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

Optimal. Leaf size=52 \[ \frac{1}{2} x \left (2 a^2-b^2\right )+\frac{2 a b \cosh (c+d x)}{d}+\frac{b^2 \sinh (c+d x) \cosh (c+d x)}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.0165329, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2644} \[ \frac{1}{2} x \left (2 a^2-b^2\right )+\frac{2 a b \cosh (c+d x)}{d}+\frac{b^2 \sinh (c+d x) \cosh (c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sinh[c + d*x])^2,x]

[Out]

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

Rule 2644

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^2, x_Symbol] :> Simp[((2*a^2 + b^2)*x)/2, x] + (-Simp[(2*a*b*Cos[c
+ d*x])/d, x] - Simp[(b^2*Cos[c + d*x]*Sin[c + d*x])/(2*d), x]) /; FreeQ[{a, b, c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0809233, size = 48, normalized size = 0.92 \[ \frac{2 \left (2 a^2-b^2\right ) (c+d x)+8 a b \cosh (c+d x)+b^2 \sinh (2 (c+d x))}{4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sinh[c + d*x])^2,x]

[Out]

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

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Maple [A]  time = 0.012, size = 51, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({b}^{2} \left ({\frac{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{2}}-{\frac{dx}{2}}-{\frac{c}{2}} \right ) +2\,ab\cosh \left ( dx+c \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,x)

[Out]

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

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Maxima [A]  time = 1.09643, size = 74, normalized size = 1.42 \begin{align*} -\frac{1}{8} \, b^{2}{\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 + \frac{2 \, a b \cosh \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.99934, size = 112, normalized size = 2.15 \begin{align*} \frac{b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) +{\left (2 \, a^{2} - b^{2}\right )} d x + 4 \, a b \cosh \left (d x + c\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.345343, size = 78, normalized size = 1.5 \begin{align*} \begin{cases} a^{2} x + \frac{2 a b \cosh{\left (c + d x \right )}}{d} + \frac{b^{2} x \sinh ^{2}{\left (c + d x \right )}}{2} - \frac{b^{2} x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac{b^{2} \sinh{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh{\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,x)

[Out]

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

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Giac [A]  time = 1.25374, size = 99, normalized size = 1.9 \begin{align*} \frac{b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a b e^{\left (d x + c\right )} + 4 \,{\left (2 \, a^{2} - b^{2}\right )}{\left (d x + c\right )} +{\left (8 \, a b e^{\left (d x + c\right )} - b^{2}\right )} e^{\left (-2 \, d x - 2 \, 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,x, algorithm="giac")

[Out]

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