3.4.6 \(\int \cosh ^2(c+d x) (a+b \sinh ^2(c+d x))^3 \, dx\) [306]

Optimal. Leaf size=203 \[ \frac {1}{128} \left (64 a^3-48 a^2 b+24 a b^2-5 b^3\right ) x+\frac {\left (64 a^3-48 a^2 b+24 a b^2-5 b^3\right ) \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac {b \left (88 a^2-68 a b+15 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}+\frac {b \cosh ^7(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{8 d}+\frac {b \cosh ^5(c+d x) \sinh (c+d x) \left (a (8 a-b)-(8 a-5 b) (a-b) \tanh ^2(c+d x)\right )}{48 d} \]

[Out]

1/128*(64*a^3-48*a^2*b+24*a*b^2-5*b^3)*x+1/128*(64*a^3-48*a^2*b+24*a*b^2-5*b^3)*cosh(d*x+c)*sinh(d*x+c)/d+1/19
2*b*(88*a^2-68*a*b+15*b^2)*cosh(d*x+c)^3*sinh(d*x+c)/d+1/8*b*cosh(d*x+c)^7*sinh(d*x+c)*(a-(a-b)*tanh(d*x+c)^2)
^2/d+1/48*b*cosh(d*x+c)^5*sinh(d*x+c)*(a*(8*a-b)-(8*a-5*b)*(a-b)*tanh(d*x+c)^2)/d

________________________________________________________________________________________

Rubi [A]
time = 0.19, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 6, 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 (88 a^2-68 a b+15 b^2\right ) \sinh (c+d x) \cosh ^3(c+d x)}{192 d}+\frac {\left (64 a^3-48 a^2 b+24 a b^2-5 b^3\right ) \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac {1}{128} x \left (64 a^3-48 a^2 b+24 a b^2-5 b^3\right )+\frac {b \sinh (c+d x) \cosh ^7(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{8 d}+\frac {b \sinh (c+d x) \cosh ^5(c+d x) \left (a (8 a-b)-(8 a-5 b) (a-b) \tanh ^2(c+d x)\right )}{48 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((64*a^3 - 48*a^2*b + 24*a*b^2 - 5*b^3)*x)/128 + ((64*a^3 - 48*a^2*b + 24*a*b^2 - 5*b^3)*Cosh[c + d*x]*Sinh[c
+ d*x])/(128*d) + (b*(88*a^2 - 68*a*b + 15*b^2)*Cosh[c + d*x]^3*Sinh[c + d*x])/(192*d) + (b*Cosh[c + d*x]^7*Si
nh[c + d*x]*(a - (a - b)*Tanh[c + d*x]^2)^2)/(8*d) + (b*Cosh[c + d*x]^5*Sinh[c + d*x]*(a*(8*a - b) - (8*a - 5*
b)*(a - b)*Tanh[c + d*x]^2))/(48*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 ^2(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 )^5} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b \cosh ^7(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{8 d}-\frac {\text {Subst}\left (\int \frac {\left (-a (8 a-b)+(8 a-5 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 )}{8 d}\\ &=\frac {b \cosh ^7(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{8 d}+\frac {b \cosh ^5(c+d x) \sinh (c+d x) \left (a (8 a-b)-(8 a-5 b) (a-b) \tanh ^2(c+d x)\right )}{48 d}-\frac {\text {Subst}\left (\int \frac {-a (6 a-b) (8 a-b)+3 (8 a-5 b) (a-b) (2 a-b) x^2}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{48 d}\\ &=\frac {b \left (88 a^2-68 a b+15 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}+\frac {b \cosh ^7(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{8 d}+\frac {b \cosh ^5(c+d x) \sinh (c+d x) \left (a (8 a-b)-(8 a-5 b) (a-b) \tanh ^2(c+d x)\right )}{48 d}+\frac {\left (64 a^3-48 a^2 b+24 a b^2-5 b^3\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{64 d}\\ &=\frac {\left (64 a^3-48 a^2 b+24 a b^2-5 b^3\right ) \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac {b \left (88 a^2-68 a b+15 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}+\frac {b \cosh ^7(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{8 d}+\frac {b \cosh ^5(c+d x) \sinh (c+d x) \left (a (8 a-b)-(8 a-5 b) (a-b) \tanh ^2(c+d x)\right )}{48 d}+\frac {\left (64 a^3-48 a^2 b+24 a b^2-5 b^3\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{128 d}\\ &=\frac {1}{128} \left (64 a^3-48 a^2 b+24 a b^2-5 b^3\right ) x+\frac {\left (64 a^3-48 a^2 b+24 a b^2-5 b^3\right ) \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac {b \left (88 a^2-68 a b+15 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}+\frac {b \cosh ^7(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{8 d}+\frac {b \cosh ^5(c+d x) \sinh (c+d x) \left (a (8 a-b)-(8 a-5 b) (a-b) \tanh ^2(c+d x)\right )}{48 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.24, size = 120, normalized size = 0.59 \begin {gather*} \frac {24 \left (64 a^3-48 a^2 b+24 a b^2-5 b^3\right ) (c+d x)+48 \left (16 a^3-3 a b^2+b^3\right ) \sinh (2 (c+d x))+24 b \left (12 a^2-6 a b+b^2\right ) \sinh (4 (c+d x))+16 (3 a-b) b^2 \sinh (6 (c+d x))+3 b^3 \sinh (8 (c+d x))}{3072 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(24*(64*a^3 - 48*a^2*b + 24*a*b^2 - 5*b^3)*(c + d*x) + 48*(16*a^3 - 3*a*b^2 + b^3)*Sinh[2*(c + d*x)] + 24*b*(1
2*a^2 - 6*a*b + b^2)*Sinh[4*(c + d*x)] + 16*(3*a - b)*b^2*Sinh[6*(c + d*x)] + 3*b^3*Sinh[8*(c + d*x)])/(3072*d
)

________________________________________________________________________________________

Maple [A]
time = 1.46, size = 216, normalized size = 1.06

method	result	size
derivativedivides	\(\frac {b^{3} \left (\frac {\left (\sinh ^{5}\left (d x +c \right )\right ) \left (\cosh ^{3}\left (d x +c \right )\right )}{8}-\frac {5 \left (\sinh ^{3}\left (d x +c \right )\right ) \left (\cosh ^{3}\left (d x +c \right )\right )}{48}+\frac {5 \sinh \left (d x +c \right ) \left (\cosh ^{3}\left (d x +c \right )\right )}{64}-\frac {5 \cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{128}-\frac {5 d x}{128}-\frac {5 c}{128}\right )+3 a \,b^{2} \left (\frac {\left (\sinh ^{3}\left (d x +c \right )\right ) \left (\cosh ^{3}\left (d x +c \right )\right )}{6}-\frac {\sinh \left (d x +c \right ) \left (\cosh ^{3}\left (d x +c \right )\right )}{8}+\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+3 a^{2} b \left (\frac {\sinh \left (d x +c \right ) \left (\cosh ^{3}\left (d x +c \right )\right )}{4}-\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{8}-\frac {d x}{8}-\frac {c}{8}\right )+a^{3} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\)	\(216\)
default	\(\frac {b^{3} \left (\frac {\left (\sinh ^{5}\left (d x +c \right )\right ) \left (\cosh ^{3}\left (d x +c \right )\right )}{8}-\frac {5 \left (\sinh ^{3}\left (d x +c \right )\right ) \left (\cosh ^{3}\left (d x +c \right )\right )}{48}+\frac {5 \sinh \left (d x +c \right ) \left (\cosh ^{3}\left (d x +c \right )\right )}{64}-\frac {5 \cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{128}-\frac {5 d x}{128}-\frac {5 c}{128}\right )+3 a \,b^{2} \left (\frac {\left (\sinh ^{3}\left (d x +c \right )\right ) \left (\cosh ^{3}\left (d x +c \right )\right )}{6}-\frac {\sinh \left (d x +c \right ) \left (\cosh ^{3}\left (d x +c \right )\right )}{8}+\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+3 a^{2} b \left (\frac {\sinh \left (d x +c \right ) \left (\cosh ^{3}\left (d x +c \right )\right )}{4}-\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{8}-\frac {d x}{8}-\frac {c}{8}\right )+a^{3} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\)	\(216\)
risch	\(\frac {a^{3} x}{2}-\frac {3 a^{2} b x}{8}+\frac {3 a \,b^{2} x}{16}-\frac {5 b^{3} x}{128}+\frac {b^{3} {\mathrm e}^{8 d x +8 c}}{2048 d}+\frac {b^{2} {\mathrm e}^{6 d x +6 c} a}{128 d}-\frac {b^{3} {\mathrm e}^{6 d x +6 c}}{384 d}+\frac {3 \,{\mathrm e}^{4 d x +4 c} a^{2} b}{64 d}-\frac {3 b^{2} {\mathrm e}^{4 d x +4 c} a}{128 d}+\frac {b^{3} {\mathrm e}^{4 d x +4 c}}{256 d}+\frac {{\mathrm e}^{2 d x +2 c} a^{3}}{8 d}-\frac {3 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}}{128 d}+\frac {b^{3} {\mathrm e}^{2 d x +2 c}}{128 d}-\frac {{\mathrm e}^{-2 d x -2 c} a^{3}}{8 d}+\frac {3 \,{\mathrm e}^{-2 d x -2 c} a \,b^{2}}{128 d}-\frac {b^{3} {\mathrm e}^{-2 d x -2 c}}{128 d}-\frac {3 \,{\mathrm e}^{-4 d x -4 c} a^{2} b}{64 d}+\frac {3 b^{2} {\mathrm e}^{-4 d x -4 c} a}{128 d}-\frac {b^{3} {\mathrm e}^{-4 d x -4 c}}{256 d}-\frac {b^{2} {\mathrm e}^{-6 d x -6 c} a}{128 d}+\frac {b^{3} {\mathrm e}^{-6 d x -6 c}}{384 d}-\frac {b^{3} {\mathrm e}^{-8 d x -8 c}}{2048 d}\)	\(342\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(b^3*(1/8*sinh(d*x+c)^5*cosh(d*x+c)^3-5/48*sinh(d*x+c)^3*cosh(d*x+c)^3+5/64*sinh(d*x+c)*cosh(d*x+c)^3-5/12
8*cosh(d*x+c)*sinh(d*x+c)-5/128*d*x-5/128*c)+3*a*b^2*(1/6*sinh(d*x+c)^3*cosh(d*x+c)^3-1/8*sinh(d*x+c)*cosh(d*x
+c)^3+1/16*cosh(d*x+c)*sinh(d*x+c)+1/16*d*x+1/16*c)+3*a^2*b*(1/4*sinh(d*x+c)*cosh(d*x+c)^3-1/8*cosh(d*x+c)*sin
h(d*x+c)-1/8*d*x-1/8*c)+a^3*(1/2*cosh(d*x+c)*sinh(d*x+c)+1/2*d*x+1/2*c))

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 287, normalized size = 1.41 \begin {gather*} \frac {1}{8} \, a^{3} {\left (4 \, x + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{6144} \, b^{3} {\left (\frac {{\left (16 \, e^{\left (-2 \, d x - 2 \, c\right )} - 24 \, e^{\left (-4 \, d x - 4 \, c\right )} - 48 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3\right )} e^{\left (8 \, d x + 8 \, c\right )}}{d} + \frac {240 \, {\left (d x + c\right )}}{d} + \frac {48 \, e^{\left (-2 \, d x - 2 \, c\right )} + 24 \, e^{\left (-4 \, d x - 4 \, c\right )} - 16 \, e^{\left (-6 \, d x - 6 \, c\right )} + 3 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d}\right )} - \frac {1}{128} \, a b^{2} {\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 )} - \frac {3}{64} \, a^{2} b {\left (\frac {8 \, {\left (d x + c\right )}}{d} - \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a^3*(4*x + e^(2*d*x + 2*c)/d - e^(-2*d*x - 2*c)/d) - 1/6144*b^3*((16*e^(-2*d*x - 2*c) - 24*e^(-4*d*x - 4*c
) - 48*e^(-6*d*x - 6*c) - 3)*e^(8*d*x + 8*c)/d + 240*(d*x + c)/d + (48*e^(-2*d*x - 2*c) + 24*e^(-4*d*x - 4*c)
- 16*e^(-6*d*x - 6*c) + 3*e^(-8*d*x - 8*c))/d) - 1/128*a*b^2*((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) - 3/64*a^2*
b*(8*(d*x + c)/d - e^(4*d*x + 4*c)/d + e^(-4*d*x - 4*c)/d)

________________________________________________________________________________________

Fricas [A]
time = 0.39, size = 257, normalized size = 1.27 \begin {gather*} \frac {3 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + 3 \, {\left (7 \, b^{3} \cosh \left (d x + c\right )^{3} + 4 \, {\left (3 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + {\left (21 \, b^{3} \cosh \left (d x + c\right )^{5} + 40 \, {\left (3 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )^{3} + 12 \, {\left (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} + 3 \, {\left (64 \, a^{3} - 48 \, a^{2} b + 24 \, a b^{2} - 5 \, b^{3}\right )} d x + 3 \, {\left (b^{3} \cosh \left (d x + c\right )^{7} + 4 \, {\left (3 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )^{5} + 4 \, {\left (12 \, a^{2} b - 6 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{3} + 4 \, {\left (16 \, a^{3} - 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{384 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(3*b^3*cosh(d*x + c)*sinh(d*x + c)^7 + 3*(7*b^3*cosh(d*x + c)^3 + 4*(3*a*b^2 - b^3)*cosh(d*x + c))*sinh(
d*x + c)^5 + (21*b^3*cosh(d*x + c)^5 + 40*(3*a*b^2 - b^3)*cosh(d*x + c)^3 + 12*(12*a^2*b - 6*a*b^2 + b^3)*cosh
(d*x + c))*sinh(d*x + c)^3 + 3*(64*a^3 - 48*a^2*b + 24*a*b^2 - 5*b^3)*d*x + 3*(b^3*cosh(d*x + c)^7 + 4*(3*a*b^
2 - b^3)*cosh(d*x + c)^5 + 4*(12*a^2*b - 6*a*b^2 + b^3)*cosh(d*x + c)^3 + 4*(16*a^3 - 3*a*b^2 + b^3)*cosh(d*x
+ c))*sinh(d*x + c))/d

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 559 vs. \(2 (190) = 380\).
time = 1.04, size = 559, normalized size = 2.75 \begin {gather*} \begin {cases} - \frac {a^{3} x \sinh ^{2}{\left (c + d x \right )}}{2} + \frac {a^{3} x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac {a^{3} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2 d} - \frac {3 a^{2} b x \sinh ^{4}{\left (c + d x \right )}}{8} + \frac {3 a^{2} b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} - \frac {3 a^{2} b x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac {3 a^{2} b \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} + \frac {3 a^{2} b \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} - \frac {3 a b^{2} x \sinh ^{6}{\left (c + d x \right )}}{16} + \frac {9 a b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{16} - \frac {9 a b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{16} + \frac {3 a b^{2} x \cosh ^{6}{\left (c + d x \right )}}{16} + \frac {3 a b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{16 d} + \frac {a b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{2 d} - \frac {3 a b^{2} \sinh {\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{16 d} - \frac {5 b^{3} x \sinh ^{8}{\left (c + d x \right )}}{128} + \frac {5 b^{3} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{32} - \frac {15 b^{3} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{64} + \frac {5 b^{3} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{32} - \frac {5 b^{3} x \cosh ^{8}{\left (c + d x \right )}}{128} + \frac {5 b^{3} \sinh ^{7}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{128 d} + \frac {73 b^{3} \sinh ^{5}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{384 d} - \frac {55 b^{3} \sinh ^{3}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{384 d} + \frac {5 b^{3} \sinh {\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{128 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right )^{3} \cosh ^{2}{\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)**2*(a+b*sinh(d*x+c)**2)**3,x)

[Out]

Piecewise((-a**3*x*sinh(c + d*x)**2/2 + a**3*x*cosh(c + d*x)**2/2 + a**3*sinh(c + d*x)*cosh(c + d*x)/(2*d) - 3
*a**2*b*x*sinh(c + d*x)**4/8 + 3*a**2*b*x*sinh(c + d*x)**2*cosh(c + d*x)**2/4 - 3*a**2*b*x*cosh(c + d*x)**4/8
+ 3*a**2*b*sinh(c + d*x)**3*cosh(c + d*x)/(8*d) + 3*a**2*b*sinh(c + d*x)*cosh(c + d*x)**3/(8*d) - 3*a*b**2*x*s
inh(c + d*x)**6/16 + 9*a*b**2*x*sinh(c + d*x)**4*cosh(c + d*x)**2/16 - 9*a*b**2*x*sinh(c + d*x)**2*cosh(c + d*
x)**4/16 + 3*a*b**2*x*cosh(c + d*x)**6/16 + 3*a*b**2*sinh(c + d*x)**5*cosh(c + d*x)/(16*d) + a*b**2*sinh(c + d
*x)**3*cosh(c + d*x)**3/(2*d) - 3*a*b**2*sinh(c + d*x)*cosh(c + d*x)**5/(16*d) - 5*b**3*x*sinh(c + d*x)**8/128
+ 5*b**3*x*sinh(c + d*x)**6*cosh(c + d*x)**2/32 - 15*b**3*x*sinh(c + d*x)**4*cosh(c + d*x)**4/64 + 5*b**3*x*s
inh(c + d*x)**2*cosh(c + d*x)**6/32 - 5*b**3*x*cosh(c + d*x)**8/128 + 5*b**3*sinh(c + d*x)**7*cosh(c + d*x)/(1
28*d) + 73*b**3*sinh(c + d*x)**5*cosh(c + d*x)**3/(384*d) - 55*b**3*sinh(c + d*x)**3*cosh(c + d*x)**5/(384*d)
+ 5*b**3*sinh(c + d*x)*cosh(c + d*x)**7/(128*d), Ne(d, 0)), (x*(a + b*sinh(c)**2)**3*cosh(c)**2, True))

________________________________________________________________________________________

Giac [A]
time = 0.43, size = 231, normalized size = 1.14 \begin {gather*} \frac {b^{3} e^{\left (8 \, d x + 8 \, c\right )}}{2048 \, d} - \frac {b^{3} e^{\left (-8 \, d x - 8 \, c\right )}}{2048 \, d} + \frac {1}{128} \, {\left (64 \, a^{3} - 48 \, a^{2} b + 24 \, a b^{2} - 5 \, b^{3}\right )} x + \frac {{\left (3 \, a b^{2} - b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )}}{384 \, d} + \frac {{\left (12 \, a^{2} b - 6 \, a b^{2} + b^{3}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{256 \, d} + \frac {{\left (16 \, a^{3} - 3 \, a b^{2} + b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{128 \, d} - \frac {{\left (16 \, a^{3} - 3 \, a b^{2} + b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{128 \, d} - \frac {{\left (12 \, a^{2} b - 6 \, a b^{2} + b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{256 \, d} - \frac {{\left (3 \, a b^{2} - b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{384 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2048*b^3*e^(8*d*x + 8*c)/d - 1/2048*b^3*e^(-8*d*x - 8*c)/d + 1/128*(64*a^3 - 48*a^2*b + 24*a*b^2 - 5*b^3)*x
+ 1/384*(3*a*b^2 - b^3)*e^(6*d*x + 6*c)/d + 1/256*(12*a^2*b - 6*a*b^2 + b^3)*e^(4*d*x + 4*c)/d + 1/128*(16*a^3
- 3*a*b^2 + b^3)*e^(2*d*x + 2*c)/d - 1/128*(16*a^3 - 3*a*b^2 + b^3)*e^(-2*d*x - 2*c)/d - 1/256*(12*a^2*b - 6*
a*b^2 + b^3)*e^(-4*d*x - 4*c)/d - 1/384*(3*a*b^2 - b^3)*e^(-6*d*x - 6*c)/d

________________________________________________________________________________________

Mupad [B]
time = 0.42, size = 166, normalized size = 0.82 \begin {gather*} \frac {96\,a^3\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+6\,b^3\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+3\,b^3\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )-2\,b^3\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )+\frac {3\,b^3\,\mathrm {sinh}\left (8\,c+8\,d\,x\right )}{8}-18\,a\,b^2\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )-18\,a\,b^2\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )+36\,a^2\,b\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )+6\,a\,b^2\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )+192\,a^3\,d\,x-15\,b^3\,d\,x+72\,a\,b^2\,d\,x-144\,a^2\,b\,d\,x}{384\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(96*a^3*sinh(2*c + 2*d*x) + 6*b^3*sinh(2*c + 2*d*x) + 3*b^3*sinh(4*c + 4*d*x) - 2*b^3*sinh(6*c + 6*d*x) + (3*b
^3*sinh(8*c + 8*d*x))/8 - 18*a*b^2*sinh(2*c + 2*d*x) - 18*a*b^2*sinh(4*c + 4*d*x) + 36*a^2*b*sinh(4*c + 4*d*x)
+ 6*a*b^2*sinh(6*c + 6*d*x) + 192*a^3*d*x - 15*b^3*d*x + 72*a*b^2*d*x - 144*a^2*b*d*x)/(384*d)

________________________________________________________________________________________