^2 + 15*
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 - 5*b^3)*cosh(d*x + c)^5 + 5*(16*
a^2*b - 32*a*b^2 + 15*b^3)*cosh(d*x + c)^3 + 4*(8*a^3 - 36*a^2*b + 42*a*b^2 - 15*b^3)*cosh(d*x + c))*sinh(d*x
+ c)^3 + 30*(32*a^3 - 80*a^2*b + 70*a*b^2 - 21*b^3)*d*x + 5*(b^3*cosh(d*x + c)^9 + 2*(6*a*b^2 - 5*b^3)*cosh(d*
x + c)^7 + 3*(16*a^2*b - 32*a*b^2 + 15*b^3)*cosh(d*x + c)^5 + 8*(8*a^3 - 36*a^2*b + 42*a*b^2 - 15*b^3)*cosh(d*
x + c)^3 - 2*(128*a^3 - 360*a^2*b + 336*a*b^2 - 105*b^3)*cosh(d*x + c))*sinh(d*x + c))/d

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Sympy [A]  time = 37.0623, size = 777, normalized size = 2.98 \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)**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 + 5*a**3*sinh(c + d*x)**3*cosh(c + d*x)/(8*d) - 3*a**3*sinh(c + d*x)*cosh(c + d*x)**3/(8*d) + 15*a**2*b*
x*sinh(c + d*x)**6/16 - 45*a**2*b*x*sinh(c + d*x)**4*cosh(c + d*x)**2/16 + 45*a**2*b*x*sinh(c + d*x)**2*cosh(c
+ d*x)**4/16 - 15*a**2*b*x*cosh(c + d*x)**6/16 + 33*a**2*b*sinh(c + d*x)**5*cosh(c + d*x)/(16*d) - 5*a**2*b*s
inh(c + d*x)**3*cosh(c + d*x)**3/(2*d) + 15*a**2*b*sinh(c + d*x)*cosh(c + d*x)**5/(16*d) + 105*a*b**2*x*sinh(c
+ d*x)**8/128 - 105*a*b**2*x*sinh(c + d*x)**6*cosh(c + d*x)**2/32 + 315*a*b**2*x*sinh(c + d*x)**4*cosh(c + d*
x)**4/64 - 105*a*b**2*x*sinh(c + d*x)**2*cosh(c + d*x)**6/32 + 105*a*b**2*x*cosh(c + d*x)**8/128 + 279*a*b**2*
sinh(c + d*x)**7*cosh(c + d*x)/(128*d) - 511*a*b**2*sinh(c + d*x)**5*cosh(c + d*x)**3/(128*d) + 385*a*b**2*sin
h(c + d*x)**3*cosh(c + d*x)**5/(128*d) - 105*a*b**2*sinh(c + d*x)*cosh(c + d*x)**7/(128*d) + 63*b**3*x*sinh(c
+ d*x)**10/256 - 315*b**3*x*sinh(c + d*x)**8*cosh(c + d*x)**2/256 + 315*b**3*x*sinh(c + d*x)**6*cosh(c + d*x)*
*4/128 - 315*b**3*x*sinh(c + d*x)**4*cosh(c + d*x)**6/128 + 315*b**3*x*sinh(c + d*x)**2*cosh(c + d*x)**8/256 -
63*b**3*x*cosh(c + d*x)**10/256 + 193*b**3*sinh(c + d*x)**9*cosh(c + d*x)/(256*d) - 237*b**3*sinh(c + d*x)**7
*cosh(c + d*x)**3/(128*d) + 21*b**3*sinh(c + d*x)**5*cosh(c + d*x)**5/(10*d) - 147*b**3*sinh(c + d*x)**3*cosh(
c + d*x)**7/(128*d) + 63*b**3*sinh(c + d*x)*cosh(c + d*x)**9/(256*d), Ne(d, 0)), (x*(a + b*sinh(c)**2)**3*sinh
(c)**4, True))

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Giac [B]  time = 1.39929, size = 679, normalized size = 2.6 \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 )} - 25 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 160 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 320 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 150 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 320 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 1440 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 1680 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 600 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 2560 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 7200 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 6720 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2100 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 240 \,{\left (32 \, a^{3} - 80 \, a^{2} b + 70 \, a b^{2} - 21 \, b^{3}\right )}{\left (d x + c\right )} -{\left (8768 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} - 21920 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 19180 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} - 5754 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} - 2560 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 7200 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} - 6720 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 2100 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 320 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 1440 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 1680 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 600 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 160 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 320 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 150 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 30 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 25 \, 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(sinh(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) - 25*b^3*e^(8*d*x + 8*c) + 160*a^2*b*e^(6*d*x + 6*
c) - 320*a*b^2*e^(6*d*x + 6*c) + 150*b^3*e^(6*d*x + 6*c) + 320*a^3*e^(4*d*x + 4*c) - 1440*a^2*b*e^(4*d*x + 4*c
) + 1680*a*b^2*e^(4*d*x + 4*c) - 600*b^3*e^(4*d*x + 4*c) - 2560*a^3*e^(2*d*x + 2*c) + 7200*a^2*b*e^(2*d*x + 2*
c) - 6720*a*b^2*e^(2*d*x + 2*c) + 2100*b^3*e^(2*d*x + 2*c) + 240*(32*a^3 - 80*a^2*b + 70*a*b^2 - 21*b^3)*(d*x
+ c) - (8768*a^3*e^(10*d*x + 10*c) - 21920*a^2*b*e^(10*d*x + 10*c) + 19180*a*b^2*e^(10*d*x + 10*c) - 5754*b^3*
e^(10*d*x + 10*c) - 2560*a^3*e^(8*d*x + 8*c) + 7200*a^2*b*e^(8*d*x + 8*c) - 6720*a*b^2*e^(8*d*x + 8*c) + 2100*
b^3*e^(8*d*x + 8*c) + 320*a^3*e^(6*d*x + 6*c) - 1440*a^2*b*e^(6*d*x + 6*c) + 1680*a*b^2*e^(6*d*x + 6*c) - 600*
b^3*e^(6*d*x + 6*c) + 160*a^2*b*e^(4*d*x + 4*c) - 320*a*b^2*e^(4*d*x + 4*c) + 150*b^3*e^(4*d*x + 4*c) + 30*a*b
^2*e^(2*d*x + 2*c) - 25*b^3*e^(2*d*x + 2*c) + 2*b^3)*e^(-10*d*x - 10*c))/d