3.2.96 \(\int \sinh ^3(c+d x) (a+b \sinh ^4(c+d x))^2 \, dx\) [196]

Optimal. Leaf size=120 \[ -\frac {(a+b)^2 \cosh (c+d x)}{d}+\frac {(a+b) (a+5 b) \cosh ^3(c+d x)}{3 d}-\frac {2 b (3 a+5 b) \cosh ^5(c+d x)}{5 d}+\frac {2 b (a+5 b) \cosh ^7(c+d x)}{7 d}-\frac {5 b^2 \cosh ^9(c+d x)}{9 d}+\frac {b^2 \cosh ^{11}(c+d x)}{11 d} \]

[Out]

-(a+b)^2*cosh(d*x+c)/d+1/3*(a+b)*(a+5*b)*cosh(d*x+c)^3/d-2/5*b*(3*a+5*b)*cosh(d*x+c)^5/d+2/7*b*(a+5*b)*cosh(d*
x+c)^7/d-5/9*b^2*cosh(d*x+c)^9/d+1/11*b^2*cosh(d*x+c)^11/d

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Rubi [A]
time = 0.09, antiderivative size = 120, 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 = {3294, 1167} \begin {gather*} \frac {2 b (a+5 b) \cosh ^7(c+d x)}{7 d}-\frac {2 b (3 a+5 b) \cosh ^5(c+d x)}{5 d}+\frac {(a+b) (a+5 b) \cosh ^3(c+d x)}{3 d}-\frac {(a+b)^2 \cosh (c+d x)}{d}+\frac {b^2 \cosh ^{11}(c+d x)}{11 d}-\frac {5 b^2 \cosh ^9(c+d x)}{9 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^4)^2,x]

[Out]

-(((a + b)^2*Cosh[c + d*x])/d) + ((a + b)*(a + 5*b)*Cosh[c + d*x]^3)/(3*d) - (2*b*(3*a + 5*b)*Cosh[c + d*x]^5)
/(5*d) + (2*b*(a + 5*b)*Cosh[c + d*x]^7)/(7*d) - (5*b^2*Cosh[c + d*x]^9)/(9*d) + (b^2*Cosh[c + d*x]^11)/(11*d)

Rule 1167

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, 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] && IGtQ[q, -2]

Rule 3294

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \sinh ^3(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx &=-\frac {\text {Subst}\left (\int \left (1-x^2\right ) \left (a+b-2 b x^2+b x^4\right )^2 \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \left ((a+b)^2+(-a-5 b) (a+b) x^2+2 b (3 a+5 b) x^4-2 b (a+5 b) x^6+5 b^2 x^8-b^2 x^{10}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {(a+b)^2 \cosh (c+d x)}{d}+\frac {(a+b) (a+5 b) \cosh ^3(c+d x)}{3 d}-\frac {2 b (3 a+5 b) \cosh ^5(c+d x)}{5 d}+\frac {2 b (a+5 b) \cosh ^7(c+d x)}{7 d}-\frac {5 b^2 \cosh ^9(c+d x)}{9 d}+\frac {b^2 \cosh ^{11}(c+d x)}{11 d}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 207, normalized size = 1.72 \begin {gather*} -\frac {3 a^2 \cosh (c+d x)}{4 d}-\frac {35 a b \cosh (c+d x)}{32 d}-\frac {231 b^2 \cosh (c+d x)}{512 d}+\frac {a^2 \cosh (3 (c+d x))}{12 d}+\frac {7 a b \cosh (3 (c+d x))}{32 d}+\frac {55 b^2 \cosh (3 (c+d x))}{512 d}-\frac {7 a b \cosh (5 (c+d x))}{160 d}-\frac {33 b^2 \cosh (5 (c+d x))}{1024 d}+\frac {a b \cosh (7 (c+d x))}{224 d}+\frac {55 b^2 \cosh (7 (c+d x))}{7168 d}-\frac {11 b^2 \cosh (9 (c+d x))}{9216 d}+\frac {b^2 \cosh (11 (c+d x))}{11264 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^4)^2,x]

[Out]

(-3*a^2*Cosh[c + d*x])/(4*d) - (35*a*b*Cosh[c + d*x])/(32*d) - (231*b^2*Cosh[c + d*x])/(512*d) + (a^2*Cosh[3*(
c + d*x)])/(12*d) + (7*a*b*Cosh[3*(c + d*x)])/(32*d) + (55*b^2*Cosh[3*(c + d*x)])/(512*d) - (7*a*b*Cosh[5*(c +
d*x)])/(160*d) - (33*b^2*Cosh[5*(c + d*x)])/(1024*d) + (a*b*Cosh[7*(c + d*x)])/(224*d) + (55*b^2*Cosh[7*(c +
d*x)])/(7168*d) - (11*b^2*Cosh[9*(c + d*x)])/(9216*d) + (b^2*Cosh[11*(c + d*x)])/(11264*d)

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Maple [A]
time = 0.85, size = 138, normalized size = 1.15

method	result	size
default	\(\frac {\left (-\frac {165}{1024} b^{2}-\frac {7}{32} a b \right ) \cosh \left (5 d x +5 c \right )}{5 d}+\frac {\left (\frac {55}{1024} b^{2}+\frac {1}{32} a b \right ) \cosh \left (7 d x +7 c \right )}{7 d}+\frac {\left (-\frac {231}{512} b^{2}-\frac {35}{32} a b -\frac {3}{4} a^{2}\right ) \cosh \left (d x +c \right )}{d}+\frac {\left (\frac {165}{512} b^{2}+\frac {21}{32} a b +\frac {1}{4} a^{2}\right ) \cosh \left (3 d x +3 c \right )}{3 d}-\frac {11 b^{2} \cosh \left (9 d x +9 c \right )}{9216 d}+\frac {b^{2} \cosh \left (11 d x +11 c \right )}{11264 d}\)	\(138\)
risch	\(\frac {b^{2} {\mathrm e}^{11 d x +11 c}}{22528 d}-\frac {11 b^{2} {\mathrm e}^{9 d x +9 c}}{18432 d}+\frac {55 b^{2} {\mathrm e}^{7 d x +7 c}}{14336 d}+\frac {b \,{\mathrm e}^{7 d x +7 c} a}{448 d}-\frac {33 b^{2} {\mathrm e}^{5 d x +5 c}}{2048 d}-\frac {7 b \,{\mathrm e}^{5 d x +5 c} a}{320 d}+\frac {{\mathrm e}^{3 d x +3 c} a^{2}}{24 d}+\frac {7 \,{\mathrm e}^{3 d x +3 c} a b}{64 d}+\frac {55 \,{\mathrm e}^{3 d x +3 c} b^{2}}{1024 d}-\frac {3 \,{\mathrm e}^{d x +c} a^{2}}{8 d}-\frac {35 a b \,{\mathrm e}^{d x +c}}{64 d}-\frac {231 \,{\mathrm e}^{d x +c} b^{2}}{1024 d}-\frac {3 \,{\mathrm e}^{-d x -c} a^{2}}{8 d}-\frac {35 \,{\mathrm e}^{-d x -c} a b}{64 d}-\frac {231 \,{\mathrm e}^{-d x -c} b^{2}}{1024 d}+\frac {{\mathrm e}^{-3 d x -3 c} a^{2}}{24 d}+\frac {7 \,{\mathrm e}^{-3 d x -3 c} a b}{64 d}+\frac {55 \,{\mathrm e}^{-3 d x -3 c} b^{2}}{1024 d}-\frac {33 b^{2} {\mathrm e}^{-5 d x -5 c}}{2048 d}-\frac {7 b \,{\mathrm e}^{-5 d x -5 c} a}{320 d}+\frac {55 b^{2} {\mathrm e}^{-7 d x -7 c}}{14336 d}+\frac {b \,{\mathrm e}^{-7 d x -7 c} a}{448 d}-\frac {11 b^{2} {\mathrm e}^{-9 d x -9 c}}{18432 d}+\frac {b^{2} {\mathrm e}^{-11 d x -11 c}}{22528 d}\)	\(393\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(-165/1024*b^2-7/32*a*b)/d*cosh(5*d*x+5*c)+1/7*(55/1024*b^2+1/32*a*b)/d*cosh(7*d*x+7*c)+(-231/512*b^2-35/3
2*a*b-3/4*a^2)/d*cosh(d*x+c)+1/3*(165/512*b^2+21/32*a*b+1/4*a^2)/d*cosh(3*d*x+3*c)-11/9216*b^2/d*cosh(9*d*x+9*
c)+1/11264*b^2/d*cosh(11*d*x+11*c)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 307 vs. \(2 (110) = 220\).
time = 0.27, size = 307, normalized size = 2.56 \begin {gather*} -\frac {1}{1419264} \, b^{2} {\left (\frac {{\left (847 \, e^{\left (-2 \, d x - 2 \, c\right )} - 5445 \, e^{\left (-4 \, d x - 4 \, c\right )} + 22869 \, e^{\left (-6 \, d x - 6 \, c\right )} - 76230 \, e^{\left (-8 \, d x - 8 \, c\right )} + 320166 \, e^{\left (-10 \, d x - 10 \, c\right )} - 63\right )} e^{\left (11 \, d x + 11 \, c\right )}}{d} + \frac {320166 \, e^{\left (-d x - c\right )} - 76230 \, e^{\left (-3 \, d x - 3 \, c\right )} + 22869 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5445 \, e^{\left (-7 \, d x - 7 \, c\right )} + 847 \, e^{\left (-9 \, d x - 9 \, c\right )} - 63 \, e^{\left (-11 \, d x - 11 \, c\right )}}{d}\right )} - \frac {1}{2240} \, a b {\left (\frac {{\left (49 \, e^{\left (-2 \, d x - 2 \, c\right )} - 245 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1225 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac {1225 \, e^{\left (-d x - c\right )} - 245 \, e^{\left (-3 \, d x - 3 \, c\right )} + 49 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac {1}{24} \, a^{2} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^3*(a+b*sinh(d*x+c)^4)^2,x, algorithm="maxima")

[Out]

-1/1419264*b^2*((847*e^(-2*d*x - 2*c) - 5445*e^(-4*d*x - 4*c) + 22869*e^(-6*d*x - 6*c) - 76230*e^(-8*d*x - 8*c
) + 320166*e^(-10*d*x - 10*c) - 63)*e^(11*d*x + 11*c)/d + (320166*e^(-d*x - c) - 76230*e^(-3*d*x - 3*c) + 2286
9*e^(-5*d*x - 5*c) - 5445*e^(-7*d*x - 7*c) + 847*e^(-9*d*x - 9*c) - 63*e^(-11*d*x - 11*c))/d) - 1/2240*a*b*((4
9*e^(-2*d*x - 2*c) - 245*e^(-4*d*x - 4*c) + 1225*e^(-6*d*x - 6*c) - 5)*e^(7*d*x + 7*c)/d + (1225*e^(-d*x - c)
- 245*e^(-3*d*x - 3*c) + 49*e^(-5*d*x - 5*c) - 5*e^(-7*d*x - 7*c))/d) + 1/24*a^2*(e^(3*d*x + 3*c)/d - 9*e^(d*x
+ c)/d - 9*e^(-d*x - c)/d + e^(-3*d*x - 3*c)/d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 404 vs. \(2 (110) = 220\).
time = 0.38, size = 404, normalized size = 3.37 \begin {gather*} \frac {315 \, b^{2} \cosh \left (d x + c\right )^{11} + 3465 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{10} - 4235 \, b^{2} \cosh \left (d x + c\right )^{9} + 3465 \, {\left (15 \, b^{2} \cosh \left (d x + c\right )^{3} - 11 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{8} + 495 \, {\left (32 \, a b + 55 \, b^{2}\right )} \cosh \left (d x + c\right )^{7} + 1155 \, {\left (126 \, b^{2} \cosh \left (d x + c\right )^{5} - 308 \, b^{2} \cosh \left (d x + c\right )^{3} + 3 \, {\left (32 \, a b + 55 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} - 693 \, {\left (224 \, a b + 165 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} + 3465 \, {\left (30 \, b^{2} \cosh \left (d x + c\right )^{7} - 154 \, b^{2} \cosh \left (d x + c\right )^{5} + 5 \, {\left (32 \, a b + 55 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - {\left (224 \, a b + 165 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 2310 \, {\left (128 \, a^{2} + 336 \, a b + 165 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3465 \, {\left (5 \, b^{2} \cosh \left (d x + c\right )^{9} - 44 \, b^{2} \cosh \left (d x + c\right )^{7} + 3 \, {\left (32 \, a b + 55 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} - 2 \, {\left (224 \, a b + 165 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 2 \, {\left (128 \, a^{2} + 336 \, a b + 165 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 6930 \, {\left (384 \, a^{2} + 560 \, a b + 231 \, b^{2}\right )} \cosh \left (d x + c\right )}{3548160 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^3*(a+b*sinh(d*x+c)^4)^2,x, algorithm="fricas")

[Out]

1/3548160*(315*b^2*cosh(d*x + c)^11 + 3465*b^2*cosh(d*x + c)*sinh(d*x + c)^10 - 4235*b^2*cosh(d*x + c)^9 + 346
5*(15*b^2*cosh(d*x + c)^3 - 11*b^2*cosh(d*x + c))*sinh(d*x + c)^8 + 495*(32*a*b + 55*b^2)*cosh(d*x + c)^7 + 11
55*(126*b^2*cosh(d*x + c)^5 - 308*b^2*cosh(d*x + c)^3 + 3*(32*a*b + 55*b^2)*cosh(d*x + c))*sinh(d*x + c)^6 - 6
93*(224*a*b + 165*b^2)*cosh(d*x + c)^5 + 3465*(30*b^2*cosh(d*x + c)^7 - 154*b^2*cosh(d*x + c)^5 + 5*(32*a*b +
55*b^2)*cosh(d*x + c)^3 - (224*a*b + 165*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 + 2310*(128*a^2 + 336*a*b + 165*b
^2)*cosh(d*x + c)^3 + 3465*(5*b^2*cosh(d*x + c)^9 - 44*b^2*cosh(d*x + c)^7 + 3*(32*a*b + 55*b^2)*cosh(d*x + c)
^5 - 2*(224*a*b + 165*b^2)*cosh(d*x + c)^3 + 2*(128*a^2 + 336*a*b + 165*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 -
6930*(384*a^2 + 560*a*b + 231*b^2)*cosh(d*x + c))/d

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (109) = 218\).
time = 2.99, size = 280, normalized size = 2.33 \begin {gather*} \begin {cases} \frac {a^{2} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {2 a^{2} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {2 a b \sinh ^{6}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {4 a b \sinh ^{4}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {16 a b \sinh ^{2}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac {32 a b \cosh ^{7}{\left (c + d x \right )}}{35 d} + \frac {b^{2} \sinh ^{10}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {10 b^{2} \sinh ^{8}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {16 b^{2} \sinh ^{6}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{3 d} - \frac {32 b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{7 d} + \frac {128 b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{63 d} - \frac {256 b^{2} \cosh ^{11}{\left (c + d x \right )}}{693 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{4}{\left (c \right )}\right )^{2} \sinh ^{3}{\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)**3*(a+b*sinh(d*x+c)**4)**2,x)

[Out]

Piecewise((a**2*sinh(c + d*x)**2*cosh(c + d*x)/d - 2*a**2*cosh(c + d*x)**3/(3*d) + 2*a*b*sinh(c + d*x)**6*cosh
(c + d*x)/d - 4*a*b*sinh(c + d*x)**4*cosh(c + d*x)**3/d + 16*a*b*sinh(c + d*x)**2*cosh(c + d*x)**5/(5*d) - 32*
a*b*cosh(c + d*x)**7/(35*d) + b**2*sinh(c + d*x)**10*cosh(c + d*x)/d - 10*b**2*sinh(c + d*x)**8*cosh(c + d*x)*
*3/(3*d) + 16*b**2*sinh(c + d*x)**6*cosh(c + d*x)**5/(3*d) - 32*b**2*sinh(c + d*x)**4*cosh(c + d*x)**7/(7*d) +
128*b**2*sinh(c + d*x)**2*cosh(c + d*x)**9/(63*d) - 256*b**2*cosh(c + d*x)**11/(693*d), Ne(d, 0)), (x*(a + b*
sinh(c)**4)**2*sinh(c)**3, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (110) = 220\).
time = 0.43, size = 278, normalized size = 2.32 \begin {gather*} \frac {b^{2} e^{\left (11 \, d x + 11 \, c\right )}}{22528 \, d} - \frac {11 \, b^{2} e^{\left (9 \, d x + 9 \, c\right )}}{18432 \, d} - \frac {11 \, b^{2} e^{\left (-9 \, d x - 9 \, c\right )}}{18432 \, d} + \frac {b^{2} e^{\left (-11 \, d x - 11 \, c\right )}}{22528 \, d} + \frac {{\left (32 \, a b + 55 \, b^{2}\right )} e^{\left (7 \, d x + 7 \, c\right )}}{14336 \, d} - \frac {{\left (224 \, a b + 165 \, b^{2}\right )} e^{\left (5 \, d x + 5 \, c\right )}}{10240 \, d} + \frac {{\left (128 \, a^{2} + 336 \, a b + 165 \, b^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{3072 \, d} - \frac {{\left (384 \, a^{2} + 560 \, a b + 231 \, b^{2}\right )} e^{\left (d x + c\right )}}{1024 \, d} - \frac {{\left (384 \, a^{2} + 560 \, a b + 231 \, b^{2}\right )} e^{\left (-d x - c\right )}}{1024 \, d} + \frac {{\left (128 \, a^{2} + 336 \, a b + 165 \, b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{3072 \, d} - \frac {{\left (224 \, a b + 165 \, b^{2}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{10240 \, d} + \frac {{\left (32 \, a b + 55 \, b^{2}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{14336 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^3*(a+b*sinh(d*x+c)^4)^2,x, algorithm="giac")

[Out]

1/22528*b^2*e^(11*d*x + 11*c)/d - 11/18432*b^2*e^(9*d*x + 9*c)/d - 11/18432*b^2*e^(-9*d*x - 9*c)/d + 1/22528*b
^2*e^(-11*d*x - 11*c)/d + 1/14336*(32*a*b + 55*b^2)*e^(7*d*x + 7*c)/d - 1/10240*(224*a*b + 165*b^2)*e^(5*d*x +
5*c)/d + 1/3072*(128*a^2 + 336*a*b + 165*b^2)*e^(3*d*x + 3*c)/d - 1/1024*(384*a^2 + 560*a*b + 231*b^2)*e^(d*x
+ c)/d - 1/1024*(384*a^2 + 560*a*b + 231*b^2)*e^(-d*x - c)/d + 1/3072*(128*a^2 + 336*a*b + 165*b^2)*e^(-3*d*x
- 3*c)/d - 1/10240*(224*a*b + 165*b^2)*e^(-5*d*x - 5*c)/d + 1/14336*(32*a*b + 55*b^2)*e^(-7*d*x - 7*c)/d

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Mupad [B]
time = 0.36, size = 150, normalized size = 1.25 \begin {gather*} -\frac {-\frac {a^2\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}+a^2\,\mathrm {cosh}\left (c+d\,x\right )-\frac {2\,a\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}+\frac {6\,a\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-2\,a\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^3+2\,a\,b\,\mathrm {cosh}\left (c+d\,x\right )-\frac {b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^{11}}{11}+\frac {5\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^9}{9}-\frac {10\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}+2\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^5-\frac {5\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}+b^2\,\mathrm {cosh}\left (c+d\,x\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^2*cosh(c + d*x) + b^2*cosh(c + d*x) - (a^2*cosh(c + d*x)^3)/3 - (5*b^2*cosh(c + d*x)^3)/3 + 2*b^2*cosh(c +
d*x)^5 - (10*b^2*cosh(c + d*x)^7)/7 + (5*b^2*cosh(c + d*x)^9)/9 - (b^2*cosh(c + d*x)^11)/11 + 2*a*b*cosh(c +
d*x) - 2*a*b*cosh(c + d*x)^3 + (6*a*b*cosh(c + d*x)^5)/5 - (2*a*b*cosh(c + d*x)^7)/7)/d

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