3.105 \(\int (c+d x)^3 (a+a \cosh (e+f x))^2 \, dx\)

Optimal. Leaf size=237 \[ \frac {12 a^2 d^2 (c+d x) \sinh (e+f x)}{f^3}+\frac {3 a^2 d^2 (c+d x) \sinh (e+f x) \cosh (e+f x)}{4 f^3}+\frac {3 a^2 c d^2 x}{4 f^2}-\frac {3 a^2 d (c+d x)^2 \cosh ^2(e+f x)}{4 f^2}-\frac {6 a^2 d (c+d x)^2 \cosh (e+f x)}{f^2}+\frac {2 a^2 (c+d x)^3 \sinh (e+f x)}{f}+\frac {a^2 (c+d x)^3 \sinh (e+f x) \cosh (e+f x)}{2 f}+\frac {3 a^2 (c+d x)^4}{8 d}-\frac {3 a^2 d^3 \cosh ^2(e+f x)}{8 f^4}-\frac {12 a^2 d^3 \cosh (e+f x)}{f^4}+\frac {3 a^2 d^3 x^2}{8 f^2} \]

[Out]

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

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Rubi [A]  time = 0.27, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3317, 3296, 2638, 3311, 32, 3310} \[ \frac {12 a^2 d^2 (c+d x) \sinh (e+f x)}{f^3}+\frac {3 a^2 d^2 (c+d x) \sinh (e+f x) \cosh (e+f x)}{4 f^3}+\frac {3 a^2 c d^2 x}{4 f^2}-\frac {3 a^2 d (c+d x)^2 \cosh ^2(e+f x)}{4 f^2}-\frac {6 a^2 d (c+d x)^2 \cosh (e+f x)}{f^2}+\frac {2 a^2 (c+d x)^3 \sinh (e+f x)}{f}+\frac {a^2 (c+d x)^3 \sinh (e+f x) \cosh (e+f x)}{2 f}+\frac {3 a^2 (c+d x)^4}{8 d}-\frac {3 a^2 d^3 \cosh ^2(e+f x)}{8 f^4}-\frac {12 a^2 d^3 \cosh (e+f x)}{f^4}+\frac {3 a^2 d^3 x^2}{8 f^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3*(a + a*Cosh[e + f*x])^2,x]

[Out]

(3*a^2*c*d^2*x)/(4*f^2) + (3*a^2*d^3*x^2)/(8*f^2) + (3*a^2*(c + d*x)^4)/(8*d) - (12*a^2*d^3*Cosh[e + f*x])/f^4
- (6*a^2*d*(c + d*x)^2*Cosh[e + f*x])/f^2 - (3*a^2*d^3*Cosh[e + f*x]^2)/(8*f^4) - (3*a^2*d*(c + d*x)^2*Cosh[e
+ f*x]^2)/(4*f^2) + (12*a^2*d^2*(c + d*x)*Sinh[e + f*x])/f^3 + (2*a^2*(c + d*x)^3*Sinh[e + f*x])/f + (3*a^2*d
^2*(c + d*x)*Cosh[e + f*x]*Sinh[e + f*x])/(4*f^3) + (a^2*(c + d*x)^3*Cosh[e + f*x]*Sinh[e + f*x])/(2*f)

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 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]

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]

Rule 3317

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] ||
IGtQ[m, 0] || NeQ[a^2 - b^2, 0])

Rubi steps

\begin {align*} \int (c+d x)^3 (a+a \cosh (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)^3+2 a^2 (c+d x)^3 \cosh (e+f x)+a^2 (c+d x)^3 \cosh ^2(e+f x)\right ) \, dx\\ &=\frac {a^2 (c+d x)^4}{4 d}+a^2 \int (c+d x)^3 \cosh ^2(e+f x) \, dx+\left (2 a^2\right ) \int (c+d x)^3 \cosh (e+f x) \, dx\\ &=\frac {a^2 (c+d x)^4}{4 d}-\frac {3 a^2 d (c+d x)^2 \cosh ^2(e+f x)}{4 f^2}+\frac {2 a^2 (c+d x)^3 \sinh (e+f x)}{f}+\frac {a^2 (c+d x)^3 \cosh (e+f x) \sinh (e+f x)}{2 f}+\frac {1}{2} a^2 \int (c+d x)^3 \, dx+\frac {\left (3 a^2 d^2\right ) \int (c+d x) \cosh ^2(e+f x) \, dx}{2 f^2}-\frac {\left (6 a^2 d\right ) \int (c+d x)^2 \sinh (e+f x) \, dx}{f}\\ &=\frac {3 a^2 (c+d x)^4}{8 d}-\frac {6 a^2 d (c+d x)^2 \cosh (e+f x)}{f^2}-\frac {3 a^2 d^3 \cosh ^2(e+f x)}{8 f^4}-\frac {3 a^2 d (c+d x)^2 \cosh ^2(e+f x)}{4 f^2}+\frac {2 a^2 (c+d x)^3 \sinh (e+f x)}{f}+\frac {3 a^2 d^2 (c+d x) \cosh (e+f x) \sinh (e+f x)}{4 f^3}+\frac {a^2 (c+d x)^3 \cosh (e+f x) \sinh (e+f x)}{2 f}+\frac {\left (3 a^2 d^2\right ) \int (c+d x) \, dx}{4 f^2}+\frac {\left (12 a^2 d^2\right ) \int (c+d x) \cosh (e+f x) \, dx}{f^2}\\ &=\frac {3 a^2 c d^2 x}{4 f^2}+\frac {3 a^2 d^3 x^2}{8 f^2}+\frac {3 a^2 (c+d x)^4}{8 d}-\frac {6 a^2 d (c+d x)^2 \cosh (e+f x)}{f^2}-\frac {3 a^2 d^3 \cosh ^2(e+f x)}{8 f^4}-\frac {3 a^2 d (c+d x)^2 \cosh ^2(e+f x)}{4 f^2}+\frac {12 a^2 d^2 (c+d x) \sinh (e+f x)}{f^3}+\frac {2 a^2 (c+d x)^3 \sinh (e+f x)}{f}+\frac {3 a^2 d^2 (c+d x) \cosh (e+f x) \sinh (e+f x)}{4 f^3}+\frac {a^2 (c+d x)^3 \cosh (e+f x) \sinh (e+f x)}{2 f}-\frac {\left (12 a^2 d^3\right ) \int \sinh (e+f x) \, dx}{f^3}\\ &=\frac {3 a^2 c d^2 x}{4 f^2}+\frac {3 a^2 d^3 x^2}{8 f^2}+\frac {3 a^2 (c+d x)^4}{8 d}-\frac {12 a^2 d^3 \cosh (e+f x)}{f^4}-\frac {6 a^2 d (c+d x)^2 \cosh (e+f x)}{f^2}-\frac {3 a^2 d^3 \cosh ^2(e+f x)}{8 f^4}-\frac {3 a^2 d (c+d x)^2 \cosh ^2(e+f x)}{4 f^2}+\frac {12 a^2 d^2 (c+d x) \sinh (e+f x)}{f^3}+\frac {2 a^2 (c+d x)^3 \sinh (e+f x)}{f}+\frac {3 a^2 d^2 (c+d x) \cosh (e+f x) \sinh (e+f x)}{4 f^3}+\frac {a^2 (c+d x)^3 \cosh (e+f x) \sinh (e+f x)}{2 f}\\ \end {align*}

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Mathematica [A]  time = 1.49, size = 217, normalized size = 0.92 \[ \frac {a^2 \left (-96 d \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2+2\right )\right ) \cosh (e+f x)-3 d \left (2 c^2 f^2+4 c d f^2 x+d^2 \left (2 f^2 x^2+1\right )\right ) \cosh (2 (e+f x))+2 f \left (16 (c+d x) \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2+6\right )\right ) \sinh (e+f x)+(c+d x) \left (2 c^2 f^2+4 c d f^2 x+d^2 \left (2 f^2 x^2+3\right )\right ) \sinh (2 (e+f x))+3 f^3 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )\right )\right )}{16 f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(-96*d*(c^2*f^2 + 2*c*d*f^2*x + d^2*(2 + f^2*x^2))*Cosh[e + f*x] - 3*d*(2*c^2*f^2 + 4*c*d*f^2*x + d^2*(1
+ 2*f^2*x^2))*Cosh[2*(e + f*x)] + 2*f*(3*f^3*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) + 16*(c + d*x)*(c^2
*f^2 + 2*c*d*f^2*x + d^2*(6 + f^2*x^2))*Sinh[e + f*x] + (c + d*x)*(2*c^2*f^2 + 4*c*d*f^2*x + d^2*(3 + 2*f^2*x^
2))*Sinh[2*(e + f*x)])))/(16*f^4)

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fricas [A]  time = 0.53, size = 395, normalized size = 1.67 \[ \frac {6 \, a^{2} d^{3} f^{4} x^{4} + 24 \, a^{2} c d^{2} f^{4} x^{3} + 36 \, a^{2} c^{2} d f^{4} x^{2} + 24 \, a^{2} c^{3} f^{4} x - 3 \, {\left (2 \, a^{2} d^{3} f^{2} x^{2} + 4 \, a^{2} c d^{2} f^{2} x + 2 \, a^{2} c^{2} d f^{2} + a^{2} d^{3}\right )} \cosh \left (f x + e\right )^{2} - 3 \, {\left (2 \, a^{2} d^{3} f^{2} x^{2} + 4 \, a^{2} c d^{2} f^{2} x + 2 \, a^{2} c^{2} d f^{2} + a^{2} d^{3}\right )} \sinh \left (f x + e\right )^{2} - 96 \, {\left (a^{2} d^{3} f^{2} x^{2} + 2 \, a^{2} c d^{2} f^{2} x + a^{2} c^{2} d f^{2} + 2 \, a^{2} d^{3}\right )} \cosh \left (f x + e\right ) + 4 \, {\left (8 \, a^{2} d^{3} f^{3} x^{3} + 24 \, a^{2} c d^{2} f^{3} x^{2} + 8 \, a^{2} c^{3} f^{3} + 48 \, a^{2} c d^{2} f + 24 \, {\left (a^{2} c^{2} d f^{3} + 2 \, a^{2} d^{3} f\right )} x + {\left (2 \, a^{2} d^{3} f^{3} x^{3} + 6 \, a^{2} c d^{2} f^{3} x^{2} + 2 \, a^{2} c^{3} f^{3} + 3 \, a^{2} c d^{2} f + 3 \, {\left (2 \, a^{2} c^{2} d f^{3} + a^{2} d^{3} f\right )} x\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )}{16 \, f^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(6*a^2*d^3*f^4*x^4 + 24*a^2*c*d^2*f^4*x^3 + 36*a^2*c^2*d*f^4*x^2 + 24*a^2*c^3*f^4*x - 3*(2*a^2*d^3*f^2*x^
2 + 4*a^2*c*d^2*f^2*x + 2*a^2*c^2*d*f^2 + a^2*d^3)*cosh(f*x + e)^2 - 3*(2*a^2*d^3*f^2*x^2 + 4*a^2*c*d^2*f^2*x
+ 2*a^2*c^2*d*f^2 + a^2*d^3)*sinh(f*x + e)^2 - 96*(a^2*d^3*f^2*x^2 + 2*a^2*c*d^2*f^2*x + a^2*c^2*d*f^2 + 2*a^2
*d^3)*cosh(f*x + e) + 4*(8*a^2*d^3*f^3*x^3 + 24*a^2*c*d^2*f^3*x^2 + 8*a^2*c^3*f^3 + 48*a^2*c*d^2*f + 24*(a^2*c
^2*d*f^3 + 2*a^2*d^3*f)*x + (2*a^2*d^3*f^3*x^3 + 6*a^2*c*d^2*f^3*x^2 + 2*a^2*c^3*f^3 + 3*a^2*c*d^2*f + 3*(2*a^
2*c^2*d*f^3 + a^2*d^3*f)*x)*cosh(f*x + e))*sinh(f*x + e))/f^4

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giac [B]  time = 0.13, size = 581, normalized size = 2.45 \[ \frac {3}{8} \, a^{2} d^{3} x^{4} + \frac {3}{2} \, a^{2} c d^{2} x^{3} + \frac {9}{4} \, a^{2} c^{2} d x^{2} + \frac {3}{2} \, a^{2} c^{3} x + \frac {{\left (4 \, a^{2} d^{3} f^{3} x^{3} + 12 \, a^{2} c d^{2} f^{3} x^{2} + 12 \, a^{2} c^{2} d f^{3} x - 6 \, a^{2} d^{3} f^{2} x^{2} + 4 \, a^{2} c^{3} f^{3} - 12 \, a^{2} c d^{2} f^{2} x - 6 \, a^{2} c^{2} d f^{2} + 6 \, a^{2} d^{3} f x + 6 \, a^{2} c d^{2} f - 3 \, a^{2} d^{3}\right )} e^{\left (2 \, f x + 2 \, e\right )}}{32 \, f^{4}} + \frac {{\left (a^{2} d^{3} f^{3} x^{3} + 3 \, a^{2} c d^{2} f^{3} x^{2} + 3 \, a^{2} c^{2} d f^{3} x - 3 \, a^{2} d^{3} f^{2} x^{2} + a^{2} c^{3} f^{3} - 6 \, a^{2} c d^{2} f^{2} x - 3 \, a^{2} c^{2} d f^{2} + 6 \, a^{2} d^{3} f x + 6 \, a^{2} c d^{2} f - 6 \, a^{2} d^{3}\right )} e^{\left (f x + e\right )}}{f^{4}} - \frac {{\left (a^{2} d^{3} f^{3} x^{3} + 3 \, a^{2} c d^{2} f^{3} x^{2} + 3 \, a^{2} c^{2} d f^{3} x + 3 \, a^{2} d^{3} f^{2} x^{2} + a^{2} c^{3} f^{3} + 6 \, a^{2} c d^{2} f^{2} x + 3 \, a^{2} c^{2} d f^{2} + 6 \, a^{2} d^{3} f x + 6 \, a^{2} c d^{2} f + 6 \, a^{2} d^{3}\right )} e^{\left (-f x - e\right )}}{f^{4}} - \frac {{\left (4 \, a^{2} d^{3} f^{3} x^{3} + 12 \, a^{2} c d^{2} f^{3} x^{2} + 12 \, a^{2} c^{2} d f^{3} x + 6 \, a^{2} d^{3} f^{2} x^{2} + 4 \, a^{2} c^{3} f^{3} + 12 \, a^{2} c d^{2} f^{2} x + 6 \, a^{2} c^{2} d f^{2} + 6 \, a^{2} d^{3} f x + 6 \, a^{2} c d^{2} f + 3 \, a^{2} d^{3}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{32 \, f^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*a^2*d^3*x^4 + 3/2*a^2*c*d^2*x^3 + 9/4*a^2*c^2*d*x^2 + 3/2*a^2*c^3*x + 1/32*(4*a^2*d^3*f^3*x^3 + 12*a^2*c*d
^2*f^3*x^2 + 12*a^2*c^2*d*f^3*x - 6*a^2*d^3*f^2*x^2 + 4*a^2*c^3*f^3 - 12*a^2*c*d^2*f^2*x - 6*a^2*c^2*d*f^2 + 6
*a^2*d^3*f*x + 6*a^2*c*d^2*f - 3*a^2*d^3)*e^(2*f*x + 2*e)/f^4 + (a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2
*c^2*d*f^3*x - 3*a^2*d^3*f^2*x^2 + a^2*c^3*f^3 - 6*a^2*c*d^2*f^2*x - 3*a^2*c^2*d*f^2 + 6*a^2*d^3*f*x + 6*a^2*c
*d^2*f - 6*a^2*d^3)*e^(f*x + e)/f^4 - (a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + 3*a^2*d^3*f
^2*x^2 + a^2*c^3*f^3 + 6*a^2*c*d^2*f^2*x + 3*a^2*c^2*d*f^2 + 6*a^2*d^3*f*x + 6*a^2*c*d^2*f + 6*a^2*d^3)*e^(-f*
x - e)/f^4 - 1/32*(4*a^2*d^3*f^3*x^3 + 12*a^2*c*d^2*f^3*x^2 + 12*a^2*c^2*d*f^3*x + 6*a^2*d^3*f^2*x^2 + 4*a^2*c
^3*f^3 + 12*a^2*c*d^2*f^2*x + 6*a^2*c^2*d*f^2 + 6*a^2*d^3*f*x + 6*a^2*c*d^2*f + 3*a^2*d^3)*e^(-2*f*x - 2*e)/f^
4

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maple [B]  time = 0.08, size = 1071, normalized size = 4.52 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.39, size = 527, normalized size = 2.22 \[ \frac {1}{4} \, a^{2} d^{3} x^{4} + a^{2} c d^{2} x^{3} + \frac {3}{2} \, a^{2} c^{2} d x^{2} + \frac {3}{16} \, {\left (4 \, x^{2} + \frac {{\left (2 \, f x e^{\left (2 \, e\right )} - e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{2}} - \frac {{\left (2 \, f x + 1\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{2}}\right )} a^{2} c^{2} d + \frac {1}{16} \, {\left (8 \, x^{3} + \frac {3 \, {\left (2 \, f^{2} x^{2} e^{\left (2 \, e\right )} - 2 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{3}} - \frac {3 \, {\left (2 \, f^{2} x^{2} + 2 \, f x + 1\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{3}}\right )} a^{2} c d^{2} + \frac {1}{32} \, {\left (4 \, x^{4} + \frac {{\left (4 \, f^{3} x^{3} e^{\left (2 \, e\right )} - 6 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 6 \, f x e^{\left (2 \, e\right )} - 3 \, e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{4}} - \frac {{\left (4 \, f^{3} x^{3} + 6 \, f^{2} x^{2} + 6 \, f x + 3\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{4}}\right )} a^{2} d^{3} + \frac {1}{8} \, a^{2} c^{3} {\left (4 \, x + \frac {e^{\left (2 \, f x + 2 \, e\right )}}{f} - \frac {e^{\left (-2 \, f x - 2 \, e\right )}}{f}\right )} + a^{2} c^{3} x + 3 \, a^{2} c^{2} d {\left (\frac {{\left (f x e^{e} - e^{e}\right )} e^{\left (f x\right )}}{f^{2}} - \frac {{\left (f x + 1\right )} e^{\left (-f x - e\right )}}{f^{2}}\right )} + 3 \, a^{2} c d^{2} {\left (\frac {{\left (f^{2} x^{2} e^{e} - 2 \, f x e^{e} + 2 \, e^{e}\right )} e^{\left (f x\right )}}{f^{3}} - \frac {{\left (f^{2} x^{2} + 2 \, f x + 2\right )} e^{\left (-f x - e\right )}}{f^{3}}\right )} + a^{2} d^{3} {\left (\frac {{\left (f^{3} x^{3} e^{e} - 3 \, f^{2} x^{2} e^{e} + 6 \, f x e^{e} - 6 \, e^{e}\right )} e^{\left (f x\right )}}{f^{4}} - \frac {{\left (f^{3} x^{3} + 3 \, f^{2} x^{2} + 6 \, f x + 6\right )} e^{\left (-f x - e\right )}}{f^{4}}\right )} + \frac {2 \, a^{2} c^{3} \sinh \left (f x + e\right )}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.25, size = 452, normalized size = 1.91 \[ \frac {16\,a^2\,c^3\,f^3\,\mathrm {sinh}\left (e+f\,x\right )-\frac {3\,a^2\,d^3\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )}{2}-96\,a^2\,d^3\,\mathrm {cosh}\left (e+f\,x\right )+12\,a^2\,c^3\,f^4\,x+2\,a^2\,c^3\,f^3\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )+3\,a^2\,d^3\,f^4\,x^4+96\,a^2\,c\,d^2\,f\,\mathrm {sinh}\left (e+f\,x\right )+96\,a^2\,d^3\,f\,x\,\mathrm {sinh}\left (e+f\,x\right )-3\,a^2\,d^3\,f^2\,x^2\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )+2\,a^2\,d^3\,f^3\,x^3\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )-48\,a^2\,c^2\,d\,f^2\,\mathrm {cosh}\left (e+f\,x\right )+3\,a^2\,c\,d^2\,f\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )+3\,a^2\,d^3\,f\,x\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )-3\,a^2\,c^2\,d\,f^2\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )+18\,a^2\,c^2\,d\,f^4\,x^2+12\,a^2\,c\,d^2\,f^4\,x^3-48\,a^2\,d^3\,f^2\,x^2\,\mathrm {cosh}\left (e+f\,x\right )+16\,a^2\,d^3\,f^3\,x^3\,\mathrm {sinh}\left (e+f\,x\right )-6\,a^2\,c\,d^2\,f^2\,x\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )+6\,a^2\,c^2\,d\,f^3\,x\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )+48\,a^2\,c\,d^2\,f^3\,x^2\,\mathrm {sinh}\left (e+f\,x\right )+6\,a^2\,c\,d^2\,f^3\,x^2\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )-96\,a^2\,c\,d^2\,f^2\,x\,\mathrm {cosh}\left (e+f\,x\right )+48\,a^2\,c^2\,d\,f^3\,x\,\mathrm {sinh}\left (e+f\,x\right )}{8\,f^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(16*a^2*c^3*f^3*sinh(e + f*x) - (3*a^2*d^3*cosh(2*e + 2*f*x))/2 - 96*a^2*d^3*cosh(e + f*x) + 12*a^2*c^3*f^4*x
+ 2*a^2*c^3*f^3*sinh(2*e + 2*f*x) + 3*a^2*d^3*f^4*x^4 + 96*a^2*c*d^2*f*sinh(e + f*x) + 96*a^2*d^3*f*x*sinh(e +
f*x) - 3*a^2*d^3*f^2*x^2*cosh(2*e + 2*f*x) + 2*a^2*d^3*f^3*x^3*sinh(2*e + 2*f*x) - 48*a^2*c^2*d*f^2*cosh(e +
f*x) + 3*a^2*c*d^2*f*sinh(2*e + 2*f*x) + 3*a^2*d^3*f*x*sinh(2*e + 2*f*x) - 3*a^2*c^2*d*f^2*cosh(2*e + 2*f*x) +
18*a^2*c^2*d*f^4*x^2 + 12*a^2*c*d^2*f^4*x^3 - 48*a^2*d^3*f^2*x^2*cosh(e + f*x) + 16*a^2*d^3*f^3*x^3*sinh(e +
f*x) - 6*a^2*c*d^2*f^2*x*cosh(2*e + 2*f*x) + 6*a^2*c^2*d*f^3*x*sinh(2*e + 2*f*x) + 48*a^2*c*d^2*f^3*x^2*sinh(e
+ f*x) + 6*a^2*c*d^2*f^3*x^2*sinh(2*e + 2*f*x) - 96*a^2*c*d^2*f^2*x*cosh(e + f*x) + 48*a^2*c^2*d*f^3*x*sinh(e
+ f*x))/(8*f^4)

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sympy [A]  time = 4.08, size = 779, normalized size = 3.29 \[ \begin {cases} - \frac {a^{2} c^{3} x \sinh ^{2}{\left (e + f x \right )}}{2} + \frac {a^{2} c^{3} x \cosh ^{2}{\left (e + f x \right )}}{2} + a^{2} c^{3} x + \frac {a^{2} c^{3} \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{2 f} + \frac {2 a^{2} c^{3} \sinh {\left (e + f x \right )}}{f} - \frac {3 a^{2} c^{2} d x^{2} \sinh ^{2}{\left (e + f x \right )}}{4} + \frac {3 a^{2} c^{2} d x^{2} \cosh ^{2}{\left (e + f x \right )}}{4} + \frac {3 a^{2} c^{2} d x^{2}}{2} + \frac {3 a^{2} c^{2} d x \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{2 f} + \frac {6 a^{2} c^{2} d x \sinh {\left (e + f x \right )}}{f} - \frac {3 a^{2} c^{2} d \sinh ^{2}{\left (e + f x \right )}}{4 f^{2}} - \frac {6 a^{2} c^{2} d \cosh {\left (e + f x \right )}}{f^{2}} - \frac {a^{2} c d^{2} x^{3} \sinh ^{2}{\left (e + f x \right )}}{2} + \frac {a^{2} c d^{2} x^{3} \cosh ^{2}{\left (e + f x \right )}}{2} + a^{2} c d^{2} x^{3} + \frac {3 a^{2} c d^{2} x^{2} \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{2 f} + \frac {6 a^{2} c d^{2} x^{2} \sinh {\left (e + f x \right )}}{f} - \frac {3 a^{2} c d^{2} x \sinh ^{2}{\left (e + f x \right )}}{4 f^{2}} - \frac {3 a^{2} c d^{2} x \cosh ^{2}{\left (e + f x \right )}}{4 f^{2}} - \frac {12 a^{2} c d^{2} x \cosh {\left (e + f x \right )}}{f^{2}} + \frac {3 a^{2} c d^{2} \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{4 f^{3}} + \frac {12 a^{2} c d^{2} \sinh {\left (e + f x \right )}}{f^{3}} - \frac {a^{2} d^{3} x^{4} \sinh ^{2}{\left (e + f x \right )}}{8} + \frac {a^{2} d^{3} x^{4} \cosh ^{2}{\left (e + f x \right )}}{8} + \frac {a^{2} d^{3} x^{4}}{4} + \frac {a^{2} d^{3} x^{3} \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{2 f} + \frac {2 a^{2} d^{3} x^{3} \sinh {\left (e + f x \right )}}{f} - \frac {3 a^{2} d^{3} x^{2} \sinh ^{2}{\left (e + f x \right )}}{8 f^{2}} - \frac {3 a^{2} d^{3} x^{2} \cosh ^{2}{\left (e + f x \right )}}{8 f^{2}} - \frac {6 a^{2} d^{3} x^{2} \cosh {\left (e + f x \right )}}{f^{2}} + \frac {3 a^{2} d^{3} x \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{4 f^{3}} + \frac {12 a^{2} d^{3} x \sinh {\left (e + f x \right )}}{f^{3}} - \frac {3 a^{2} d^{3} \sinh ^{2}{\left (e + f x \right )}}{8 f^{4}} - \frac {12 a^{2} d^{3} \cosh {\left (e + f x \right )}}{f^{4}} & \text {for}\: f \neq 0 \\\left (a \cosh {\relax (e )} + a\right )^{2} \left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**2*c**3*x*sinh(e + f*x)**2/2 + a**2*c**3*x*cosh(e + f*x)**2/2 + a**2*c**3*x + a**2*c**3*sinh(e +
f*x)*cosh(e + f*x)/(2*f) + 2*a**2*c**3*sinh(e + f*x)/f - 3*a**2*c**2*d*x**2*sinh(e + f*x)**2/4 + 3*a**2*c**2*
d*x**2*cosh(e + f*x)**2/4 + 3*a**2*c**2*d*x**2/2 + 3*a**2*c**2*d*x*sinh(e + f*x)*cosh(e + f*x)/(2*f) + 6*a**2*
c**2*d*x*sinh(e + f*x)/f - 3*a**2*c**2*d*sinh(e + f*x)**2/(4*f**2) - 6*a**2*c**2*d*cosh(e + f*x)/f**2 - a**2*c
*d**2*x**3*sinh(e + f*x)**2/2 + a**2*c*d**2*x**3*cosh(e + f*x)**2/2 + a**2*c*d**2*x**3 + 3*a**2*c*d**2*x**2*si
nh(e + f*x)*cosh(e + f*x)/(2*f) + 6*a**2*c*d**2*x**2*sinh(e + f*x)/f - 3*a**2*c*d**2*x*sinh(e + f*x)**2/(4*f**
2) - 3*a**2*c*d**2*x*cosh(e + f*x)**2/(4*f**2) - 12*a**2*c*d**2*x*cosh(e + f*x)/f**2 + 3*a**2*c*d**2*sinh(e +
f*x)*cosh(e + f*x)/(4*f**3) + 12*a**2*c*d**2*sinh(e + f*x)/f**3 - a**2*d**3*x**4*sinh(e + f*x)**2/8 + a**2*d**
3*x**4*cosh(e + f*x)**2/8 + a**2*d**3*x**4/4 + a**2*d**3*x**3*sinh(e + f*x)*cosh(e + f*x)/(2*f) + 2*a**2*d**3*
x**3*sinh(e + f*x)/f - 3*a**2*d**3*x**2*sinh(e + f*x)**2/(8*f**2) - 3*a**2*d**3*x**2*cosh(e + f*x)**2/(8*f**2)
- 6*a**2*d**3*x**2*cosh(e + f*x)/f**2 + 3*a**2*d**3*x*sinh(e + f*x)*cosh(e + f*x)/(4*f**3) + 12*a**2*d**3*x*s
inh(e + f*x)/f**3 - 3*a**2*d**3*sinh(e + f*x)**2/(8*f**4) - 12*a**2*d**3*cosh(e + f*x)/f**4, Ne(f, 0)), ((a*co
sh(e) + a)**2*(c**3*x + 3*c**2*d*x**2/2 + c*d**2*x**3 + d**3*x**4/4), True))

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