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

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

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 70, normalized size of antiderivative = 1.00, 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, 2638} \[ \frac {6 d^2 (c+d x) \sinh (a+b x)}{b^3}-\frac {3 d (c+d x)^2 \cosh (a+b x)}{b^2}-\frac {6 d^3 \cosh (a+b x)}{b^4}+\frac {(c+d x)^3 \sinh (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3*Cosh[a + b*x],x]

[Out]

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

Rule 2638

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

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]

Rubi steps

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

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Mathematica [A]  time = 0.20, size = 61, normalized size = 0.87 \[ \frac {b (c+d x) \sinh (a+b x) \left (b^2 (c+d x)^2+6 d^2\right )-3 d \cosh (a+b x) \left (b^2 (c+d x)^2+2 d^2\right )}{b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.66, size = 111, normalized size = 1.59 \[ -\frac {3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d + 2 \, d^{3}\right )} \cosh \left (b x + a\right ) - {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + b^{3} c^{3} + 6 \, b c d^{2} + 3 \, {\left (b^{3} c^{2} d + 2 \, b d^{3}\right )} x\right )} \sinh \left (b x + a\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.14, size = 204, normalized size = 2.91 \[ \frac {{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x - 3 \, b^{2} d^{3} x^{2} + b^{3} c^{3} - 6 \, b^{2} c d^{2} x - 3 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} - 6 \, d^{3}\right )} e^{\left (b x + a\right )}}{2 \, b^{4}} - \frac {{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + 3 \, b^{2} d^{3} x^{2} + b^{3} c^{3} + 6 \, b^{2} c d^{2} x + 3 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} + 6 \, d^{3}\right )} e^{\left (-b x - a\right )}}{2 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*cosh(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

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maple [B]  time = 0.06, size = 308, normalized size = 4.40 \[ \frac {\frac {d^{3} \left (\left (b x +a \right )^{3} \sinh \left (b x +a \right )-3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )+6 \left (b x +a \right ) \sinh \left (b x +a \right )-6 \cosh \left (b x +a \right )\right )}{b^{3}}-\frac {3 d^{3} a \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )}{b^{3}}+\frac {3 d^{2} c \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )}{b^{2}}+\frac {3 d^{3} a^{2} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b^{3}}-\frac {6 d^{2} a c \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b^{2}}+\frac {3 d \,c^{2} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b}-\frac {d^{3} a^{3} \sinh \left (b x +a \right )}{b^{3}}+\frac {3 d^{2} a^{2} c \sinh \left (b x +a \right )}{b^{2}}-\frac {3 d a \,c^{2} \sinh \left (b x +a \right )}{b}+c^{3} \sinh \left (b x +a \right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.41, size = 222, normalized size = 3.17 \[ \frac {c^{3} e^{\left (b x + a\right )}}{2 \, b} + \frac {3 \, {\left (b x e^{a} - e^{a}\right )} c^{2} d e^{\left (b x\right )}}{2 \, b^{2}} - \frac {c^{3} e^{\left (-b x - a\right )}}{2 \, b} - \frac {3 \, {\left (b x + 1\right )} c^{2} d e^{\left (-b x - a\right )}}{2 \, b^{2}} + \frac {3 \, {\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} c d^{2} e^{\left (b x\right )}}{2 \, b^{3}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} c d^{2} e^{\left (-b x - a\right )}}{2 \, b^{3}} + \frac {{\left (b^{3} x^{3} e^{a} - 3 \, b^{2} x^{2} e^{a} + 6 \, b x e^{a} - 6 \, e^{a}\right )} d^{3} e^{\left (b x\right )}}{2 \, b^{4}} - \frac {{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} d^{3} e^{\left (-b x - a\right )}}{2 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*cosh(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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mupad [B]  time = 0.94, size = 143, normalized size = 2.04 \[ \frac {\mathrm {sinh}\left (a+b\,x\right )\,\left (b^2\,c^3+6\,c\,d^2\right )}{b^3}-\frac {3\,\mathrm {cosh}\left (a+b\,x\right )\,\left (b^2\,c^2\,d+2\,d^3\right )}{b^4}-\frac {3\,d^3\,x^2\,\mathrm {cosh}\left (a+b\,x\right )}{b^2}+\frac {d^3\,x^3\,\mathrm {sinh}\left (a+b\,x\right )}{b}+\frac {3\,x\,\mathrm {sinh}\left (a+b\,x\right )\,\left (b^2\,c^2\,d+2\,d^3\right )}{b^3}-\frac {6\,c\,d^2\,x\,\mathrm {cosh}\left (a+b\,x\right )}{b^2}+\frac {3\,c\,d^2\,x^2\,\mathrm {sinh}\left (a+b\,x\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)*(c + d*x)^3,x)

[Out]

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

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sympy [A]  time = 1.26, size = 202, normalized size = 2.89 \[ \begin {cases} \frac {c^{3} \sinh {\left (a + b x \right )}}{b} + \frac {3 c^{2} d x \sinh {\left (a + b x \right )}}{b} + \frac {3 c d^{2} x^{2} \sinh {\left (a + b x \right )}}{b} + \frac {d^{3} x^{3} \sinh {\left (a + b x \right )}}{b} - \frac {3 c^{2} d \cosh {\left (a + b x \right )}}{b^{2}} - \frac {6 c d^{2} x \cosh {\left (a + b x \right )}}{b^{2}} - \frac {3 d^{3} x^{2} \cosh {\left (a + b x \right )}}{b^{2}} + \frac {6 c d^{2} \sinh {\left (a + b x \right )}}{b^{3}} + \frac {6 d^{3} x \sinh {\left (a + b x \right )}}{b^{3}} - \frac {6 d^{3} \cosh {\left (a + b x \right )}}{b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \cosh {\relax (a )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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