3.1.21 \(\int \sinh ^2(c+d x) (a+b \sinh ^2(c+d x))^3 \, dx\) [21]

Optimal. Leaf size=181 \[ -\frac {1}{128} \left (64 a^3-144 a^2 b+120 a b^2-35 b^3\right ) x+\frac {\left (96 a^3-376 a^2 b+360 a b^2-105 b^3\right ) \cosh (c+d x) \sinh (c+d x)}{384 d}+\frac {b \left (24 a^2-64 a b+35 b^2\right ) \cosh (c+d x) \sinh ^3(c+d x)}{192 d}+\frac {(6 a-7 b) \cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{48 d}+\frac {\cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^3}{8 d} \]

[Out]

-1/128*(64*a^3-144*a^2*b+120*a*b^2-35*b^3)*x+1/384*(96*a^3-376*a^2*b+360*a*b^2-105*b^3)*cosh(d*x+c)*sinh(d*x+c
)/d+1/192*b*(24*a^2-64*a*b+35*b^2)*cosh(d*x+c)*sinh(d*x+c)^3/d+1/48*(6*a-7*b)*cosh(d*x+c)*sinh(d*x+c)*(a+b*sin
h(d*x+c)^2)^2/d+1/8*cosh(d*x+c)*sinh(d*x+c)*(a+b*sinh(d*x+c)^2)^3/d

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Rubi [A]
time = 0.13, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3249, 3248} \begin {gather*} \frac {b \left (24 a^2-64 a b+35 b^2\right ) \sinh ^3(c+d x) \cosh (c+d x)}{192 d}+\frac {\left (96 a^3-376 a^2 b+360 a b^2-105 b^3\right ) \sinh (c+d x) \cosh (c+d x)}{384 d}-\frac {1}{128} x \left (64 a^3-144 a^2 b+120 a b^2-35 b^3\right )+\frac {\sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^3}{8 d}+\frac {(6 a-7 b) \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{48 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/128*((64*a^3 - 144*a^2*b + 120*a*b^2 - 35*b^3)*x) + ((96*a^3 - 376*a^2*b + 360*a*b^2 - 105*b^3)*Cosh[c + d*
x]*Sinh[c + d*x])/(384*d) + (b*(24*a^2 - 64*a*b + 35*b^2)*Cosh[c + d*x]*Sinh[c + d*x]^3)/(192*d) + ((6*a - 7*b
)*Cosh[c + d*x]*Sinh[c + d*x]*(a + b*Sinh[c + d*x]^2)^2)/(48*d) + (Cosh[c + d*x]*Sinh[c + d*x]*(a + b*Sinh[c +
d*x]^2)^3)/(8*d)

Rule 3248

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(4*A
*(2*a + b) + B*(4*a + 3*b))*(x/8), x] + (-Simp[b*B*Cos[e + f*x]*(Sin[e + f*x]^3/(4*f)), x] - Simp[(4*A*b + B*(
4*a + 3*b))*Cos[e + f*x]*(Sin[e + f*x]/(8*f)), x]) /; FreeQ[{a, b, e, f, A, B}, x]

Rule 3249

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp
[(-B)*Cos[e + f*x]*Sin[e + f*x]*((a + b*Sin[e + f*x]^2)^p/(2*f*(p + 1))), x] + Dist[1/(2*(p + 1)), Int[(a + b*
Sin[e + f*x]^2)^(p - 1)*Simp[a*B + 2*a*A*(p + 1) + (2*A*b*(p + 1) + B*(b + 2*a*p + 2*b*p))*Sin[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, e, f, A, B}, x] && GtQ[p, 0]

Rubi steps

\begin {align*} \int \sinh ^2(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx &=\frac {\cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^3}{8 d}-\frac {1}{8} \int \left (a-(6 a-7 b) \sinh ^2(c+d x)\right ) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx\\ &=\frac {(6 a-7 b) \cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{48 d}+\frac {\cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^3}{8 d}-\frac {1}{48} \int \left (a+b \sinh ^2(c+d x)\right ) \left (a (12 a-7 b)-\left (24 a^2-64 a b+35 b^2\right ) \sinh ^2(c+d x)\right ) \, dx\\ &=-\frac {1}{128} \left (64 a^3-144 a^2 b+120 a b^2-35 b^3\right ) x+\frac {\left (96 a^3-376 a^2 b+360 a b^2-105 b^3\right ) \cosh (c+d x) \sinh (c+d x)}{384 d}+\frac {b \left (24 a^2-64 a b+35 b^2\right ) \cosh (c+d x) \sinh ^3(c+d x)}{192 d}+\frac {(6 a-7 b) \cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{48 d}+\frac {\cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^3}{8 d}\\ \end {align*}

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Mathematica [A]
time = 0.21, size = 130, normalized size = 0.72 \begin {gather*} \frac {-24 \left (64 a^3-144 a^2 b+120 a b^2-35 b^3\right ) (c+d x)+48 \left (16 a^3-48 a^2 b+45 a b^2-14 b^3\right ) \sinh (2 (c+d x))+24 b \left (12 a^2-18 a b+7 b^2\right ) \sinh (4 (c+d x))+16 (3 a-2 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[Sinh[c + d*x]^2*(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

(-24*(64*a^3 - 144*a^2*b + 120*a*b^2 - 35*b^3)*(c + d*x) + 48*(16*a^3 - 48*a^2*b + 45*a*b^2 - 14*b^3)*Sinh[2*(
c + d*x)] + 24*b*(12*a^2 - 18*a*b + 7*b^2)*Sinh[4*(c + d*x)] + 16*(3*a - 2*b)*b^2*Sinh[6*(c + d*x)] + 3*b^3*Si
nh[8*(c + d*x)])/(3072*d)

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Maple [A]
time = 1.19, size = 140, normalized size = 0.77

method	result	size
default	\(\frac {\left (-\frac {1}{16} b^{3}+\frac {3}{32} a \,b^{2}\right ) \sinh \left (6 d x +6 c \right )}{6 d}+\frac {\left (\frac {7}{32} b^{3}-\frac {9}{16} a \,b^{2}+\frac {3}{8} a^{2} b \right ) \sinh \left (4 d x +4 c \right )}{4 d}+\frac {\left (-\frac {7}{16} b^{3}+\frac {45}{32} a \,b^{2}-\frac {3}{2} a^{2} b +\frac {1}{2} a^{3}\right ) \sinh \left (2 d x +2 c \right )}{2 d}-\frac {a^{3} x}{2}+\frac {35 b^{3} x}{128}-\frac {15 a \,b^{2} x}{16}+\frac {9 a^{2} b x}{8}+\frac {b^{3} \sinh \left (8 d x +8 c \right )}{1024 d}\)	\(140\)
risch	\(\frac {35 b^{3} x}{128}-\frac {15 a \,b^{2} x}{16}+\frac {9 a^{2} b x}{8}-\frac {a^{3} x}{2}+\frac {b^{3} {\mathrm e}^{8 d x +8 c}}{2048 d}+\frac {b^{2} {\mathrm e}^{6 d x +6 c} a}{128 d}-\frac {b^{3} {\mathrm e}^{6 d x +6 c}}{192 d}+\frac {3 \,{\mathrm e}^{4 d x +4 c} a^{2} b}{64 d}-\frac {9 \,{\mathrm e}^{4 d x +4 c} a \,b^{2}}{128 d}+\frac {7 \,{\mathrm e}^{4 d x +4 c} b^{3}}{256 d}+\frac {{\mathrm e}^{2 d x +2 c} a^{3}}{8 d}-\frac {3 \,{\mathrm e}^{2 d x +2 c} a^{2} b}{8 d}+\frac {45 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}}{128 d}-\frac {7 \,{\mathrm e}^{2 d x +2 c} b^{3}}{64 d}-\frac {{\mathrm e}^{-2 d x -2 c} a^{3}}{8 d}+\frac {3 \,{\mathrm e}^{-2 d x -2 c} a^{2} b}{8 d}-\frac {45 \,{\mathrm e}^{-2 d x -2 c} a \,b^{2}}{128 d}+\frac {7 \,{\mathrm e}^{-2 d x -2 c} b^{3}}{64 d}-\frac {3 \,{\mathrm e}^{-4 d x -4 c} a^{2} b}{64 d}+\frac {9 \,{\mathrm e}^{-4 d x -4 c} a \,b^{2}}{128 d}-\frac {7 \,{\mathrm e}^{-4 d x -4 c} b^{3}}{256 d}-\frac {b^{2} {\mathrm e}^{-6 d x -6 c} a}{128 d}+\frac {b^{3} {\mathrm e}^{-6 d x -6 c}}{192 d}-\frac {b^{3} {\mathrm e}^{-8 d x -8 c}}{2048 d}\)	\(378\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(-1/16*b^3+3/32*a*b^2)/d*sinh(6*d*x+6*c)+1/4*(7/32*b^3-9/16*a*b^2+3/8*a^2*b)/d*sinh(4*d*x+4*c)+1/2*(-7/16*
b^3+45/32*a*b^2-3/2*a^2*b+1/2*a^3)*sinh(2*d*x+2*c)/d-1/2*a^3*x+35/128*b^3*x-15/16*a*b^2*x+9/8*a^2*b*x+1/1024*b
^3/d*sinh(8*d*x+8*c)

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Maxima [A]
time = 0.28, size = 306, normalized size = 1.69 \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 )} - \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 (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 b^{2} {\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^2*(a+b*sinh(d*x+c)^2)^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) - 1/8*
a^3*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) - 1/6144*b^3*((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*b^2*((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 = 0.40, size = 269, normalized size = 1.49 \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} - 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} - 2 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 12 \, {\left (12 \, a^{2} b - 18 \, a b^{2} + 7 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 3 \, {\left (64 \, a^{3} - 144 \, a^{2} b + 120 \, a b^{2} - 35 \, b^{3}\right )} d x + 3 \, {\left (b^{3} \cosh \left (d x + c\right )^{7} + 4 \, {\left (3 \, a b^{2} - 2 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 4 \, {\left (12 \, a^{2} b - 18 \, a b^{2} + 7 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 4 \, {\left (16 \, a^{3} - 48 \, a^{2} b + 45 \, a b^{2} - 14 \, 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(sinh(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 - 2*b^3)*cosh(d*x + c))*sin
h(d*x + c)^5 + (21*b^3*cosh(d*x + c)^5 + 40*(3*a*b^2 - 2*b^3)*cosh(d*x + c)^3 + 12*(12*a^2*b - 18*a*b^2 + 7*b^
3)*cosh(d*x + c))*sinh(d*x + c)^3 - 3*(64*a^3 - 144*a^2*b + 120*a*b^2 - 35*b^3)*d*x + 3*(b^3*cosh(d*x + c)^7 +
4*(3*a*b^2 - 2*b^3)*cosh(d*x + c)^5 + 4*(12*a^2*b - 18*a*b^2 + 7*b^3)*cosh(d*x + c)^3 + 4*(16*a^3 - 48*a^2*b
+ 45*a*b^2 - 14*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. 561 vs. \(2 (177) = 354\).
time = 1.05, size = 561, normalized size = 3.10 \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 {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 {15 a b^{2} x \sinh ^{6}{\left (c + d x \right )}}{16} - \frac {45 a b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{16} + \frac {45 a b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{16} - \frac {15 a b^{2} x \cosh ^{6}{\left (c + d x \right )}}{16} + \frac {33 a b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{16 d} - \frac {5 a b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{2 d} + \frac {15 a b^{2} \sinh {\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{16 d} + \frac {35 b^{3} x \sinh ^{8}{\left (c + d x \right )}}{128} - \frac {35 b^{3} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{32} + \frac {105 b^{3} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{64} - \frac {35 b^{3} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{32} + \frac {35 b^{3} x \cosh ^{8}{\left (c + d x \right )}}{128} + \frac {93 b^{3} \sinh ^{7}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{128 d} - \frac {511 b^{3} \sinh ^{5}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{384 d} + \frac {385 b^{3} \sinh ^{3}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{384 d} - \frac {35 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} \sinh ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(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) + 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) + 15*a*b**2*x*
sinh(c + d*x)**6/16 - 45*a*b**2*x*sinh(c + d*x)**4*cosh(c + d*x)**2/16 + 45*a*b**2*x*sinh(c + d*x)**2*cosh(c +
d*x)**4/16 - 15*a*b**2*x*cosh(c + d*x)**6/16 + 33*a*b**2*sinh(c + d*x)**5*cosh(c + d*x)/(16*d) - 5*a*b**2*sin
h(c + d*x)**3*cosh(c + d*x)**3/(2*d) + 15*a*b**2*sinh(c + d*x)*cosh(c + d*x)**5/(16*d) + 35*b**3*x*sinh(c + d*
x)**8/128 - 35*b**3*x*sinh(c + d*x)**6*cosh(c + d*x)**2/32 + 105*b**3*x*sinh(c + d*x)**4*cosh(c + d*x)**4/64 -
35*b**3*x*sinh(c + d*x)**2*cosh(c + d*x)**6/32 + 35*b**3*x*cosh(c + d*x)**8/128 + 93*b**3*sinh(c + d*x)**7*co
sh(c + d*x)/(128*d) - 511*b**3*sinh(c + d*x)**5*cosh(c + d*x)**3/(384*d) + 385*b**3*sinh(c + d*x)**3*cosh(c +
d*x)**5/(384*d) - 35*b**3*sinh(c + d*x)*cosh(c + d*x)**7/(128*d), Ne(d, 0)), (x*(a + b*sinh(c)**2)**3*sinh(c)*
*2, True))

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Giac [A]
time = 0.45, size = 251, normalized size = 1.39 \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} - 144 \, a^{2} b + 120 \, a b^{2} - 35 \, b^{3}\right )} x + \frac {{\left (3 \, a b^{2} - 2 \, b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )}}{384 \, d} + \frac {{\left (12 \, a^{2} b - 18 \, a b^{2} + 7 \, b^{3}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{256 \, d} + \frac {{\left (16 \, a^{3} - 48 \, a^{2} b + 45 \, a b^{2} - 14 \, b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{128 \, d} - \frac {{\left (16 \, a^{3} - 48 \, a^{2} b + 45 \, a b^{2} - 14 \, b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{128 \, d} - \frac {{\left (12 \, a^{2} b - 18 \, a b^{2} + 7 \, b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{256 \, d} - \frac {{\left (3 \, a b^{2} - 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(sinh(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 - 144*a^2*b + 120*a*b^2 - 35*b^3)
*x + 1/384*(3*a*b^2 - 2*b^3)*e^(6*d*x + 6*c)/d + 1/256*(12*a^2*b - 18*a*b^2 + 7*b^3)*e^(4*d*x + 4*c)/d + 1/128
*(16*a^3 - 48*a^2*b + 45*a*b^2 - 14*b^3)*e^(2*d*x + 2*c)/d - 1/128*(16*a^3 - 48*a^2*b + 45*a*b^2 - 14*b^3)*e^(
-2*d*x - 2*c)/d - 1/256*(12*a^2*b - 18*a*b^2 + 7*b^3)*e^(-4*d*x - 4*c)/d - 1/384*(3*a*b^2 - 2*b^3)*e^(-6*d*x -
6*c)/d

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Mupad [B]
time = 0.97, size = 181, normalized size = 1.00 \begin {gather*} \frac {96\,a^3\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )-84\,b^3\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+21\,b^3\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )-4\,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}+270\,a\,b^2\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )-288\,a^2\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )-54\,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+105\,b^3\,d\,x-360\,a\,b^2\,d\,x+432\,a^2\,b\,d\,x}{384\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(96*a^3*sinh(2*c + 2*d*x) - 84*b^3*sinh(2*c + 2*d*x) + 21*b^3*sinh(4*c + 4*d*x) - 4*b^3*sinh(6*c + 6*d*x) + (3
*b^3*sinh(8*c + 8*d*x))/8 + 270*a*b^2*sinh(2*c + 2*d*x) - 288*a^2*b*sinh(2*c + 2*d*x) - 54*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 + 105*b^3*d*x - 360*a*b^2*d*x + 43
2*a^2*b*d*x)/(384*d)

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