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

Optimal. Leaf size=67 \[ \frac{a^2 b \sinh ^3(c+d x)}{d}+\frac{a^3 \sinh (c+d x)}{d}+\frac{3 a b^2 \sinh ^5(c+d x)}{5 d}+\frac{b^3 \sinh ^7(c+d x)}{7 d} \]

[Out]

(a^3*Sinh[c + d*x])/d + (a^2*b*Sinh[c + d*x]^3)/d + (3*a*b^2*Sinh[c + d*x]^5)/(5*d) + (b^3*Sinh[c + d*x]^7)/(7
*d)

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Rubi [A]  time = 0.0454635, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3190, 194} \[ \frac{a^2 b \sinh ^3(c+d x)}{d}+\frac{a^3 \sinh (c+d x)}{d}+\frac{3 a b^2 \sinh ^5(c+d x)}{5 d}+\frac{b^3 \sinh ^7(c+d x)}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*Sinh[c + d*x])/d + (a^2*b*Sinh[c + d*x]^3)/d + (3*a*b^2*Sinh[c + d*x]^5)/(5*d) + (b^3*Sinh[c + d*x]^7)/(7
*d)

Rule 3190

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

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.11578, size = 59, normalized size = 0.88 \[ \frac{a^2 b \sinh ^3(c+d x)+a^3 \sinh (c+d x)+\frac{3}{5} a b^2 \sinh ^5(c+d x)+\frac{1}{7} b^3 \sinh ^7(c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*Sinh[c + d*x] + a^2*b*Sinh[c + d*x]^3 + (3*a*b^2*Sinh[c + d*x]^5)/5 + (b^3*Sinh[c + d*x]^7)/7)/d

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Maple [A]  time = 0.011, size = 56, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({\frac{{b}^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{7}}{7}}+{\frac{3\,a{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{5}}{5}}+{a}^{2}b \left ( \sinh \left ( dx+c \right ) \right ) ^{3}+{a}^{3}\sinh \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/7*b^3*sinh(d*x+c)^7+3/5*a*b^2*sinh(d*x+c)^5+a^2*b*sinh(d*x+c)^3+a^3*sinh(d*x+c))

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Maxima [A]  time = 1.05374, size = 85, normalized size = 1.27 \begin{align*} \frac{b^{3} \sinh \left (d x + c\right )^{7}}{7 \, d} + \frac{3 \, a b^{2} \sinh \left (d x + c\right )^{5}}{5 \, d} + \frac{a^{2} b \sinh \left (d x + c\right )^{3}}{d} + \frac{a^{3} \sinh \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^3*sinh(d*x + c)^7/d + 3/5*a*b^2*sinh(d*x + c)^5/d + a^2*b*sinh(d*x + c)^3/d + a^3*sinh(d*x + c)/d

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Fricas [B]  time = 1.54785, size = 512, normalized size = 7.64 \begin{align*} \frac{5 \, b^{3} \sinh \left (d x + c\right )^{7} + 7 \,{\left (15 \, b^{3} \cosh \left (d x + c\right )^{2} + 12 \, a b^{2} - 5 \, b^{3}\right )} \sinh \left (d x + c\right )^{5} + 35 \,{\left (5 \, b^{3} \cosh \left (d x + c\right )^{4} + 16 \, a^{2} b - 12 \, a b^{2} + 3 \, b^{3} + 2 \,{\left (12 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 35 \,{\left (b^{3} \cosh \left (d x + c\right )^{6} +{\left (12 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 64 \, a^{3} - 48 \, a^{2} b + 24 \, a b^{2} - 5 \, b^{3} + 3 \,{\left (16 \, a^{2} b - 12 \, a b^{2} + 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{2240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2240*(5*b^3*sinh(d*x + c)^7 + 7*(15*b^3*cosh(d*x + c)^2 + 12*a*b^2 - 5*b^3)*sinh(d*x + c)^5 + 35*(5*b^3*cosh
(d*x + c)^4 + 16*a^2*b - 12*a*b^2 + 3*b^3 + 2*(12*a*b^2 - 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 35*(b^3*co
sh(d*x + c)^6 + (12*a*b^2 - 5*b^3)*cosh(d*x + c)^4 + 64*a^3 - 48*a^2*b + 24*a*b^2 - 5*b^3 + 3*(16*a^2*b - 12*a
*b^2 + 3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/d

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Sympy [A]  time = 6.6694, size = 75, normalized size = 1.12 \begin{align*} \begin{cases} \frac{a^{3} \sinh{\left (c + d x \right )}}{d} + \frac{a^{2} b \sinh ^{3}{\left (c + d x \right )}}{d} + \frac{3 a b^{2} \sinh ^{5}{\left (c + d x \right )}}{5 d} + \frac{b^{3} \sinh ^{7}{\left (c + d x \right )}}{7 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right )^{3} \cosh{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**3*sinh(c + d*x)/d + a**2*b*sinh(c + d*x)**3/d + 3*a*b**2*sinh(c + d*x)**5/(5*d) + b**3*sinh(c +
d*x)**7/(7*d), Ne(d, 0)), (x*(a + b*sinh(c)**2)**3*cosh(c), True))

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Giac [B]  time = 1.2839, size = 387, normalized size = 5.78 \begin{align*} \frac{5 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} + 84 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 35 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 560 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} - 420 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 105 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 2240 \, a^{3} e^{\left (d x + c\right )} - 1680 \, a^{2} b e^{\left (d x + c\right )} + 840 \, a b^{2} e^{\left (d x + c\right )} - 175 \, b^{3} e^{\left (d x + c\right )} -{\left (2240 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 1680 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 840 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 175 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 560 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 420 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 105 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 84 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 35 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 5 \, b^{3}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{4480 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4480*(5*b^3*e^(7*d*x + 7*c) + 84*a*b^2*e^(5*d*x + 5*c) - 35*b^3*e^(5*d*x + 5*c) + 560*a^2*b*e^(3*d*x + 3*c)
- 420*a*b^2*e^(3*d*x + 3*c) + 105*b^3*e^(3*d*x + 3*c) + 2240*a^3*e^(d*x + c) - 1680*a^2*b*e^(d*x + c) + 840*a*
b^2*e^(d*x + c) - 175*b^3*e^(d*x + c) - (2240*a^3*e^(6*d*x + 6*c) - 1680*a^2*b*e^(6*d*x + 6*c) + 840*a*b^2*e^(
6*d*x + 6*c) - 175*b^3*e^(6*d*x + 6*c) + 560*a^2*b*e^(4*d*x + 4*c) - 420*a*b^2*e^(4*d*x + 4*c) + 105*b^3*e^(4*
d*x + 4*c) + 84*a*b^2*e^(2*d*x + 2*c) - 35*b^3*e^(2*d*x + 2*c) + 5*b^3)*e^(-7*d*x - 7*c))/d