3.2 \(\int (c+d x)^3 \sinh (a+b x) \, dx\)

Optimal. Leaf size=70 \[ \frac{6 d^2 (c+d x) \cosh (a+b x)}{b^3}-\frac{3 d (c+d x)^2 \sinh (a+b x)}{b^2}-\frac{6 d^3 \sinh (a+b x)}{b^4}+\frac{(c+d x)^3 \cosh (a+b x)}{b} \]

[Out]

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

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Rubi [A]  time = 0.0790137, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3296, 2637} \[ \frac{6 d^2 (c+d x) \cosh (a+b x)}{b^3}-\frac{3 d (c+d x)^2 \sinh (a+b x)}{b^2}-\frac{6 d^3 \sinh (a+b x)}{b^4}+\frac{(c+d x)^3 \cosh (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3*Sinh[a + b*x],x]

[Out]

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

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.21966, size = 61, normalized size = 0.87 \[ \frac{b (c+d x) \cosh (a+b x) \left (b^2 (c+d x)^2+6 d^2\right )-3 d \sinh (a+b x) \left (b^2 (c+d x)^2+2 d^2\right )}{b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.007, size = 308, normalized size = 4.4 \begin{align*}{\frac{1}{b} \left ({\frac{{d}^{3} \left ( \left ( bx+a \right ) ^{3}\cosh \left ( bx+a \right ) -3\, \left ( bx+a \right ) ^{2}\sinh \left ( bx+a \right ) +6\, \left ( bx+a \right ) \cosh \left ( bx+a \right ) -6\,\sinh \left ( bx+a \right ) \right ) }{{b}^{3}}}-3\,{\frac{{d}^{3}a \left ( \left ( bx+a \right ) ^{2}\cosh \left ( bx+a \right ) -2\, \left ( bx+a \right ) \sinh \left ( bx+a \right ) +2\,\cosh \left ( bx+a \right ) \right ) }{{b}^{3}}}+3\,{\frac{{d}^{2}c \left ( \left ( bx+a \right ) ^{2}\cosh \left ( bx+a \right ) -2\, \left ( bx+a \right ) \sinh \left ( bx+a \right ) +2\,\cosh \left ( bx+a \right ) \right ) }{{b}^{2}}}+3\,{\frac{{d}^{3}{a}^{2} \left ( \left ( bx+a \right ) \cosh \left ( bx+a \right ) -\sinh \left ( bx+a \right ) \right ) }{{b}^{3}}}-6\,{\frac{{d}^{2}ac \left ( \left ( bx+a \right ) \cosh \left ( bx+a \right ) -\sinh \left ( bx+a \right ) \right ) }{{b}^{2}}}+3\,{\frac{d{c}^{2} \left ( \left ( bx+a \right ) \cosh \left ( bx+a \right ) -\sinh \left ( bx+a \right ) \right ) }{b}}-{\frac{{d}^{3}{a}^{3}\cosh \left ( bx+a \right ) }{{b}^{3}}}+3\,{\frac{{d}^{2}{a}^{2}c\cosh \left ( bx+a \right ) }{{b}^{2}}}-3\,{\frac{da{c}^{2}\cosh \left ( bx+a \right ) }{b}}+{c}^{3}\cosh \left ( bx+a \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.0586, size = 300, normalized size = 4.29 \begin{align*} \frac{c^{3} e^{\left (b x + a\right )}}{2 \, b} + \frac{3 \,{\left (b x e^{a} - e^{a}\right )} c^{2} d e^{\left (b x\right )}}{2 \, b^{2}} + \frac{c^{3} e^{\left (-b x - a\right )}}{2 \, b} + \frac{3 \,{\left (b x + 1\right )} c^{2} d e^{\left (-b x - a\right )}}{2 \, b^{2}} + \frac{3 \,{\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} c d^{2} e^{\left (b x\right )}}{2 \, b^{3}} + \frac{3 \,{\left (b^{2} x^{2} + 2 \, b x + 2\right )} c d^{2} e^{\left (-b x - a\right )}}{2 \, b^{3}} + \frac{{\left (b^{3} x^{3} e^{a} - 3 \, b^{2} x^{2} e^{a} + 6 \, b x e^{a} - 6 \, e^{a}\right )} d^{3} e^{\left (b x\right )}}{2 \, b^{4}} + \frac{{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} d^{3} e^{\left (-b x - a\right )}}{2 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.53654, size = 231, normalized size = 3.3 \begin{align*} \frac{{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + b^{3} c^{3} + 6 \, b c d^{2} + 3 \,{\left (b^{3} c^{2} d + 2 \, b d^{3}\right )} x\right )} \cosh \left (b x + a\right ) - 3 \,{\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d + 2 \, d^{3}\right )} \sinh \left (b x + a\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.50237, size = 202, normalized size = 2.89 \begin{align*} \begin{cases} \frac{c^{3} \cosh{\left (a + b x \right )}}{b} + \frac{3 c^{2} d x \cosh{\left (a + b x \right )}}{b} + \frac{3 c d^{2} x^{2} \cosh{\left (a + b x \right )}}{b} + \frac{d^{3} x^{3} \cosh{\left (a + b x \right )}}{b} - \frac{3 c^{2} d \sinh{\left (a + b x \right )}}{b^{2}} - \frac{6 c d^{2} x \sinh{\left (a + b x \right )}}{b^{2}} - \frac{3 d^{3} x^{2} \sinh{\left (a + b x \right )}}{b^{2}} + \frac{6 c d^{2} \cosh{\left (a + b x \right )}}{b^{3}} + \frac{6 d^{3} x \cosh{\left (a + b x \right )}}{b^{3}} - \frac{6 d^{3} \sinh{\left (a + b x \right )}}{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 ) \sinh{\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*sinh(b*x+a),x)

[Out]

Piecewise((c**3*cosh(a + b*x)/b + 3*c**2*d*x*cosh(a + b*x)/b + 3*c*d**2*x**2*cosh(a + b*x)/b + d**3*x**3*cosh(
a + b*x)/b - 3*c**2*d*sinh(a + b*x)/b**2 - 6*c*d**2*x*sinh(a + b*x)/b**2 - 3*d**3*x**2*sinh(a + b*x)/b**2 + 6*
c*d**2*cosh(a + b*x)/b**3 + 6*d**3*x*cosh(a + b*x)/b**3 - 6*d**3*sinh(a + b*x)/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)*sinh(a), True))

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Giac [B]  time = 1.19993, size = 275, normalized size = 3.93 \begin{align*} \frac{{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x - 3 \, b^{2} d^{3} x^{2} + b^{3} c^{3} - 6 \, b^{2} c d^{2} x - 3 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} - 6 \, d^{3}\right )} e^{\left (b x + a\right )}}{2 \, b^{4}} + \frac{{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + 3 \, b^{2} d^{3} x^{2} + b^{3} c^{3} + 6 \, b^{2} c d^{2} x + 3 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} + 6 \, d^{3}\right )} e^{\left (-b x - a\right )}}{2 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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