3.196 \(\int \sinh ^3(c+d x) (a+b \sinh ^4(c+d x))^2 \, dx\)

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

[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)

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Rubi [A]  time = 0.129451, antiderivative size = 120, normalized size of antiderivative = 1., 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 = {3215, 1153} \[ \frac{2 b (a+5 b) \cosh ^7(c+d x)}{7 d}-\frac{2 b (3 a+5 b) \cosh ^5(c+d x)}{5 d}+\frac{(a+b) (a+5 b) \cosh ^3(c+d x)}{3 d}-\frac{(a+b)^2 \cosh (c+d x)}{d}+\frac{b^2 \cosh ^{11}(c+d x)}{11 d}-\frac{5 b^2 \cosh ^9(c+d x)}{9 d} \]

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 3215

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]

Rule 1153

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]

Rubi steps

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

Mathematica [A]  time = 0.0635944, size = 207, normalized size = 1.72 \[ -\frac{3 a^2 \cosh (c+d x)}{4 d}+\frac{a^2 \cosh (3 (c+d x))}{12 d}-\frac{35 a b \cosh (c+d x)}{32 d}+\frac{7 a b \cosh (3 (c+d x))}{32 d}-\frac{7 a b \cosh (5 (c+d x))}{160 d}+\frac{a b \cosh (7 (c+d x))}{224 d}-\frac{231 b^2 \cosh (c+d x)}{512 d}+\frac{55 b^2 \cosh (3 (c+d x))}{512 d}-\frac{33 b^2 \cosh (5 (c+d x))}{1024 d}+\frac{55 b^2 \cosh (7 (c+d x))}{7168 d}-\frac{11 b^2 \cosh (9 (c+d x))}{9216 d}+\frac{b^2 \cosh (11 (c+d x))}{11264 d} \]

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 = 0.023, size = 132, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({b}^{2} \left ( -{\frac{256}{693}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{10}}{11}}-{\frac{10\, \left ( \sinh \left ( dx+c \right ) \right ) ^{8}}{99}}+{\frac{80\, \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}{693}}-{\frac{32\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{231}}+{\frac{128\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{693}} \right ) \cosh \left ( dx+c \right ) +2\,ab \left ( -{\frac{16}{35}}+1/7\, \left ( \sinh \left ( dx+c \right ) \right ) ^{6}-{\frac{6\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{35}}+{\frac{8\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{35}} \right ) \cosh \left ( dx+c \right ) +{a}^{2} \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \cosh \left ( dx+c \right ) \right ) } \end{align*}

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)

[Out]

1/d*(b^2*(-256/693+1/11*sinh(d*x+c)^10-10/99*sinh(d*x+c)^8+80/693*sinh(d*x+c)^6-32/231*sinh(d*x+c)^4+128/693*s
inh(d*x+c)^2)*cosh(d*x+c)+2*a*b*(-16/35+1/7*sinh(d*x+c)^6-6/35*sinh(d*x+c)^4+8/35*sinh(d*x+c)^2)*cosh(d*x+c)+a
^2*(-2/3+1/3*sinh(d*x+c)^2)*cosh(d*x+c))

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Maxima [B]  time = 1.04615, size = 414, normalized size = 3.45 \begin{align*} -\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{align*}

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]  time = 1.63243, size = 1111, normalized size = 9.26 \begin{align*} \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{align*}

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 [A]  time = 74.8908, size = 280, normalized size = 2.33 \begin{align*} \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{align*}

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]  time = 1.31993, size = 443, normalized size = 3.69 \begin{align*} \frac{315 \, b^{2} e^{\left (11 \, d x + 11 \, c\right )} - 4235 \, b^{2} e^{\left (9 \, d x + 9 \, c\right )} + 15840 \, a b e^{\left (7 \, d x + 7 \, c\right )} + 27225 \, b^{2} e^{\left (7 \, d x + 7 \, c\right )} - 155232 \, a b e^{\left (5 \, d x + 5 \, c\right )} - 114345 \, b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 295680 \, a^{2} e^{\left (3 \, d x + 3 \, c\right )} + 776160 \, a b e^{\left (3 \, d x + 3 \, c\right )} + 381150 \, b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 2661120 \, a^{2} e^{\left (d x + c\right )} - 3880800 \, a b e^{\left (d x + c\right )} - 1600830 \, b^{2} e^{\left (d x + c\right )} -{\left (2661120 \, a^{2} e^{\left (10 \, d x + 10 \, c\right )} + 3880800 \, a b e^{\left (10 \, d x + 10 \, c\right )} + 1600830 \, b^{2} e^{\left (10 \, d x + 10 \, c\right )} - 295680 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} - 776160 \, a b e^{\left (8 \, d x + 8 \, c\right )} - 381150 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 155232 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 114345 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 15840 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 27225 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 4235 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 315 \, b^{2}\right )} e^{\left (-11 \, d x - 11 \, c\right )}}{7096320 \, d} \end{align*}

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]

1/7096320*(315*b^2*e^(11*d*x + 11*c) - 4235*b^2*e^(9*d*x + 9*c) + 15840*a*b*e^(7*d*x + 7*c) + 27225*b^2*e^(7*d
*x + 7*c) - 155232*a*b*e^(5*d*x + 5*c) - 114345*b^2*e^(5*d*x + 5*c) + 295680*a^2*e^(3*d*x + 3*c) + 776160*a*b*
e^(3*d*x + 3*c) + 381150*b^2*e^(3*d*x + 3*c) - 2661120*a^2*e^(d*x + c) - 3880800*a*b*e^(d*x + c) - 1600830*b^2
*e^(d*x + c) - (2661120*a^2*e^(10*d*x + 10*c) + 3880800*a*b*e^(10*d*x + 10*c) + 1600830*b^2*e^(10*d*x + 10*c)
- 295680*a^2*e^(8*d*x + 8*c) - 776160*a*b*e^(8*d*x + 8*c) - 381150*b^2*e^(8*d*x + 8*c) + 155232*a*b*e^(6*d*x +
6*c) + 114345*b^2*e^(6*d*x + 6*c) - 15840*a*b*e^(4*d*x + 4*c) - 27225*b^2*e^(4*d*x + 4*c) + 4235*b^2*e^(2*d*x
+ 2*c) - 315*b^2)*e^(-11*d*x - 11*c))/d