3.4.4 \(\int \cosh ^4(c+d x) (a+b \sinh ^2(c+d x))^3 \, dx\) [304]

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

[Out]

3/256*(4*a-b)*(8*a^2-2*a*b+b^2)*x+3/256*(4*a-b)*(8*a^2-2*a*b+b^2)*cosh(d*x+c)*sinh(d*x+c)/d+1/128*(4*a-b)*(8*a
^2-2*a*b+b^2)*cosh(d*x+c)^3*sinh(d*x+c)/d+1/160*b*(44*a^2-28*a*b+5*b^2)*cosh(d*x+c)^5*sinh(d*x+c)/d+1/10*b*cos
h(d*x+c)^9*sinh(d*x+c)*(a-(a-b)*tanh(d*x+c)^2)^2/d+1/80*b*cosh(d*x+c)^7*sinh(d*x+c)*(a*(10*a-b)-5*(a-b)*(2*a-b
)*tanh(d*x+c)^2)/d

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Rubi [A]
time = 0.22, antiderivative size = 238, normalized size of antiderivative = 1.00, 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 = {3270, 424, 540, 393, 205, 212} \begin {gather*} \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} \end {gather*}

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 205

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] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

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 424

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 540

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 3270

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]

Rubi steps

\begin {align*} \int \cosh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx &=\frac {\text {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 {\text {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 {\text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.39, size = 144, normalized size = 0.61 \begin {gather*} \frac {120 (4 a-b) \left (8 a^2-2 a b+b^2\right ) (c+d x)+20 \left (128 a^3-24 a^2 b+b^3\right ) \sinh (2 (c+d x))+40 \left (8 a^3+12 a^2 b-6 a b^2+b^3\right ) \sinh (4 (c+d x))-10 b \left (-16 a^2+b^2\right ) \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} \end {gather*}

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 = 2.36, size = 165, normalized size = 0.69

method	result	size
default	\(\frac {\left (-\frac {3}{512} b^{3}+\frac {3}{32} a^{2} b \right ) \sinh \left (6 d x +6 c \right )}{6 d}+\frac {\left (-\frac {1}{256} b^{3}+\frac {3}{128} a \,b^{2}\right ) \sinh \left (8 d x +8 c \right )}{8 d}+\frac {\left (\frac {1}{256} b^{3}-\frac {3}{32} a^{2} b +\frac {1}{2} a^{3}\right ) \sinh \left (2 d x +2 c \right )}{2 d}+\frac {\left (\frac {1}{64} b^{3}-\frac {3}{32} a \,b^{2}+\frac {3}{16} a^{2} b +\frac {1}{8} a^{3}\right ) \sinh \left (4 d x +4 c \right )}{4 d}+\frac {3 a^{3} x}{8}-\frac {3 b^{3} x}{256}+\frac {9 a \,b^{2} x}{128}-\frac {3 a^{2} b x}{16}+\frac {b^{3} \sinh \left (10 d x +10 c \right )}{5120 d}\)	\(165\)
risch	\(-\frac {3 a^{2} b x}{16}-\frac {3 b \,{\mathrm e}^{2 d x +2 c} a^{2}}{128 d}+\frac {{\mathrm e}^{4 d x +4 c} a^{3}}{64 d}+\frac {{\mathrm e}^{2 d x +2 c} a^{3}}{8 d}-\frac {{\mathrm e}^{-2 d x -2 c} a^{3}}{8 d}-\frac {b^{3} {\mathrm e}^{-10 d x -10 c}}{10240 d}-\frac {b^{3} {\mathrm e}^{-2 d x -2 c}}{1024 d}-\frac {b^{3} {\mathrm e}^{-4 d x -4 c}}{512 d}-\frac {3 b^{3} x}{256}+\frac {3 b^{2} {\mathrm e}^{-4 d x -4 c} a}{256 d}+\frac {b^{3} {\mathrm e}^{10 d x +10 c}}{10240 d}+\frac {3 a^{3} x}{8}+\frac {9 a \,b^{2} x}{128}+\frac {3 b \,{\mathrm e}^{-2 d x -2 c} a^{2}}{128 d}+\frac {b^{3} {\mathrm e}^{-6 d x -6 c}}{2048 d}+\frac {b^{3} {\mathrm e}^{-8 d x -8 c}}{4096 d}-\frac {b^{3} {\mathrm e}^{8 d x +8 c}}{4096 d}-\frac {b^{3} {\mathrm e}^{6 d x +6 c}}{2048 d}+\frac {b^{3} {\mathrm e}^{4 d x +4 c}}{512 d}+\frac {b^{3} {\mathrm e}^{2 d x +2 c}}{1024 d}-\frac {{\mathrm e}^{-4 d x -4 c} a^{3}}{64 d}-\frac {3 b^{2} {\mathrm e}^{4 d x +4 c} a}{256 d}-\frac {3 \,{\mathrm e}^{-4 d x -4 c} a^{2} b}{128 d}-\frac {b \,{\mathrm e}^{-6 d x -6 c} a^{2}}{128 d}-\frac {3 b^{2} {\mathrm e}^{-8 d x -8 c} a}{2048 d}+\frac {3 b^{2} {\mathrm e}^{8 d x +8 c} a}{2048 d}+\frac {b \,{\mathrm e}^{6 d x +6 c} a^{2}}{128 d}+\frac {3 \,{\mathrm e}^{4 d x +4 c} a^{2} b}{128 d}\)	\(446\)

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,method=_RETURNVERBOSE)

[Out]

1/6*(-3/512*b^3+3/32*a^2*b)*sinh(6*d*x+6*c)/d+1/8*(-1/256*b^3+3/128*a*b^2)*sinh(8*d*x+8*c)/d+1/2*(1/256*b^3-3/
32*a^2*b+1/2*a^3)*sinh(2*d*x+2*c)/d+1/4*(1/64*b^3-3/32*a*b^2+3/16*a^2*b+1/8*a^3)*sinh(4*d*x+4*c)/d+3/8*a^3*x-3
/256*b^3*x+9/128*a*b^2*x-3/16*a^2*b*x+1/5120*b^3*sinh(10*d*x+10*c)/d

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Maxima [A]
time = 0.27, size = 363, normalized size = 1.53 \begin {gather*} \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 {gather*}

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 = 0.41, size = 376, normalized size = 1.58 \begin {gather*} \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 774 vs. \(2 (219) = 438\).
time = 1.99, size = 774, normalized size = 3.25 \begin {gather*} \begin {cases} \frac {3 a^{3} x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac {3 a^{3} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {3 a^{3} x \cosh ^{4}{\left (c + d x \right )}}{8} - \frac {3 a^{3} \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} + \frac {5 a^{3} \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac {3 a^{2} b x \sinh ^{6}{\left (c + d x \right )}}{16} - \frac {9 a^{2} b x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{16} + \frac {9 a^{2} b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{16} - \frac {3 a^{2} b x \cosh ^{6}{\left (c + d x \right )}}{16} - \frac {3 a^{2} b \sinh ^{5}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{16 d} + \frac {a^{2} b \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{2 d} + \frac {3 a^{2} b \sinh {\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{16 d} + \frac {9 a b^{2} x \sinh ^{8}{\left (c + d x \right )}}{128} - \frac {9 a b^{2} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{32} + \frac {27 a b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{64} - \frac {9 a b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{32} + \frac {9 a b^{2} x \cosh ^{8}{\left (c + d x \right )}}{128} - \frac {9 a b^{2} \sinh ^{7}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{128 d} + \frac {33 a b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{128 d} + \frac {33 a b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{128 d} - \frac {9 a b^{2} \sinh {\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{128 d} + \frac {3 b^{3} x \sinh ^{10}{\left (c + d x \right )}}{256} - \frac {15 b^{3} x \sinh ^{8}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{256} + \frac {15 b^{3} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{128} - \frac {15 b^{3} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{128} + \frac {15 b^{3} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{8}{\left (c + d x \right )}}{256} - \frac {3 b^{3} x \cosh ^{10}{\left (c + d x \right )}}{256} - \frac {3 b^{3} \sinh ^{9}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{256 d} + \frac {7 b^{3} \sinh ^{7}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{128 d} + \frac {b^{3} \sinh ^{5}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{10 d} - \frac {7 b^{3} \sinh ^{3}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{128 d} + \frac {3 b^{3} \sinh {\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{256 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right )^{3} \cosh ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}

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 = 0.44, size = 293, normalized size = 1.23 \begin {gather*} \frac {b^{3} e^{\left (10 \, d x + 10 \, c\right )}}{10240 \, d} - \frac {b^{3} e^{\left (-10 \, d x - 10 \, c\right )}}{10240 \, d} + \frac {3}{256} \, {\left (32 \, a^{3} - 16 \, a^{2} b + 6 \, a b^{2} - b^{3}\right )} x + \frac {{\left (6 \, a b^{2} - b^{3}\right )} e^{\left (8 \, d x + 8 \, c\right )}}{4096 \, d} + \frac {{\left (16 \, a^{2} b - b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )}}{2048 \, d} + \frac {{\left (8 \, a^{3} + 12 \, a^{2} b - 6 \, a b^{2} + b^{3}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{512 \, d} + \frac {{\left (128 \, a^{3} - 24 \, a^{2} b + b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{1024 \, d} - \frac {{\left (128 \, a^{3} - 24 \, a^{2} b + b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{1024 \, d} - \frac {{\left (8 \, a^{3} + 12 \, a^{2} b - 6 \, a b^{2} + b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{512 \, d} - \frac {{\left (16 \, a^{2} b - b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{2048 \, d} - \frac {{\left (6 \, a b^{2} - b^{3}\right )} e^{\left (-8 \, d x - 8 \, c\right )}}{4096 \, d} \end {gather*}

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]

1/10240*b^3*e^(10*d*x + 10*c)/d - 1/10240*b^3*e^(-10*d*x - 10*c)/d + 3/256*(32*a^3 - 16*a^2*b + 6*a*b^2 - b^3)
*x + 1/4096*(6*a*b^2 - b^3)*e^(8*d*x + 8*c)/d + 1/2048*(16*a^2*b - b^3)*e^(6*d*x + 6*c)/d + 1/512*(8*a^3 + 12*
a^2*b - 6*a*b^2 + b^3)*e^(4*d*x + 4*c)/d + 1/1024*(128*a^3 - 24*a^2*b + b^3)*e^(2*d*x + 2*c)/d - 1/1024*(128*a
^3 - 24*a^2*b + b^3)*e^(-2*d*x - 2*c)/d - 1/512*(8*a^3 + 12*a^2*b - 6*a*b^2 + b^3)*e^(-4*d*x - 4*c)/d - 1/2048
*(16*a^2*b - b^3)*e^(-6*d*x - 6*c)/d - 1/4096*(6*a*b^2 - b^3)*e^(-8*d*x - 8*c)/d

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Mupad [B]
time = 0.58, size = 209, normalized size = 0.88 \begin {gather*} \frac {320\,a^3\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+40\,a^3\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )+\frac {5\,b^3\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{2}+5\,b^3\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )-\frac {5\,b^3\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )}{4}-\frac {5\,b^3\,\mathrm {sinh}\left (8\,c+8\,d\,x\right )}{8}+\frac {b^3\,\mathrm {sinh}\left (10\,c+10\,d\,x\right )}{4}-60\,a^2\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )-30\,a\,b^2\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )+60\,a^2\,b\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )+20\,a^2\,b\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )+\frac {15\,a\,b^2\,\mathrm {sinh}\left (8\,c+8\,d\,x\right )}{4}+480\,a^3\,d\,x-15\,b^3\,d\,x+90\,a\,b^2\,d\,x-240\,a^2\,b\,d\,x}{1280\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(320*a^3*sinh(2*c + 2*d*x) + 40*a^3*sinh(4*c + 4*d*x) + (5*b^3*sinh(2*c + 2*d*x))/2 + 5*b^3*sinh(4*c + 4*d*x)
- (5*b^3*sinh(6*c + 6*d*x))/4 - (5*b^3*sinh(8*c + 8*d*x))/8 + (b^3*sinh(10*c + 10*d*x))/4 - 60*a^2*b*sinh(2*c
+ 2*d*x) - 30*a*b^2*sinh(4*c + 4*d*x) + 60*a^2*b*sinh(4*c + 4*d*x) + 20*a^2*b*sinh(6*c + 6*d*x) + (15*a*b^2*si
nh(8*c + 8*d*x))/4 + 480*a^3*d*x - 15*b^3*d*x + 90*a*b^2*d*x - 240*a^2*b*d*x)/(1280*d)

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