3.163 \(\int (c+d x)^3 (a+b \sinh (e+f x))^2 \, dx\)

Optimal. Leaf size=250 \[ \frac{a^2 (c+d x)^4}{4 d}+\frac{12 a b d^2 (c+d x) \cosh (e+f x)}{f^3}-\frac{6 a b d (c+d x)^2 \sinh (e+f x)}{f^2}+\frac{2 a b (c+d x)^3 \cosh (e+f x)}{f}-\frac{12 a b d^3 \sinh (e+f x)}{f^4}+\frac{3 b^2 d^2 (c+d x) \sinh (e+f x) \cosh (e+f x)}{4 f^3}-\frac{3 b^2 c d^2 x}{4 f^2}-\frac{3 b^2 d (c+d x)^2 \sinh ^2(e+f x)}{4 f^2}+\frac{b^2 (c+d x)^3 \sinh (e+f x) \cosh (e+f x)}{2 f}-\frac{b^2 (c+d x)^4}{8 d}-\frac{3 b^2 d^3 \sinh ^2(e+f x)}{8 f^4}-\frac{3 b^2 d^3 x^2}{8 f^2} \]

[Out]

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

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Rubi [A]  time = 0.289963, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3317, 3296, 2637, 3311, 32, 3310} \[ \frac{a^2 (c+d x)^4}{4 d}+\frac{12 a b d^2 (c+d x) \cosh (e+f x)}{f^3}-\frac{6 a b d (c+d x)^2 \sinh (e+f x)}{f^2}+\frac{2 a b (c+d x)^3 \cosh (e+f x)}{f}-\frac{12 a b d^3 \sinh (e+f x)}{f^4}+\frac{3 b^2 d^2 (c+d x) \sinh (e+f x) \cosh (e+f x)}{4 f^3}-\frac{3 b^2 c d^2 x}{4 f^2}-\frac{3 b^2 d (c+d x)^2 \sinh ^2(e+f x)}{4 f^2}+\frac{b^2 (c+d x)^3 \sinh (e+f x) \cosh (e+f x)}{2 f}-\frac{b^2 (c+d x)^4}{8 d}-\frac{3 b^2 d^3 \sinh ^2(e+f x)}{8 f^4}-\frac{3 b^2 d^3 x^2}{8 f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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])

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]

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

Mathematica [A]  time = 1.38262, size = 235, normalized size = 0.94 \[ \frac{2 \left (f^4 x \left (2 a^2-b^2\right ) \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right )-48 a b d \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2+2\right )\right ) \sinh (e+f x)+b^2 f (c+d x) \left (2 c^2 f^2+4 c d f^2 x+d^2 \left (2 f^2 x^2+3\right )\right ) \sinh (2 (e+f x))\right )+32 a b f (c+d x) \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2+6\right )\right ) \cosh (e+f x)-3 b^2 d \left (2 c^2 f^2+4 c d f^2 x+d^2 \left (2 f^2 x^2+1\right )\right ) \cosh (2 (e+f x))}{16 f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(32*a*b*f*(c + d*x)*(c^2*f^2 + 2*c*d*f^2*x + d^2*(6 + f^2*x^2))*Cosh[e + f*x] - 3*b^2*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*((2*a^2 - b^2)*f^4*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^
3) - 48*a*b*d*(c^2*f^2 + 2*c*d*f^2*x + d^2*(2 + f^2*x^2))*Sinh[e + f*x] + b^2*f*(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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Maple [B]  time = 0.014, size = 1061, normalized size = 4.2 \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*(a+b*sinh(f*x+e))^2,x)

[Out]

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

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Maxima [B]  time = 1.3345, size = 702, normalized size = 2.81 \begin{align*} \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 )} b^{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 )} b^{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 )} b^{2} d^{3} - \frac{1}{8} \, b^{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 b 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 b 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 b 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 b c^{3} \cosh \left (f x + e\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [A]  time = 2.4951, size = 880, normalized size = 3.52 \begin{align*} \frac{2 \,{\left (2 \, a^{2} - b^{2}\right )} d^{3} f^{4} x^{4} + 8 \,{\left (2 \, a^{2} - b^{2}\right )} c d^{2} f^{4} x^{3} + 12 \,{\left (2 \, a^{2} - b^{2}\right )} c^{2} d f^{4} x^{2} + 8 \,{\left (2 \, a^{2} - b^{2}\right )} c^{3} f^{4} x - 3 \,{\left (2 \, b^{2} d^{3} f^{2} x^{2} + 4 \, b^{2} c d^{2} f^{2} x + 2 \, b^{2} c^{2} d f^{2} + b^{2} d^{3}\right )} \cosh \left (f x + e\right )^{2} - 3 \,{\left (2 \, b^{2} d^{3} f^{2} x^{2} + 4 \, b^{2} c d^{2} f^{2} x + 2 \, b^{2} c^{2} d f^{2} + b^{2} d^{3}\right )} \sinh \left (f x + e\right )^{2} + 32 \,{\left (a b d^{3} f^{3} x^{3} + 3 \, a b c d^{2} f^{3} x^{2} + a b c^{3} f^{3} + 6 \, a b c d^{2} f + 3 \,{\left (a b c^{2} d f^{3} + 2 \, a b d^{3} f\right )} x\right )} \cosh \left (f x + e\right ) - 4 \,{\left (24 \, a b d^{3} f^{2} x^{2} + 48 \, a b c d^{2} f^{2} x + 24 \, a b c^{2} d f^{2} + 48 \, a b d^{3} -{\left (2 \, b^{2} d^{3} f^{3} x^{3} + 6 \, b^{2} c d^{2} f^{3} x^{2} + 2 \, b^{2} c^{3} f^{3} + 3 \, b^{2} c d^{2} f + 3 \,{\left (2 \, b^{2} c^{2} d f^{3} + b^{2} d^{3} f\right )} x\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )}{16 \, f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(2*(2*a^2 - b^2)*d^3*f^4*x^4 + 8*(2*a^2 - b^2)*c*d^2*f^4*x^3 + 12*(2*a^2 - b^2)*c^2*d*f^4*x^2 + 8*(2*a^2
- b^2)*c^3*f^4*x - 3*(2*b^2*d^3*f^2*x^2 + 4*b^2*c*d^2*f^2*x + 2*b^2*c^2*d*f^2 + b^2*d^3)*cosh(f*x + e)^2 - 3*(
2*b^2*d^3*f^2*x^2 + 4*b^2*c*d^2*f^2*x + 2*b^2*c^2*d*f^2 + b^2*d^3)*sinh(f*x + e)^2 + 32*(a*b*d^3*f^3*x^3 + 3*a
*b*c*d^2*f^3*x^2 + a*b*c^3*f^3 + 6*a*b*c*d^2*f + 3*(a*b*c^2*d*f^3 + 2*a*b*d^3*f)*x)*cosh(f*x + e) - 4*(24*a*b*
d^3*f^2*x^2 + 48*a*b*c*d^2*f^2*x + 24*a*b*c^2*d*f^2 + 48*a*b*d^3 - (2*b^2*d^3*f^3*x^3 + 6*b^2*c*d^2*f^3*x^2 +
2*b^2*c^3*f^3 + 3*b^2*c*d^2*f + 3*(2*b^2*c^2*d*f^3 + b^2*d^3*f)*x)*cosh(f*x + e))*sinh(f*x + e))/f^4

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Sympy [A]  time = 5.31346, size = 779, normalized size = 3.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*c**3*x + 3*a**2*c**2*d*x**2/2 + a**2*c*d**2*x**3 + a**2*d**3*x**4/4 + 2*a*b*c**3*cosh(e + f*x)
/f + 6*a*b*c**2*d*x*cosh(e + f*x)/f - 6*a*b*c**2*d*sinh(e + f*x)/f**2 + 6*a*b*c*d**2*x**2*cosh(e + f*x)/f - 12
*a*b*c*d**2*x*sinh(e + f*x)/f**2 + 12*a*b*c*d**2*cosh(e + f*x)/f**3 + 2*a*b*d**3*x**3*cosh(e + f*x)/f - 6*a*b*
d**3*x**2*sinh(e + f*x)/f**2 + 12*a*b*d**3*x*cosh(e + f*x)/f**3 - 12*a*b*d**3*sinh(e + f*x)/f**4 + b**2*c**3*x
*sinh(e + f*x)**2/2 - b**2*c**3*x*cosh(e + f*x)**2/2 + b**2*c**3*sinh(e + f*x)*cosh(e + f*x)/(2*f) + 3*b**2*c*
*2*d*x**2*sinh(e + f*x)**2/4 - 3*b**2*c**2*d*x**2*cosh(e + f*x)**2/4 + 3*b**2*c**2*d*x*sinh(e + f*x)*cosh(e +
f*x)/(2*f) - 3*b**2*c**2*d*sinh(e + f*x)**2/(4*f**2) + b**2*c*d**2*x**3*sinh(e + f*x)**2/2 - b**2*c*d**2*x**3*
cosh(e + f*x)**2/2 + 3*b**2*c*d**2*x**2*sinh(e + f*x)*cosh(e + f*x)/(2*f) - 3*b**2*c*d**2*x*sinh(e + f*x)**2/(
4*f**2) - 3*b**2*c*d**2*x*cosh(e + f*x)**2/(4*f**2) + 3*b**2*c*d**2*sinh(e + f*x)*cosh(e + f*x)/(4*f**3) + b**
2*d**3*x**4*sinh(e + f*x)**2/8 - b**2*d**3*x**4*cosh(e + f*x)**2/8 + b**2*d**3*x**3*sinh(e + f*x)*cosh(e + f*x
)/(2*f) - 3*b**2*d**3*x**2*sinh(e + f*x)**2/(8*f**2) - 3*b**2*d**3*x**2*cosh(e + f*x)**2/(8*f**2) + 3*b**2*d**
3*x*sinh(e + f*x)*cosh(e + f*x)/(4*f**3) - 3*b**2*d**3*sinh(e + f*x)**2/(8*f**4), Ne(f, 0)), ((a + b*sinh(e))*
*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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Giac [B]  time = 1.26267, size = 813, normalized size = 3.25 \begin{align*} \frac{1}{4} \, a^{2} d^{3} x^{4} - \frac{1}{8} \, b^{2} d^{3} x^{4} + a^{2} c d^{2} x^{3} - \frac{1}{2} \, b^{2} c d^{2} x^{3} + \frac{3}{2} \, a^{2} c^{2} d x^{2} - \frac{3}{4} \, b^{2} c^{2} d x^{2} + a^{2} c^{3} x - \frac{1}{2} \, b^{2} c^{3} x + \frac{{\left (4 \, b^{2} d^{3} f^{3} x^{3} + 12 \, b^{2} c d^{2} f^{3} x^{2} + 12 \, b^{2} c^{2} d f^{3} x - 6 \, b^{2} d^{3} f^{2} x^{2} + 4 \, b^{2} c^{3} f^{3} - 12 \, b^{2} c d^{2} f^{2} x - 6 \, b^{2} c^{2} d f^{2} + 6 \, b^{2} d^{3} f x + 6 \, b^{2} c d^{2} f - 3 \, b^{2} d^{3}\right )} e^{\left (2 \, f x + 2 \, e\right )}}{32 \, f^{4}} + \frac{{\left (a b d^{3} f^{3} x^{3} + 3 \, a b c d^{2} f^{3} x^{2} + 3 \, a b c^{2} d f^{3} x - 3 \, a b d^{3} f^{2} x^{2} + a b c^{3} f^{3} - 6 \, a b c d^{2} f^{2} x - 3 \, a b c^{2} d f^{2} + 6 \, a b d^{3} f x + 6 \, a b c d^{2} f - 6 \, a b d^{3}\right )} e^{\left (f x + e\right )}}{f^{4}} + \frac{{\left (a b d^{3} f^{3} x^{3} + 3 \, a b c d^{2} f^{3} x^{2} + 3 \, a b c^{2} d f^{3} x + 3 \, a b d^{3} f^{2} x^{2} + a b c^{3} f^{3} + 6 \, a b c d^{2} f^{2} x + 3 \, a b c^{2} d f^{2} + 6 \, a b d^{3} f x + 6 \, a b c d^{2} f + 6 \, a b d^{3}\right )} e^{\left (-f x - e\right )}}{f^{4}} - \frac{{\left (4 \, b^{2} d^{3} f^{3} x^{3} + 12 \, b^{2} c d^{2} f^{3} x^{2} + 12 \, b^{2} c^{2} d f^{3} x + 6 \, b^{2} d^{3} f^{2} x^{2} + 4 \, b^{2} c^{3} f^{3} + 12 \, b^{2} c d^{2} f^{2} x + 6 \, b^{2} c^{2} d f^{2} + 6 \, b^{2} d^{3} f x + 6 \, b^{2} c d^{2} f + 3 \, b^{2} d^{3}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{32 \, f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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