3.3.18 \(\int (a+b \sinh ^4(c+d x))^3 \, dx\) [218]

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

[Out]

1/1024*(1024*a^3+1152*a^2*b+840*a*b^2+231*b^3)*x-1/1024*b*(1920*a^2+2232*a*b+793*b^2)*cosh(d*x+c)*sinh(d*x+c)/
d+1/1536*b*(1152*a^2+3912*a*b+2279*b^2)*cosh(d*x+c)^3*sinh(d*x+c)/d-1/1920*b^2*(3000*a+3481*b)*cosh(d*x+c)^5*s
inh(d*x+c)/d+3/320*b^2*(40*a+139*b)*cosh(d*x+c)^7*sinh(d*x+c)/d-61/120*b^3*cosh(d*x+c)^9*sinh(d*x+c)/d+1/12*b^
3*cosh(d*x+c)^11*sinh(d*x+c)/d

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Rubi [A]
time = 0.28, antiderivative size = 211, normalized size of antiderivative = 1.00, 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 = {3288, 1171, 1828, 393, 212} \begin {gather*} \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 (1024 a^3+1152 a^2 b+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} \end {gather*}

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 393

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 1171

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] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 1828

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 3288

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]

Rubi steps

\begin {align*} \int \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 ) \text {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*}

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Mathematica [A]
time = 0.31, size = 156, normalized size = 0.74 \begin {gather*} \frac {120 \left (1024 a^3+1152 a^2 b+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} \end {gather*}

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 = 1.48, size = 177, normalized size = 0.84

method	result	size
default	\(a^{3} x +\frac {\left (-\frac {55}{512} b^{3}-\frac {3}{16} a \,b^{2}\right ) \sinh \left (6 d x +6 c \right )}{6 d}+\frac {\left (\frac {33}{1024} b^{3}+\frac {3}{128} a \,b^{2}\right ) \sinh \left (8 d x +8 c \right )}{8 d}+\frac {\left (-\frac {99}{256} b^{3}-\frac {21}{16} a \,b^{2}-\frac {3}{2} a^{2} b \right ) \sinh \left (2 d x +2 c \right )}{2 d}+\frac {\left (\frac {495}{2048} b^{3}+\frac {21}{32} a \,b^{2}+\frac {3}{8} a^{2} b \right ) \sinh \left (4 d x +4 c \right )}{4 d}+\frac {231 b^{3} x}{1024}+\frac {105 a \,b^{2} x}{128}+\frac {9 a^{2} b x}{8}-\frac {3 b^{3} \sinh \left (10 d x +10 c \right )}{5120 d}+\frac {b^{3} \sinh \left (12 d x +12 c \right )}{24576 d}\)	\(177\)
risch	\(\frac {9 a^{2} b x}{8}+\frac {3 b^{2} {\mathrm e}^{8 d x +8 c} a}{2048 d}-\frac {3 b \,{\mathrm e}^{2 d x +2 c} a^{2}}{8 d}+\frac {b^{3} {\mathrm e}^{12 d x +12 c}}{49152 d}-\frac {b^{3} {\mathrm e}^{-12 d x -12 c}}{49152 d}+\frac {99 b^{3} {\mathrm e}^{-2 d x -2 c}}{1024 d}-\frac {495 b^{3} {\mathrm e}^{-4 d x -4 c}}{16384 d}+\frac {231 b^{3} x}{1024}+\frac {21 \,{\mathrm e}^{-2 d x -2 c} a \,b^{2}}{64 d}-\frac {3 b \,{\mathrm e}^{-4 d x -4 c} a^{2}}{64 d}-\frac {21 b^{2} {\mathrm e}^{-4 d x -4 c} a}{256 d}+\frac {b^{2} {\mathrm e}^{-6 d x -6 c} a}{64 d}+a^{3} x +\frac {105 a \,b^{2} x}{128}+\frac {3 b \,{\mathrm e}^{-2 d x -2 c} a^{2}}{8 d}+\frac {55 b^{3} {\mathrm e}^{-6 d x -6 c}}{6144 d}-\frac {33 b^{3} {\mathrm e}^{-8 d x -8 c}}{16384 d}-\frac {3 b^{3} {\mathrm e}^{10 d x +10 c}}{10240 d}+\frac {33 b^{3} {\mathrm e}^{8 d x +8 c}}{16384 d}-\frac {55 b^{3} {\mathrm e}^{6 d x +6 c}}{6144 d}+\frac {3 b^{3} {\mathrm e}^{-10 d x -10 c}}{10240 d}+\frac {495 b^{3} {\mathrm e}^{4 d x +4 c}}{16384 d}-\frac {99 b^{3} {\mathrm e}^{2 d x +2 c}}{1024 d}-\frac {b^{2} {\mathrm e}^{6 d x +6 c} a}{64 d}+\frac {3 b \,{\mathrm e}^{4 d x +4 c} a^{2}}{64 d}+\frac {21 b^{2} {\mathrm e}^{4 d x +4 c} a}{256 d}-\frac {21 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}}{64 d}-\frac {3 b^{2} {\mathrm e}^{-8 d x -8 c} a}{2048 d}\)	\(447\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3*x+1/6*(-55/512*b^3-3/16*a*b^2)*sinh(6*d*x+6*c)/d+1/8*(33/1024*b^3+3/128*a*b^2)*sinh(8*d*x+8*c)/d+1/2*(-99/
256*b^3-21/16*a*b^2-3/2*a^2*b)*sinh(2*d*x+2*c)/d+1/4*(495/2048*b^3+21/32*a*b^2+3/8*a^2*b)*sinh(4*d*x+4*c)/d+23
1/1024*b^3*x+105/128*a*b^2*x+9/8*a^2*b*x-3/5120*b^3*sinh(10*d*x+10*c)/d+1/24576*b^3*sinh(12*d*x+12*c)/d

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Maxima [A]
time = 0.27, size = 344, normalized size = 1.63 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (197) = 394\).
time = 0.38, size = 461, normalized size = 2.18 \begin {gather*} \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 666 vs. \(2 (202) = 404\).
time = 4.39, size = 666, normalized size = 3.16 \begin {gather*} \begin {cases} a^{3} x + \frac {9 a^{2} b x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac {9 a^{2} b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {9 a^{2} b x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac {15 a^{2} b \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} - \frac {9 a^{2} b \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac {105 a b^{2} x \sinh ^{8}{\left (c + d x \right )}}{128} - \frac {105 a b^{2} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{32} + \frac {315 a b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{64} - \frac {105 a b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{32} + \frac {105 a b^{2} x \cosh ^{8}{\left (c + d x \right )}}{128} + \frac {279 a b^{2} \sinh ^{7}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{128 d} - \frac {511 a b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{128 d} + \frac {385 a b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{128 d} - \frac {105 a b^{2} \sinh {\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{128 d} + \frac {231 b^{3} x \sinh ^{12}{\left (c + d x \right )}}{1024} - \frac {693 b^{3} x \sinh ^{10}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{512} + \frac {3465 b^{3} x \sinh ^{8}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{1024} - \frac {1155 b^{3} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{256} + \frac {3465 b^{3} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{8}{\left (c + d x \right )}}{1024} - \frac {693 b^{3} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{10}{\left (c + d x \right )}}{512} + \frac {231 b^{3} x \cosh ^{12}{\left (c + d x \right )}}{1024} + \frac {793 b^{3} \sinh ^{11}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{1024 d} - \frac {7337 b^{3} \sinh ^{9}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3072 d} + \frac {9273 b^{3} \sinh ^{7}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{2560 d} - \frac {7623 b^{3} \sinh ^{5}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{2560 d} + \frac {1309 b^{3} \sinh ^{3}{\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{1024 d} - \frac {231 b^{3} \sinh {\left (c + d x \right )} \cosh ^{11}{\left (c + d x \right )}}{1024 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{4}{\left (c \right )}\right )^{3} & \text {otherwise} \end {cases} \end {gather*}

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 [A]
time = 0.42, size = 327, normalized size = 1.55 \begin {gather*} \frac {b^{3} e^{\left (12 \, d x + 12 \, c\right )}}{49152 \, d} - \frac {3 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )}}{10240 \, d} + \frac {3 \, b^{3} e^{\left (-10 \, d x - 10 \, c\right )}}{10240 \, d} - \frac {b^{3} e^{\left (-12 \, d x - 12 \, c\right )}}{49152 \, d} + \frac {1}{1024} \, {\left (1024 \, a^{3} + 1152 \, a^{2} b + 840 \, a b^{2} + 231 \, b^{3}\right )} x + \frac {3 \, {\left (8 \, a b^{2} + 11 \, b^{3}\right )} e^{\left (8 \, d x + 8 \, c\right )}}{16384 \, d} - \frac {{\left (96 \, a b^{2} + 55 \, b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )}}{6144 \, d} + \frac {3 \, {\left (256 \, a^{2} b + 448 \, a b^{2} + 165 \, b^{3}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{16384 \, d} - \frac {3 \, {\left (128 \, a^{2} b + 112 \, a b^{2} + 33 \, b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{1024 \, d} + \frac {3 \, {\left (128 \, a^{2} b + 112 \, a b^{2} + 33 \, b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{1024 \, d} - \frac {3 \, {\left (256 \, a^{2} b + 448 \, a b^{2} + 165 \, b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{16384 \, d} + \frac {{\left (96 \, a b^{2} + 55 \, b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{6144 \, d} - \frac {3 \, {\left (8 \, a b^{2} + 11 \, b^{3}\right )} e^{\left (-8 \, d x - 8 \, c\right )}}{16384 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/49152*b^3*e^(12*d*x + 12*c)/d - 3/10240*b^3*e^(10*d*x + 10*c)/d + 3/10240*b^3*e^(-10*d*x - 10*c)/d - 1/49152
*b^3*e^(-12*d*x - 12*c)/d + 1/1024*(1024*a^3 + 1152*a^2*b + 840*a*b^2 + 231*b^3)*x + 3/16384*(8*a*b^2 + 11*b^3
)*e^(8*d*x + 8*c)/d - 1/6144*(96*a*b^2 + 55*b^3)*e^(6*d*x + 6*c)/d + 3/16384*(256*a^2*b + 448*a*b^2 + 165*b^3)
*e^(4*d*x + 4*c)/d - 3/1024*(128*a^2*b + 112*a*b^2 + 33*b^3)*e^(2*d*x + 2*c)/d + 3/1024*(128*a^2*b + 112*a*b^2
+ 33*b^3)*e^(-2*d*x - 2*c)/d - 3/16384*(256*a^2*b + 448*a*b^2 + 165*b^3)*e^(-4*d*x - 4*c)/d + 1/6144*(96*a*b^
2 + 55*b^3)*e^(-6*d*x - 6*c)/d - 3/16384*(8*a*b^2 + 11*b^3)*e^(-8*d*x - 8*c)/d

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Mupad [B]
time = 0.56, size = 210, normalized size = 1.00 \begin {gather*} \frac {\frac {7425\,b^3\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{8}-2970\,b^3\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )-275\,b^3\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )+\frac {495\,b^3\,\mathrm {sinh}\left (8\,c+8\,d\,x\right )}{8}-9\,b^3\,\mathrm {sinh}\left (10\,c+10\,d\,x\right )+\frac {5\,b^3\,\mathrm {sinh}\left (12\,c+12\,d\,x\right )}{8}-10080\,a\,b^2\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )-11520\,a^2\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+2520\,a\,b^2\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )+1440\,a^2\,b\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )-480\,a\,b^2\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )+45\,a\,b^2\,\mathrm {sinh}\left (8\,c+8\,d\,x\right )+15360\,a^3\,d\,x+3465\,b^3\,d\,x+12600\,a\,b^2\,d\,x+17280\,a^2\,b\,d\,x}{15360\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sinh(c + d*x)^4)^3,x)

[Out]

((7425*b^3*sinh(4*c + 4*d*x))/8 - 2970*b^3*sinh(2*c + 2*d*x) - 275*b^3*sinh(6*c + 6*d*x) + (495*b^3*sinh(8*c +
8*d*x))/8 - 9*b^3*sinh(10*c + 10*d*x) + (5*b^3*sinh(12*c + 12*d*x))/8 - 10080*a*b^2*sinh(2*c + 2*d*x) - 11520
*a^2*b*sinh(2*c + 2*d*x) + 2520*a*b^2*sinh(4*c + 4*d*x) + 1440*a^2*b*sinh(4*c + 4*d*x) - 480*a*b^2*sinh(6*c +
6*d*x) + 45*a*b^2*sinh(8*c + 8*d*x) + 15360*a^3*d*x + 3465*b^3*d*x + 12600*a*b^2*d*x + 17280*a^2*b*d*x)/(15360
*d)

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