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

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

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 134, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.43, size = 104, normalized size = 0.78 \[ \frac {2 b (c+d x) \sinh (2 (a+b x)) \left (2 b^2 (c+d x)^2+3 d^2\right )-3 d \cosh (2 (a+b x)) \left (2 b^2 (c+d x)^2+d^2\right )+2 b^4 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )}{16 b^4} \]

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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fricas [A]  time = 0.50, size = 209, normalized size = 1.56 \[ \frac {2 \, b^{4} d^{3} x^{4} + 8 \, b^{4} c d^{2} x^{3} + 12 \, b^{4} c^{2} d x^{2} + 8 \, b^{4} c^{3} x - 3 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d + d^{3}\right )} \cosh \left (b x + a\right )^{2} + 4 \, {\left (2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 2 \, b^{3} c^{3} + 3 \, b c d^{2} + 3 \, {\left (2 \, b^{3} c^{2} d + b d^{3}\right )} x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - 3 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d + d^{3}\right )} \sinh \left (b x + a\right )^{2}}{16 \, b^{4}} \]

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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giac [B]  time = 0.13, size = 243, normalized size = 1.81 \[ \frac {1}{8} \, d^{3} x^{4} + \frac {1}{2} \, c d^{2} x^{3} + \frac {3}{4} \, c^{2} d x^{2} + \frac {1}{2} \, c^{3} x + \frac {{\left (4 \, b^{3} d^{3} x^{3} + 12 \, b^{3} c d^{2} x^{2} + 12 \, b^{3} c^{2} d x - 6 \, b^{2} d^{3} x^{2} + 4 \, b^{3} c^{3} - 12 \, b^{2} c d^{2} x - 6 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} - 3 \, d^{3}\right )} e^{\left (2 \, b x + 2 \, a\right )}}{32 \, b^{4}} - \frac {{\left (4 \, b^{3} d^{3} x^{3} + 12 \, b^{3} c d^{2} x^{2} + 12 \, b^{3} c^{2} d x + 6 \, b^{2} d^{3} x^{2} + 4 \, b^{3} c^{3} + 12 \, b^{2} c d^{2} x + 6 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} + 3 \, d^{3}\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b^{4}} \]

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

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maple [B]  time = 0.06, size = 523, normalized size = 3.90 \[ \frac {\frac {d^{3} \left (\frac {\left (b x +a \right )^{3} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{4}}{8}-\frac {3 \left (b x +a \right )^{2} \left (\cosh ^{2}\left (b x +a \right )\right )}{4}+\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {3 \left (b x +a \right )^{2}}{8}-\frac {3 \left (\cosh ^{2}\left (b x +a \right )\right )}{8}\right )}{b^{3}}-\frac {3 d^{3} a \left (\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{3}}{6}-\frac {\left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{2}+\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b^{3}}+\frac {3 d^{3} a^{2} \left (\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{2}}{4}-\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{4}\right )}{b^{3}}-\frac {d^{3} a^{3} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b^{3}}+\frac {3 c \,d^{2} \left (\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{3}}{6}-\frac {\left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{2}+\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b^{2}}-\frac {6 c \,d^{2} a \left (\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{2}}{4}-\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{4}\right )}{b^{2}}+\frac {3 c \,d^{2} a^{2} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b^{2}}+\frac {3 c^{2} d \left (\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{2}}{4}-\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{4}\right )}{b}-\frac {3 c^{2} d a \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b}+c^{3} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b} \]

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 = 0.48, size = 263, normalized size = 1.96 \[ \frac {3}{16} \, {\left (4 \, x^{2} + \frac {{\left (2 \, b x e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{2}} - \frac {{\left (2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{2}}\right )} c^{2} d + \frac {1}{16} \, {\left (8 \, x^{3} + \frac {3 \, {\left (2 \, b^{2} x^{2} e^{\left (2 \, a\right )} - 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{3}} - \frac {3 \, {\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{3}}\right )} c d^{2} + \frac {1}{32} \, {\left (4 \, x^{4} + \frac {{\left (4 \, b^{3} x^{3} e^{\left (2 \, a\right )} - 6 \, b^{2} x^{2} e^{\left (2 \, a\right )} + 6 \, b x e^{\left (2 \, a\right )} - 3 \, e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{4}} - \frac {{\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{4}}\right )} d^{3} + \frac {1}{8} \, c^{3} {\left (4 \, x + \frac {e^{\left (2 \, b x + 2 \, a\right )}}{b} - \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{b}\right )} \]

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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mupad [B]  time = 1.18, size = 229, normalized size = 1.71 \[ \frac {4\,b^4\,c^3\,x-\frac {3\,d^3\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{2}+2\,b^3\,c^3\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )+b^4\,d^3\,x^4-3\,b^2\,c^2\,d\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )+6\,b^4\,c^2\,d\,x^2+4\,b^4\,c\,d^2\,x^3-3\,b^2\,d^3\,x^2\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )+2\,b^3\,d^3\,x^3\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )+3\,b\,c\,d^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )+3\,b\,d^3\,x\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )-6\,b^2\,c\,d^2\,x\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )+6\,b^3\,c^2\,d\,x\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )+6\,b^3\,c\,d^2\,x^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{8\,b^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)^2*(c + d*x)^3,x)

[Out]

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

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sympy [A]  time = 2.58, size = 456, normalized size = 3.40 \[ \begin {cases} - \frac {c^{3} x \sinh ^{2}{\left (a + b x \right )}}{2} + \frac {c^{3} x \cosh ^{2}{\left (a + b x \right )}}{2} - \frac {3 c^{2} d x^{2} \sinh ^{2}{\left (a + b x \right )}}{4} + \frac {3 c^{2} d x^{2} \cosh ^{2}{\left (a + b x \right )}}{4} - \frac {c d^{2} x^{3} \sinh ^{2}{\left (a + b x \right )}}{2} + \frac {c d^{2} x^{3} \cosh ^{2}{\left (a + b x \right )}}{2} - \frac {d^{3} x^{4} \sinh ^{2}{\left (a + b x \right )}}{8} + \frac {d^{3} x^{4} \cosh ^{2}{\left (a + b x \right )}}{8} + \frac {c^{3} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} + \frac {3 c^{2} d x \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} + \frac {3 c d^{2} x^{2} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} + \frac {d^{3} x^{3} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} - \frac {3 c^{2} d \cosh ^{2}{\left (a + b x \right )}}{4 b^{2}} - \frac {3 c d^{2} x \sinh ^{2}{\left (a + b x \right )}}{4 b^{2}} - \frac {3 c d^{2} x \cosh ^{2}{\left (a + b x \right )}}{4 b^{2}} - \frac {3 d^{3} x^{2} \sinh ^{2}{\left (a + b x \right )}}{8 b^{2}} - \frac {3 d^{3} x^{2} \cosh ^{2}{\left (a + b x \right )}}{8 b^{2}} + \frac {3 c d^{2} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{4 b^{3}} + \frac {3 d^{3} x \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{4 b^{3}} - \frac {3 d^{3} \cosh ^{2}{\left (a + b x \right )}}{8 b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \cosh ^{2}{\relax (a )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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