3.14 \(\int (a+b \sinh ^2(c+d x))^2 \, dx\)

Optimal. Leaf size=72 \[ \frac{1}{8} x \left (8 a^2-8 a b+3 b^2\right )+\frac{b (8 a-3 b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{b^2 \sinh ^3(c+d x) \cosh (c+d x)}{4 d} \]

[Out]

((8*a^2 - 8*a*b + 3*b^2)*x)/8 + ((8*a - 3*b)*b*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (b^2*Cosh[c + d*x]*Sinh[c
+ d*x]^3)/(4*d)

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Rubi [A]  time = 0.0217836, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {3179} \[ \frac{1}{8} x \left (8 a^2-8 a b+3 b^2\right )+\frac{b (8 a-3 b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{b^2 \sinh ^3(c+d x) \cosh (c+d x)}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sinh[c + d*x]^2)^2,x]

[Out]

((8*a^2 - 8*a*b + 3*b^2)*x)/8 + ((8*a - 3*b)*b*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (b^2*Cosh[c + d*x]*Sinh[c
+ d*x]^3)/(4*d)

Rule 3179

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

Rubi steps

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

Mathematica [A]  time = 0.126727, size = 60, normalized size = 0.83 \[ \frac{4 \left (8 a^2-8 a b+3 b^2\right ) (c+d x)+8 b (2 a-b) \sinh (2 (c+d x))+b^2 \sinh (4 (c+d x))}{32 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(8*a^2 - 8*a*b + 3*b^2)*(c + d*x) + 8*(2*a - b)*b*Sinh[2*(c + d*x)] + b^2*Sinh[4*(c + d*x)])/(32*d)

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Maple [A]  time = 0.013, size = 79, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({b}^{2} \left ( \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{4}}-{\frac{3\,\sinh \left ( dx+c \right ) }{8}} \right ) \cosh \left ( dx+c \right ) +{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) +2\,ab \left ( 1/2\,\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) -1/2\,dx-c/2 \right ) +{a}^{2} \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(b^2*((1/4*sinh(d*x+c)^3-3/8*sinh(d*x+c))*cosh(d*x+c)+3/8*d*x+3/8*c)+2*a*b*(1/2*cosh(d*x+c)*sinh(d*x+c)-1/
2*d*x-1/2*c)+a^2*(d*x+c))

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Maxima [A]  time = 1.00849, size = 142, normalized size = 1.97 \begin{align*} \frac{1}{64} \, b^{2}{\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}{4} \, a b{\left (4 \, x - \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + a^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*b^2*(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) - 1/4*a*
b*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) + a^2*x

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Fricas [A]  time = 1.87806, size = 193, normalized size = 2.68 \begin{align*} \frac{b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} +{\left (8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right )} d x +{\left (b^{2} \cosh \left (d x + c\right )^{3} + 4 \,{\left (2 \, a b - b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(b^2*cosh(d*x + c)*sinh(d*x + c)^3 + (8*a^2 - 8*a*b + 3*b^2)*d*x + (b^2*cosh(d*x + c)^3 + 4*(2*a*b - b^2)*
cosh(d*x + c))*sinh(d*x + c))/d

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Sympy [A]  time = 1.27699, size = 168, normalized size = 2.33 \begin{align*} \begin{cases} a^{2} x + a b x \sinh ^{2}{\left (c + d x \right )} - a b x \cosh ^{2}{\left (c + d x \right )} + \frac{a b \sinh{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} + \frac{3 b^{2} x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac{3 b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac{3 b^{2} x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac{5 b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{8 d} - \frac{3 b^{2} \sinh{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right )^{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*x + a*b*x*sinh(c + d*x)**2 - a*b*x*cosh(c + d*x)**2 + a*b*sinh(c + d*x)*cosh(c + d*x)/d + 3*b*
*2*x*sinh(c + d*x)**4/8 - 3*b**2*x*sinh(c + d*x)**2*cosh(c + d*x)**2/4 + 3*b**2*x*cosh(c + d*x)**4/8 + 5*b**2*
sinh(c + d*x)**3*cosh(c + d*x)/(8*d) - 3*b**2*sinh(c + d*x)*cosh(c + d*x)**3/(8*d), Ne(d, 0)), (x*(a + b*sinh(
c)**2)**2, True))

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Giac [B]  time = 1.22594, size = 204, normalized size = 2.83 \begin{align*} \frac{b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 16 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 8 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 8 \,{\left (8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right )}{\left (d x + c\right )} -{\left (48 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 48 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 18 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 16 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 8 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(b^2*e^(4*d*x + 4*c) + 16*a*b*e^(2*d*x + 2*c) - 8*b^2*e^(2*d*x + 2*c) + 8*(8*a^2 - 8*a*b + 3*b^2)*(d*x +
c) - (48*a^2*e^(4*d*x + 4*c) - 48*a*b*e^(4*d*x + 4*c) + 18*b^2*e^(4*d*x + 4*c) + 16*a*b*e^(2*d*x + 2*c) - 8*b^
2*e^(2*d*x + 2*c) + b^2)*e^(-4*d*x - 4*c))/d