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

Optimal. Leaf size=134 \[ \frac{3 d^2 (c+d x) \sinh (a+b x) \cosh (a+b x)}{4 b^3}-\frac{3 d (c+d x)^2 \cosh ^2(a+b x)}{4 b^2}-\frac{3 d^3 \cosh ^2(a+b x)}{8 b^4}+\frac{(c+d x)^3 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac{3 c d^2 x}{4 b^2}+\frac{3 d^3 x^2}{8 b^2}+\frac{(c+d x)^4}{8 d} \]

[Out]

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

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Rubi [A]  time = 0.0727015, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {3311, 32, 3310} \[ \frac{3 d^2 (c+d x) \sinh (a+b x) \cosh (a+b x)}{4 b^3}-\frac{3 d (c+d x)^2 \cosh ^2(a+b x)}{4 b^2}-\frac{3 d^3 \cosh ^2(a+b x)}{8 b^4}+\frac{(c+d x)^3 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac{3 c d^2 x}{4 b^2}+\frac{3 d^3 x^2}{8 b^2}+\frac{(c+d x)^4}{8 d} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3*Cosh[a + b*x]^2,x]

[Out]

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

Rule 3311

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*m*(c + d*x)^(m - 1)*(
b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*(c + d*x)^m*Cos[e +
f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 3310

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*(b*Sin[e + f*x])^n)/(f^2*n
^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[(b*(c + d*x)*Cos[e + f*
x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

Rubi steps

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

Mathematica [A]  time = 0.410089, size = 104, normalized size = 0.78 \[ \frac{2 b (c+d x) \sinh (2 (a+b x)) \left (2 b^2 (c+d x)^2+3 d^2\right )-3 d \cosh (2 (a+b x)) \left (2 b^2 (c+d x)^2+d^2\right )+2 b^4 x \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right )}{16 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b^4*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) - 3*d*(d^2 + 2*b^2*(c + d*x)^2)*Cosh[2*(a + b*x)] + 2*b*(
c + d*x)*(3*d^2 + 2*b^2*(c + d*x)^2)*Sinh[2*(a + b*x)])/(16*b^4)

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Maple [B]  time = 0.01, size = 523, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(1/b^3*d^3*(1/2*(b*x+a)^3*cosh(b*x+a)*sinh(b*x+a)+1/8*(b*x+a)^4-3/4*(b*x+a)^2*cosh(b*x+a)^2+3/4*(b*x+a)*co
sh(b*x+a)*sinh(b*x+a)+3/8*(b*x+a)^2-3/8*cosh(b*x+a)^2)-3/b^3*d^3*a*(1/2*(b*x+a)^2*cosh(b*x+a)*sinh(b*x+a)+1/6*
(b*x+a)^3-1/2*(b*x+a)*cosh(b*x+a)^2+1/4*cosh(b*x+a)*sinh(b*x+a)+1/4*b*x+1/4*a)+3/b^3*d^3*a^2*(1/2*(b*x+a)*cosh
(b*x+a)*sinh(b*x+a)+1/4*(b*x+a)^2-1/4*cosh(b*x+a)^2)-1/b^3*d^3*a^3*(1/2*cosh(b*x+a)*sinh(b*x+a)+1/2*b*x+1/2*a)
+3/b^2*c*d^2*(1/2*(b*x+a)^2*cosh(b*x+a)*sinh(b*x+a)+1/6*(b*x+a)^3-1/2*(b*x+a)*cosh(b*x+a)^2+1/4*cosh(b*x+a)*si
nh(b*x+a)+1/4*b*x+1/4*a)-6/b^2*c*d^2*a*(1/2*(b*x+a)*cosh(b*x+a)*sinh(b*x+a)+1/4*(b*x+a)^2-1/4*cosh(b*x+a)^2)+3
/b^2*c*d^2*a^2*(1/2*cosh(b*x+a)*sinh(b*x+a)+1/2*b*x+1/2*a)+3/b*c^2*d*(1/2*(b*x+a)*cosh(b*x+a)*sinh(b*x+a)+1/4*
(b*x+a)^2-1/4*cosh(b*x+a)^2)-3/b*c^2*d*a*(1/2*cosh(b*x+a)*sinh(b*x+a)+1/2*b*x+1/2*a)+c^3*(1/2*cosh(b*x+a)*sinh
(b*x+a)+1/2*b*x+1/2*a))

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Maxima [B]  time = 1.10515, size = 355, normalized size = 2.65 \begin{align*} \frac{3}{16} \,{\left (4 \, x^{2} + \frac{{\left (2 \, b x e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{2}} - \frac{{\left (2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{2}}\right )} c^{2} d + \frac{1}{16} \,{\left (8 \, x^{3} + \frac{3 \,{\left (2 \, b^{2} x^{2} e^{\left (2 \, a\right )} - 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{3}} - \frac{3 \,{\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{3}}\right )} c d^{2} + \frac{1}{32} \,{\left (4 \, x^{4} + \frac{{\left (4 \, b^{3} x^{3} e^{\left (2 \, a\right )} - 6 \, b^{2} x^{2} e^{\left (2 \, a\right )} + 6 \, b x e^{\left (2 \, a\right )} - 3 \, e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{4}} - \frac{{\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{4}}\right )} d^{3} + \frac{1}{8} \, c^{3}{\left (4 \, x + \frac{e^{\left (2 \, b x + 2 \, a\right )}}{b} - \frac{e^{\left (-2 \, b x - 2 \, a\right )}}{b}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/16*(4*x^2 + (2*b*x*e^(2*a) - e^(2*a))*e^(2*b*x)/b^2 - (2*b*x + 1)*e^(-2*b*x - 2*a)/b^2)*c^2*d + 1/16*(8*x^3
+ 3*(2*b^2*x^2*e^(2*a) - 2*b*x*e^(2*a) + e^(2*a))*e^(2*b*x)/b^3 - 3*(2*b^2*x^2 + 2*b*x + 1)*e^(-2*b*x - 2*a)/b
^3)*c*d^2 + 1/32*(4*x^4 + (4*b^3*x^3*e^(2*a) - 6*b^2*x^2*e^(2*a) + 6*b*x*e^(2*a) - 3*e^(2*a))*e^(2*b*x)/b^4 -
(4*b^3*x^3 + 6*b^2*x^2 + 6*b*x + 3)*e^(-2*b*x - 2*a)/b^4)*d^3 + 1/8*c^3*(4*x + e^(2*b*x + 2*a)/b - e^(-2*b*x -
2*a)/b)

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Fricas [A]  time = 1.79494, size = 458, normalized size = 3.42 \begin{align*} \frac{2 \, b^{4} d^{3} x^{4} + 8 \, b^{4} c d^{2} x^{3} + 12 \, b^{4} c^{2} d x^{2} + 8 \, b^{4} c^{3} x - 3 \,{\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d + d^{3}\right )} \cosh \left (b x + a\right )^{2} + 4 \,{\left (2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 2 \, b^{3} c^{3} + 3 \, b c d^{2} + 3 \,{\left (2 \, b^{3} c^{2} d + b d^{3}\right )} x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - 3 \,{\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d + d^{3}\right )} \sinh \left (b x + a\right )^{2}}{16 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(2*b^4*d^3*x^4 + 8*b^4*c*d^2*x^3 + 12*b^4*c^2*d*x^2 + 8*b^4*c^3*x - 3*(2*b^2*d^3*x^2 + 4*b^2*c*d^2*x + 2*
b^2*c^2*d + d^3)*cosh(b*x + a)^2 + 4*(2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 + 2*b^3*c^3 + 3*b*c*d^2 + 3*(2*b^3*c^2*d
+ b*d^3)*x)*cosh(b*x + a)*sinh(b*x + a) - 3*(2*b^2*d^3*x^2 + 4*b^2*c*d^2*x + 2*b^2*c^2*d + d^3)*sinh(b*x + a)
^2)/b^4

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Sympy [A]  time = 3.50572, size = 456, normalized size = 3.4 \begin{align*} \begin{cases} - \frac{c^{3} x \sinh ^{2}{\left (a + b x \right )}}{2} + \frac{c^{3} x \cosh ^{2}{\left (a + b x \right )}}{2} - \frac{3 c^{2} d x^{2} \sinh ^{2}{\left (a + b x \right )}}{4} + \frac{3 c^{2} d x^{2} \cosh ^{2}{\left (a + b x \right )}}{4} - \frac{c d^{2} x^{3} \sinh ^{2}{\left (a + b x \right )}}{2} + \frac{c d^{2} x^{3} \cosh ^{2}{\left (a + b x \right )}}{2} - \frac{d^{3} x^{4} \sinh ^{2}{\left (a + b x \right )}}{8} + \frac{d^{3} x^{4} \cosh ^{2}{\left (a + b x \right )}}{8} + \frac{c^{3} \sinh{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{2 b} + \frac{3 c^{2} d x \sinh{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{2 b} + \frac{3 c d^{2} x^{2} \sinh{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{2 b} + \frac{d^{3} x^{3} \sinh{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{2 b} - \frac{3 c^{2} d \sinh ^{2}{\left (a + b x \right )}}{4 b^{2}} - \frac{3 c d^{2} x \sinh ^{2}{\left (a + b x \right )}}{4 b^{2}} - \frac{3 c d^{2} x \cosh ^{2}{\left (a + b x \right )}}{4 b^{2}} - \frac{3 d^{3} x^{2} \sinh ^{2}{\left (a + b x \right )}}{8 b^{2}} - \frac{3 d^{3} x^{2} \cosh ^{2}{\left (a + b x \right )}}{8 b^{2}} + \frac{3 c d^{2} \sinh{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{4 b^{3}} + \frac{3 d^{3} x \sinh{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{4 b^{3}} - \frac{3 d^{3} \sinh ^{2}{\left (a + b x \right )}}{8 b^{4}} & \text{for}\: b \neq 0 \\\left (c^{3} x + \frac{3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac{d^{3} x^{4}}{4}\right ) \cosh ^{2}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.37301, size = 328, normalized size = 2.45 \begin{align*} \frac{1}{8} \, d^{3} x^{4} + \frac{1}{2} \, c d^{2} x^{3} + \frac{3}{4} \, c^{2} d x^{2} + \frac{1}{2} \, c^{3} x + \frac{{\left (4 \, b^{3} d^{3} x^{3} + 12 \, b^{3} c d^{2} x^{2} + 12 \, b^{3} c^{2} d x - 6 \, b^{2} d^{3} x^{2} + 4 \, b^{3} c^{3} - 12 \, b^{2} c d^{2} x - 6 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} - 3 \, d^{3}\right )} e^{\left (2 \, b x + 2 \, a\right )}}{32 \, b^{4}} - \frac{{\left (4 \, b^{3} d^{3} x^{3} + 12 \, b^{3} c d^{2} x^{2} + 12 \, b^{3} c^{2} d x + 6 \, b^{2} d^{3} x^{2} + 4 \, b^{3} c^{3} + 12 \, b^{2} c d^{2} x + 6 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} + 3 \, d^{3}\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*d^3*x^4 + 1/2*c*d^2*x^3 + 3/4*c^2*d*x^2 + 1/2*c^3*x + 1/32*(4*b^3*d^3*x^3 + 12*b^3*c*d^2*x^2 + 12*b^3*c^2*
d*x - 6*b^2*d^3*x^2 + 4*b^3*c^3 - 12*b^2*c*d^2*x - 6*b^2*c^2*d + 6*b*d^3*x + 6*b*c*d^2 - 3*d^3)*e^(2*b*x + 2*a
)/b^4 - 1/32*(4*b^3*d^3*x^3 + 12*b^3*c*d^2*x^2 + 12*b^3*c^2*d*x + 6*b^2*d^3*x^2 + 4*b^3*c^3 + 12*b^2*c*d^2*x +
6*b^2*c^2*d + 6*b*d^3*x + 6*b*c*d^2 + 3*d^3)*e^(-2*b*x - 2*a)/b^4