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

Optimal. Leaf size=211 \[ \frac{b \left (1152 a^2+3912 a b+2279 b^2\right ) \sinh (c+d x) \cosh ^3(c+d x)}{1536 d}-\frac{b \left (1920 a^2+2232 a b+793 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{1024 d}+\frac{x \left (1152 a^2 b+1024 a^3+840 a b^2+231 b^3\right )}{1024}+\frac{3 b^2 (40 a+139 b) \sinh (c+d x) \cosh ^7(c+d x)}{320 d}-\frac{b^2 (3000 a+3481 b) \sinh (c+d x) \cosh ^5(c+d x)}{1920 d}+\frac{b^3 \sinh (c+d x) \cosh ^{11}(c+d x)}{12 d}-\frac{61 b^3 \sinh (c+d x) \cosh ^9(c+d x)}{120 d} \]

[Out]

((1024*a^3 + 1152*a^2*b + 840*a*b^2 + 231*b^3)*x)/1024 - (b*(1920*a^2 + 2232*a*b + 793*b^2)*Cosh[c + d*x]*Sinh
[c + d*x])/(1024*d) + (b*(1152*a^2 + 3912*a*b + 2279*b^2)*Cosh[c + d*x]^3*Sinh[c + d*x])/(1536*d) - (b^2*(3000
*a + 3481*b)*Cosh[c + d*x]^5*Sinh[c + d*x])/(1920*d) + (3*b^2*(40*a + 139*b)*Cosh[c + d*x]^7*Sinh[c + d*x])/(3
20*d) - (61*b^3*Cosh[c + d*x]^9*Sinh[c + d*x])/(120*d) + (b^3*Cosh[c + d*x]^11*Sinh[c + d*x])/(12*d)

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Rubi [A]  time = 0.386844, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {3209, 1157, 1814, 385, 206} \[ \frac{b \left (1152 a^2+3912 a b+2279 b^2\right ) \sinh (c+d x) \cosh ^3(c+d x)}{1536 d}-\frac{b \left (1920 a^2+2232 a b+793 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{1024 d}+\frac{x \left (1152 a^2 b+1024 a^3+840 a b^2+231 b^3\right )}{1024}+\frac{3 b^2 (40 a+139 b) \sinh (c+d x) \cosh ^7(c+d x)}{320 d}-\frac{b^2 (3000 a+3481 b) \sinh (c+d x) \cosh ^5(c+d x)}{1920 d}+\frac{b^3 \sinh (c+d x) \cosh ^{11}(c+d x)}{12 d}-\frac{61 b^3 \sinh (c+d x) \cosh ^9(c+d x)}{120 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

((1024*a^3 + 1152*a^2*b + 840*a*b^2 + 231*b^3)*x)/1024 - (b*(1920*a^2 + 2232*a*b + 793*b^2)*Cosh[c + d*x]*Sinh
[c + d*x])/(1024*d) + (b*(1152*a^2 + 3912*a*b + 2279*b^2)*Cosh[c + d*x]^3*Sinh[c + d*x])/(1536*d) - (b^2*(3000
*a + 3481*b)*Cosh[c + d*x]^5*Sinh[c + d*x])/(1920*d) + (3*b^2*(40*a + 139*b)*Cosh[c + d*x]^7*Sinh[c + d*x])/(3
20*d) - (61*b^3*Cosh[c + d*x]^9*Sinh[c + d*x])/(120*d) + (b^3*Cosh[c + d*x]^11*Sinh[c + d*x])/(12*d)

Rule 3209

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

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 \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a-2 a x^2+(a+b) x^4\right )^3}{\left (1-x^2\right )^7} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{12 d}-\frac{\operatorname{Subst}\left (\int \frac{-12 a^3+b^3+12 \left (5 a^3+b^3\right ) x^2-12 \left (10 a^3+3 a^2 b-b^3\right ) x^4+12 \left (10 a^3+9 a^2 b+b^3\right ) x^6-12 (5 a-b) (a+b)^2 x^8+12 (a+b)^3 x^{10}}{\left (1-x^2\right )^6} \, dx,x,\tanh (c+d x)\right )}{12 d}\\ &=-\frac{61 b^3 \cosh ^9(c+d x) \sinh (c+d x)}{120 d}+\frac{b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{12 d}+\frac{\operatorname{Subst}\left (\int \frac{3 \left (40 a^3+17 b^3\right )-480 \left (a^3-b^3\right ) x^2+360 \left (2 a^3+a^2 b+b^3\right ) x^4-240 (2 a-b) (a+b)^2 x^6+120 (a+b)^3 x^8}{\left (1-x^2\right )^5} \, dx,x,\tanh (c+d x)\right )}{120 d}\\ &=\frac{3 b^2 (40 a+139 b) \cosh ^7(c+d x) \sinh (c+d x)}{320 d}-\frac{61 b^3 \cosh ^9(c+d x) \sinh (c+d x)}{120 d}+\frac{b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{12 d}-\frac{\operatorname{Subst}\left (\int \frac{-3 \left (320 a^3-120 a b^2-281 b^3\right )+2880 \left (a^3+a b^2+2 b^3\right ) x^2-2880 (a-b) (a+b)^2 x^4+960 (a+b)^3 x^6}{\left (1-x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{960 d}\\ &=-\frac{b^2 (3000 a+3481 b) \cosh ^5(c+d x) \sinh (c+d x)}{1920 d}+\frac{3 b^2 (40 a+139 b) \cosh ^7(c+d x) \sinh (c+d x)}{320 d}-\frac{61 b^3 \cosh ^9(c+d x) \sinh (c+d x)}{120 d}+\frac{b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{12 d}+\frac{\operatorname{Subst}\left (\int \frac{15 \left (384 a^3+456 a b^2+359 b^3\right )-11520 (a-2 b) (a+b)^2 x^2+5760 (a+b)^3 x^4}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{5760 d}\\ &=\frac{b \left (1152 a^2+3912 a b+2279 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{1536 d}-\frac{b^2 (3000 a+3481 b) \cosh ^5(c+d x) \sinh (c+d x)}{1920 d}+\frac{3 b^2 (40 a+139 b) \cosh ^7(c+d x) \sinh (c+d x)}{320 d}-\frac{61 b^3 \cosh ^9(c+d x) \sinh (c+d x)}{120 d}+\frac{b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{12 d}-\frac{\operatorname{Subst}\left (\int \frac{-45 \left (512 a^3-384 a^2 b-696 a b^2-281 b^3\right )+23040 (a+b)^3 x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{23040 d}\\ &=-\frac{b \left (1920 a^2+2232 a b+793 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{1024 d}+\frac{b \left (1152 a^2+3912 a b+2279 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{1536 d}-\frac{b^2 (3000 a+3481 b) \cosh ^5(c+d x) \sinh (c+d x)}{1920 d}+\frac{3 b^2 (40 a+139 b) \cosh ^7(c+d x) \sinh (c+d x)}{320 d}-\frac{61 b^3 \cosh ^9(c+d x) \sinh (c+d x)}{120 d}+\frac{b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{12 d}+\frac{\left (1024 a^3+1152 a^2 b+840 a b^2+231 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{1024 d}\\ &=\frac{\left (1024 a^3+1152 a^2 b+840 a b^2+231 b^3\right ) x}{1024}-\frac{b \left (1920 a^2+2232 a b+793 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{1024 d}+\frac{b \left (1152 a^2+3912 a b+2279 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{1536 d}-\frac{b^2 (3000 a+3481 b) \cosh ^5(c+d x) \sinh (c+d x)}{1920 d}+\frac{3 b^2 (40 a+139 b) \cosh ^7(c+d x) \sinh (c+d x)}{320 d}-\frac{61 b^3 \cosh ^9(c+d x) \sinh (c+d x)}{120 d}+\frac{b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{12 d}\\ \end{align*}

Mathematica [A]  time = 0.389919, size = 156, normalized size = 0.74 \[ \frac{120 \left (1152 a^2 b+1024 a^3+840 a b^2+231 b^3\right ) (c+d x)-720 b \left (128 a^2+112 a b+33 b^2\right ) \sinh (2 (c+d x))+45 b \left (256 a^2+448 a b+165 b^2\right ) \sinh (4 (c+d x))-40 b^2 (96 a+55 b) \sinh (6 (c+d x))+45 b^2 (8 a+11 b) \sinh (8 (c+d x))-72 b^3 \sinh (10 (c+d x))+5 b^3 \sinh (12 (c+d x))}{122880 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(120*(1024*a^3 + 1152*a^2*b + 840*a*b^2 + 231*b^3)*(c + d*x) - 720*b*(128*a^2 + 112*a*b + 33*b^2)*Sinh[2*(c +
d*x)] + 45*b*(256*a^2 + 448*a*b + 165*b^2)*Sinh[4*(c + d*x)] - 40*b^2*(96*a + 55*b)*Sinh[6*(c + d*x)] + 45*b^2
*(8*a + 11*b)*Sinh[8*(c + d*x)] - 72*b^3*Sinh[10*(c + d*x)] + 5*b^3*Sinh[12*(c + d*x)])/(122880*d)

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Maple [A]  time = 0.056, size = 193, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({b}^{3} \left ( \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{11}}{12}}-{\frac{11\, \left ( \sinh \left ( dx+c \right ) \right ) ^{9}}{120}}+{\frac{33\, \left ( \sinh \left ( dx+c \right ) \right ) ^{7}}{320}}-{\frac{77\, \left ( \sinh \left ( dx+c \right ) \right ) ^{5}}{640}}+{\frac{77\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{512}}-{\frac{231\,\sinh \left ( dx+c \right ) }{1024}} \right ) \cosh \left ( dx+c \right ) +{\frac{231\,dx}{1024}}+{\frac{231\,c}{1024}} \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 ( 1/4\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}-3/8\,\sinh \left ( dx+c \right ) \right ) \cosh \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +{a}^{3} \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(b^3*((1/12*sinh(d*x+c)^11-11/120*sinh(d*x+c)^9+33/320*sinh(d*x+c)^7-77/640*sinh(d*x+c)^5+77/512*sinh(d*x+
c)^3-231/1024*sinh(d*x+c))*cosh(d*x+c)+231/1024*d*x+231/1024*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*sinh(d*x+c))*cosh(d*x+c)+35/128*d*x+35/128*c)+3*a^2*b*((1/4*sinh(d*x+c)^3-3/8*sin
h(d*x+c))*cosh(d*x+c)+3/8*d*x+3/8*c)+a^3*(d*x+c))

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Maxima [A]  time = 1.1122, size = 464, normalized size = 2.2 \begin{align*} \frac{3}{64} \, a^{2} b{\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 )} + a^{3} x - \frac{1}{245760} \, b^{3}{\left (\frac{{\left (72 \, e^{\left (-2 \, d x - 2 \, c\right )} - 495 \, e^{\left (-4 \, d x - 4 \, c\right )} + 2200 \, e^{\left (-6 \, d x - 6 \, c\right )} - 7425 \, e^{\left (-8 \, d x - 8 \, c\right )} + 23760 \, e^{\left (-10 \, d x - 10 \, c\right )} - 5\right )} e^{\left (12 \, d x + 12 \, c\right )}}{d} - \frac{55440 \,{\left (d x + c\right )}}{d} - \frac{23760 \, e^{\left (-2 \, d x - 2 \, c\right )} - 7425 \, e^{\left (-4 \, d x - 4 \, c\right )} + 2200 \, e^{\left (-6 \, d x - 6 \, c\right )} - 495 \, e^{\left (-8 \, d x - 8 \, c\right )} + 72 \, e^{\left (-10 \, d x - 10 \, c\right )} - 5 \, e^{\left (-12 \, d x - 12 \, 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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/64*a^2*b*(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) + a^3*
x - 1/245760*b^3*((72*e^(-2*d*x - 2*c) - 495*e^(-4*d*x - 4*c) + 2200*e^(-6*d*x - 6*c) - 7425*e^(-8*d*x - 8*c)
+ 23760*e^(-10*d*x - 10*c) - 5)*e^(12*d*x + 12*c)/d - 55440*(d*x + c)/d - (23760*e^(-2*d*x - 2*c) - 7425*e^(-4
*d*x - 4*c) + 2200*e^(-6*d*x - 6*c) - 495*e^(-8*d*x - 8*c) + 72*e^(-10*d*x - 10*c) - 5*e^(-12*d*x - 12*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 - 16
80*(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)

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Fricas [B]  time = 1.48421, size = 1200, normalized size = 5.69 \begin{align*} \frac{15 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{11} + 5 \,{\left (55 \, b^{3} \cosh \left (d x + c\right )^{3} - 36 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{9} + 90 \,{\left (11 \, b^{3} \cosh \left (d x + c\right )^{5} - 24 \, b^{3} \cosh \left (d x + c\right )^{3} +{\left (8 \, a b^{2} + 11 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{7} + 6 \,{\left (165 \, b^{3} \cosh \left (d x + c\right )^{7} - 756 \, b^{3} \cosh \left (d x + c\right )^{5} + 105 \,{\left (8 \, a b^{2} + 11 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 10 \,{\left (96 \, a b^{2} + 55 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 5 \,{\left (55 \, b^{3} \cosh \left (d x + c\right )^{9} - 432 \, b^{3} \cosh \left (d x + c\right )^{7} + 126 \,{\left (8 \, a b^{2} + 11 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} - 40 \,{\left (96 \, a b^{2} + 55 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 9 \,{\left (256 \, a^{2} b + 448 \, a b^{2} + 165 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 30 \,{\left (1024 \, a^{3} + 1152 \, a^{2} b + 840 \, a b^{2} + 231 \, b^{3}\right )} d x + 15 \,{\left (b^{3} \cosh \left (d x + c\right )^{11} - 12 \, b^{3} \cosh \left (d x + c\right )^{9} + 6 \,{\left (8 \, a b^{2} + 11 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} - 4 \,{\left (96 \, a b^{2} + 55 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 3 \,{\left (256 \, a^{2} b + 448 \, a b^{2} + 165 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 24 \,{\left (128 \, a^{2} b + 112 \, a b^{2} + 33 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{30720 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30720*(15*b^3*cosh(d*x + c)*sinh(d*x + c)^11 + 5*(55*b^3*cosh(d*x + c)^3 - 36*b^3*cosh(d*x + c))*sinh(d*x +
c)^9 + 90*(11*b^3*cosh(d*x + c)^5 - 24*b^3*cosh(d*x + c)^3 + (8*a*b^2 + 11*b^3)*cosh(d*x + c))*sinh(d*x + c)^7
+ 6*(165*b^3*cosh(d*x + c)^7 - 756*b^3*cosh(d*x + c)^5 + 105*(8*a*b^2 + 11*b^3)*cosh(d*x + c)^3 - 10*(96*a*b^
2 + 55*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 5*(55*b^3*cosh(d*x + c)^9 - 432*b^3*cosh(d*x + c)^7 + 126*(8*a*b^
2 + 11*b^3)*cosh(d*x + c)^5 - 40*(96*a*b^2 + 55*b^3)*cosh(d*x + c)^3 + 9*(256*a^2*b + 448*a*b^2 + 165*b^3)*cos
h(d*x + c))*sinh(d*x + c)^3 + 30*(1024*a^3 + 1152*a^2*b + 840*a*b^2 + 231*b^3)*d*x + 15*(b^3*cosh(d*x + c)^11
- 12*b^3*cosh(d*x + c)^9 + 6*(8*a*b^2 + 11*b^3)*cosh(d*x + c)^7 - 4*(96*a*b^2 + 55*b^3)*cosh(d*x + c)^5 + 3*(2
56*a^2*b + 448*a*b^2 + 165*b^3)*cosh(d*x + c)^3 - 24*(128*a^2*b + 112*a*b^2 + 33*b^3)*cosh(d*x + c))*sinh(d*x
+ c))/d

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Sympy [A]  time = 103.123, size = 666, normalized size = 3.16 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**3*x + 9*a**2*b*x*sinh(c + d*x)**4/8 - 9*a**2*b*x*sinh(c + d*x)**2*cosh(c + d*x)**2/4 + 9*a**2*b*
x*cosh(c + d*x)**4/8 + 15*a**2*b*sinh(c + d*x)**3*cosh(c + d*x)/(8*d) - 9*a**2*b*sinh(c + d*x)*cosh(c + d*x)**
3/(8*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*sinh(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) + 231*b**3*x*sinh(c + d*x)**12/1024 - 693*b**3*x*sinh(c + d*x)**10*cosh(c + d*x)**2/512 + 3465*b**3*
x*sinh(c + d*x)**8*cosh(c + d*x)**4/1024 - 1155*b**3*x*sinh(c + d*x)**6*cosh(c + d*x)**6/256 + 3465*b**3*x*sin
h(c + d*x)**4*cosh(c + d*x)**8/1024 - 693*b**3*x*sinh(c + d*x)**2*cosh(c + d*x)**10/512 + 231*b**3*x*cosh(c +
d*x)**12/1024 + 793*b**3*sinh(c + d*x)**11*cosh(c + d*x)/(1024*d) - 7337*b**3*sinh(c + d*x)**9*cosh(c + d*x)**
3/(3072*d) + 9273*b**3*sinh(c + d*x)**7*cosh(c + d*x)**5/(2560*d) - 7623*b**3*sinh(c + d*x)**5*cosh(c + d*x)**
7/(2560*d) + 1309*b**3*sinh(c + d*x)**3*cosh(c + d*x)**9/(1024*d) - 231*b**3*sinh(c + d*x)*cosh(c + d*x)**11/(
1024*d), Ne(d, 0)), (x*(a + b*sinh(c)**4)**3, True))

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Giac [B]  time = 1.14116, size = 601, normalized size = 2.85 \begin{align*} \frac{5 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} - 72 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 360 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 495 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 3840 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 2200 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 11520 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 20160 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 7425 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 92160 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 80640 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 23760 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 240 \,{\left (1024 \, a^{3} + 1152 \, a^{2} b + 840 \, a b^{2} + 231 \, b^{3}\right )}{\left (d x + c\right )} -{\left (301056 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 338688 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 246960 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 67914 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} - 92160 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} - 80640 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} - 23760 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 11520 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 20160 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 7425 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 3840 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 2200 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 360 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 495 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 72 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 5 \, b^{3}\right )} e^{\left (-12 \, d x - 12 \, c\right )}}{245760 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/245760*(5*b^3*e^(12*d*x + 12*c) - 72*b^3*e^(10*d*x + 10*c) + 360*a*b^2*e^(8*d*x + 8*c) + 495*b^3*e^(8*d*x +
8*c) - 3840*a*b^2*e^(6*d*x + 6*c) - 2200*b^3*e^(6*d*x + 6*c) + 11520*a^2*b*e^(4*d*x + 4*c) + 20160*a*b^2*e^(4*
d*x + 4*c) + 7425*b^3*e^(4*d*x + 4*c) - 92160*a^2*b*e^(2*d*x + 2*c) - 80640*a*b^2*e^(2*d*x + 2*c) - 23760*b^3*
e^(2*d*x + 2*c) + 240*(1024*a^3 + 1152*a^2*b + 840*a*b^2 + 231*b^3)*(d*x + c) - (301056*a^3*e^(12*d*x + 12*c)
+ 338688*a^2*b*e^(12*d*x + 12*c) + 246960*a*b^2*e^(12*d*x + 12*c) + 67914*b^3*e^(12*d*x + 12*c) - 92160*a^2*b*
e^(10*d*x + 10*c) - 80640*a*b^2*e^(10*d*x + 10*c) - 23760*b^3*e^(10*d*x + 10*c) + 11520*a^2*b*e^(8*d*x + 8*c)
+ 20160*a*b^2*e^(8*d*x + 8*c) + 7425*b^3*e^(8*d*x + 8*c) - 3840*a*b^2*e^(6*d*x + 6*c) - 2200*b^3*e^(6*d*x + 6*
c) + 360*a*b^2*e^(4*d*x + 4*c) + 495*b^3*e^(4*d*x + 4*c) - 72*b^3*e^(2*d*x + 2*c) + 5*b^3)*e^(-12*d*x - 12*c))
/d