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

Optimal. Leaf size=175 \[ -\frac{40 d^2 (c+d x) \cosh (a+b x)}{9 b^3}+\frac{2 d^2 (c+d x) \sinh ^2(a+b x) \cosh (a+b x)}{9 b^3}-\frac{d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}+\frac{2 d (c+d x)^2 \sinh (a+b x)}{b^2}-\frac{2 d^3 \sinh ^3(a+b x)}{27 b^4}+\frac{40 d^3 \sinh (a+b x)}{9 b^4}-\frac{2 (c+d x)^3 \cosh (a+b x)}{3 b}+\frac{(c+d x)^3 \sinh ^2(a+b x) \cosh (a+b x)}{3 b} \]

[Out]

(-40*d^2*(c + d*x)*Cosh[a + b*x])/(9*b^3) - (2*(c + d*x)^3*Cosh[a + b*x])/(3*b) + (40*d^3*Sinh[a + b*x])/(9*b^
4) + (2*d*(c + d*x)^2*Sinh[a + b*x])/b^2 + (2*d^2*(c + d*x)*Cosh[a + b*x]*Sinh[a + b*x]^2)/(9*b^3) + ((c + d*x
)^3*Cosh[a + b*x]*Sinh[a + b*x]^2)/(3*b) - (2*d^3*Sinh[a + b*x]^3)/(27*b^4) - (d*(c + d*x)^2*Sinh[a + b*x]^3)/
(3*b^2)

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Rubi [A]  time = 0.226485, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3311, 3296, 2637, 3310} \[ -\frac{40 d^2 (c+d x) \cosh (a+b x)}{9 b^3}+\frac{2 d^2 (c+d x) \sinh ^2(a+b x) \cosh (a+b x)}{9 b^3}-\frac{d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}+\frac{2 d (c+d x)^2 \sinh (a+b x)}{b^2}-\frac{2 d^3 \sinh ^3(a+b x)}{27 b^4}+\frac{40 d^3 \sinh (a+b x)}{9 b^4}-\frac{2 (c+d x)^3 \cosh (a+b x)}{3 b}+\frac{(c+d x)^3 \sinh ^2(a+b x) \cosh (a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3*Sinh[a + b*x]^3,x]

[Out]

(-40*d^2*(c + d*x)*Cosh[a + b*x])/(9*b^3) - (2*(c + d*x)^3*Cosh[a + b*x])/(3*b) + (40*d^3*Sinh[a + b*x])/(9*b^
4) + (2*d*(c + d*x)^2*Sinh[a + b*x])/b^2 + (2*d^2*(c + d*x)*Cosh[a + b*x]*Sinh[a + b*x]^2)/(9*b^3) + ((c + d*x
)^3*Cosh[a + b*x]*Sinh[a + b*x]^2)/(3*b) - (2*d^3*Sinh[a + b*x]^3)/(27*b^4) - (d*(c + d*x)^2*Sinh[a + b*x]^3)/
(3*b^2)

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 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 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 \sinh ^3(a+b x) \, dx &=\frac{(c+d x)^3 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac{d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}-\frac{2}{3} \int (c+d x)^3 \sinh (a+b x) \, dx+\frac{\left (2 d^2\right ) \int (c+d x) \sinh ^3(a+b x) \, dx}{3 b^2}\\ &=-\frac{2 (c+d x)^3 \cosh (a+b x)}{3 b}+\frac{2 d^2 (c+d x) \cosh (a+b x) \sinh ^2(a+b x)}{9 b^3}+\frac{(c+d x)^3 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac{2 d^3 \sinh ^3(a+b x)}{27 b^4}-\frac{d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}+\frac{(2 d) \int (c+d x)^2 \cosh (a+b x) \, dx}{b}-\frac{\left (4 d^2\right ) \int (c+d x) \sinh (a+b x) \, dx}{9 b^2}\\ &=-\frac{4 d^2 (c+d x) \cosh (a+b x)}{9 b^3}-\frac{2 (c+d x)^3 \cosh (a+b x)}{3 b}+\frac{2 d (c+d x)^2 \sinh (a+b x)}{b^2}+\frac{2 d^2 (c+d x) \cosh (a+b x) \sinh ^2(a+b x)}{9 b^3}+\frac{(c+d x)^3 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac{2 d^3 \sinh ^3(a+b x)}{27 b^4}-\frac{d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}-\frac{\left (4 d^2\right ) \int (c+d x) \sinh (a+b x) \, dx}{b^2}+\frac{\left (4 d^3\right ) \int \cosh (a+b x) \, dx}{9 b^3}\\ &=-\frac{40 d^2 (c+d x) \cosh (a+b x)}{9 b^3}-\frac{2 (c+d x)^3 \cosh (a+b x)}{3 b}+\frac{4 d^3 \sinh (a+b x)}{9 b^4}+\frac{2 d (c+d x)^2 \sinh (a+b x)}{b^2}+\frac{2 d^2 (c+d x) \cosh (a+b x) \sinh ^2(a+b x)}{9 b^3}+\frac{(c+d x)^3 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac{2 d^3 \sinh ^3(a+b x)}{27 b^4}-\frac{d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}+\frac{\left (4 d^3\right ) \int \cosh (a+b x) \, dx}{b^3}\\ &=-\frac{40 d^2 (c+d x) \cosh (a+b x)}{9 b^3}-\frac{2 (c+d x)^3 \cosh (a+b x)}{3 b}+\frac{40 d^3 \sinh (a+b x)}{9 b^4}+\frac{2 d (c+d x)^2 \sinh (a+b x)}{b^2}+\frac{2 d^2 (c+d x) \cosh (a+b x) \sinh ^2(a+b x)}{9 b^3}+\frac{(c+d x)^3 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac{2 d^3 \sinh ^3(a+b x)}{27 b^4}-\frac{d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}\\ \end{align*}

Mathematica [A]  time = 0.992243, size = 127, normalized size = 0.73 \[ \frac{-162 b (c+d x) \cosh (a+b x) \left (b^2 (c+d x)^2+6 d^2\right )+6 b (c+d x) \cosh (3 (a+b x)) \left (3 b^2 (c+d x)^2+2 d^2\right )-4 d \sinh (a+b x) \left (\cosh (2 (a+b x)) \left (9 b^2 (c+d x)^2+2 d^2\right )-117 b^2 (c+d x)^2-242 d^2\right )}{216 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-162*b*(c + d*x)*(6*d^2 + b^2*(c + d*x)^2)*Cosh[a + b*x] + 6*b*(c + d*x)*(2*d^2 + 3*b^2*(c + d*x)^2)*Cosh[3*(
a + b*x)] - 4*d*(-242*d^2 - 117*b^2*(c + d*x)^2 + (2*d^2 + 9*b^2*(c + d*x)^2)*Cosh[2*(a + b*x)])*Sinh[a + b*x]
)/(216*b^4)

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Maple [B]  time = 0.007, size = 676, normalized size = 3.9 \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*sinh(b*x+a)^3,x)

[Out]

1/b*(1/b^3*d^3*(-2/3*(b*x+a)^3*cosh(b*x+a)+1/3*(b*x+a)^3*cosh(b*x+a)*sinh(b*x+a)^2+7/3*(b*x+a)^2*sinh(b*x+a)-4
0/9*(b*x+a)*cosh(b*x+a)+122/27*sinh(b*x+a)-1/3*(b*x+a)^2*sinh(b*x+a)*cosh(b*x+a)^2+2/9*(b*x+a)*sinh(b*x+a)^2*c
osh(b*x+a)-2/27*sinh(b*x+a)*cosh(b*x+a)^2)-3/b^3*d^3*a*(1/3*(b*x+a)^2*sinh(b*x+a)^2*cosh(b*x+a)-2/3*(b*x+a)^2*
cosh(b*x+a)-2/9*(b*x+a)*sinh(b*x+a)*cosh(b*x+a)^2+14/9*(b*x+a)*sinh(b*x+a)+2/27*sinh(b*x+a)^2*cosh(b*x+a)-40/2
7*cosh(b*x+a))+3/b^3*d^3*a^2*(-2/3*(b*x+a)*cosh(b*x+a)+1/3*(b*x+a)*sinh(b*x+a)^2*cosh(b*x+a)+7/9*sinh(b*x+a)-1
/9*sinh(b*x+a)*cosh(b*x+a)^2)-1/b^3*d^3*a^3*(-2/3+1/3*sinh(b*x+a)^2)*cosh(b*x+a)+3/b^2*c*d^2*(1/3*(b*x+a)^2*si
nh(b*x+a)^2*cosh(b*x+a)-2/3*(b*x+a)^2*cosh(b*x+a)-2/9*(b*x+a)*sinh(b*x+a)*cosh(b*x+a)^2+14/9*(b*x+a)*sinh(b*x+
a)+2/27*sinh(b*x+a)^2*cosh(b*x+a)-40/27*cosh(b*x+a))-6/b^2*c*d^2*a*(-2/3*(b*x+a)*cosh(b*x+a)+1/3*(b*x+a)*sinh(
b*x+a)^2*cosh(b*x+a)+7/9*sinh(b*x+a)-1/9*sinh(b*x+a)*cosh(b*x+a)^2)+3/b^2*c*d^2*a^2*(-2/3+1/3*sinh(b*x+a)^2)*c
osh(b*x+a)+3/b*c^2*d*(-2/3*(b*x+a)*cosh(b*x+a)+1/3*(b*x+a)*sinh(b*x+a)^2*cosh(b*x+a)+7/9*sinh(b*x+a)-1/9*sinh(
b*x+a)*cosh(b*x+a)^2)-3/b*c^2*d*a*(-2/3+1/3*sinh(b*x+a)^2)*cosh(b*x+a)+c^3*(-2/3+1/3*sinh(b*x+a)^2)*cosh(b*x+a
))

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Maxima [B]  time = 1.37749, size = 587, normalized size = 3.35 \begin{align*} \frac{1}{24} \, c^{2} d{\left (\frac{{\left (3 \, b x e^{\left (3 \, a\right )} - e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{2}} - \frac{27 \,{\left (b x e^{a} - e^{a}\right )} e^{\left (b x\right )}}{b^{2}} - \frac{27 \,{\left (b x + 1\right )} e^{\left (-b x - a\right )}}{b^{2}} + \frac{{\left (3 \, b x + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{2}}\right )} + \frac{1}{24} \, c^{3}{\left (\frac{e^{\left (3 \, b x + 3 \, a\right )}}{b} - \frac{9 \, e^{\left (b x + a\right )}}{b} - \frac{9 \, e^{\left (-b x - a\right )}}{b} + \frac{e^{\left (-3 \, b x - 3 \, a\right )}}{b}\right )} + \frac{1}{72} \, c d^{2}{\left (\frac{{\left (9 \, b^{2} x^{2} e^{\left (3 \, a\right )} - 6 \, b x e^{\left (3 \, a\right )} + 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{3}} - \frac{81 \,{\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} e^{\left (b x\right )}}{b^{3}} - \frac{81 \,{\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x - a\right )}}{b^{3}} + \frac{{\left (9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{3}}\right )} + \frac{1}{216} \, d^{3}{\left (\frac{{\left (9 \, b^{3} x^{3} e^{\left (3 \, a\right )} - 9 \, b^{2} x^{2} e^{\left (3 \, a\right )} + 6 \, b x e^{\left (3 \, a\right )} - 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{4}} - \frac{81 \,{\left (b^{3} x^{3} e^{a} - 3 \, b^{2} x^{2} e^{a} + 6 \, b x e^{a} - 6 \, e^{a}\right )} e^{\left (b x\right )}}{b^{4}} - \frac{81 \,{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{b^{4}} + \frac{{\left (9 \, b^{3} x^{3} + 9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{4}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*c^2*d*((3*b*x*e^(3*a) - e^(3*a))*e^(3*b*x)/b^2 - 27*(b*x*e^a - e^a)*e^(b*x)/b^2 - 27*(b*x + 1)*e^(-b*x -
a)/b^2 + (3*b*x + 1)*e^(-3*b*x - 3*a)/b^2) + 1/24*c^3*(e^(3*b*x + 3*a)/b - 9*e^(b*x + a)/b - 9*e^(-b*x - a)/b
+ e^(-3*b*x - 3*a)/b) + 1/72*c*d^2*((9*b^2*x^2*e^(3*a) - 6*b*x*e^(3*a) + 2*e^(3*a))*e^(3*b*x)/b^3 - 81*(b^2*x^
2*e^a - 2*b*x*e^a + 2*e^a)*e^(b*x)/b^3 - 81*(b^2*x^2 + 2*b*x + 2)*e^(-b*x - a)/b^3 + (9*b^2*x^2 + 6*b*x + 2)*e
^(-3*b*x - 3*a)/b^3) + 1/216*d^3*((9*b^3*x^3*e^(3*a) - 9*b^2*x^2*e^(3*a) + 6*b*x*e^(3*a) - 2*e^(3*a))*e^(3*b*x
)/b^4 - 81*(b^3*x^3*e^a - 3*b^2*x^2*e^a + 6*b*x*e^a - 6*e^a)*e^(b*x)/b^4 - 81*(b^3*x^3 + 3*b^2*x^2 + 6*b*x + 6
)*e^(-b*x - a)/b^4 + (9*b^3*x^3 + 9*b^2*x^2 + 6*b*x + 2)*e^(-3*b*x - 3*a)/b^4)

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Fricas [B]  time = 2.57785, size = 759, normalized size = 4.34 \begin{align*} \frac{3 \,{\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{3} + 2 \, b c d^{2} +{\left (9 \, b^{3} c^{2} d + 2 \, b d^{3}\right )} x\right )} \cosh \left (b x + a\right )^{3} + 9 \,{\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{3} + 2 \, b c d^{2} +{\left (9 \, b^{3} c^{2} d + 2 \, b d^{3}\right )} x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} -{\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d + 2 \, d^{3}\right )} \sinh \left (b x + a\right )^{3} - 81 \,{\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 (81 \, b^{2} d^{3} x^{2} + 162 \, b^{2} c d^{2} x + 81 \, b^{2} c^{2} d + 162 \, d^{3} -{\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d + 2 \, d^{3}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )}{108 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/108*(3*(3*b^3*d^3*x^3 + 9*b^3*c*d^2*x^2 + 3*b^3*c^3 + 2*b*c*d^2 + (9*b^3*c^2*d + 2*b*d^3)*x)*cosh(b*x + a)^3
+ 9*(3*b^3*d^3*x^3 + 9*b^3*c*d^2*x^2 + 3*b^3*c^3 + 2*b*c*d^2 + (9*b^3*c^2*d + 2*b*d^3)*x)*cosh(b*x + a)*sinh(
b*x + a)^2 - (9*b^2*d^3*x^2 + 18*b^2*c*d^2*x + 9*b^2*c^2*d + 2*d^3)*sinh(b*x + a)^3 - 81*(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*(81*b^2*d^3*x^2 + 162*b^2*c*d^2
*x + 81*b^2*c^2*d + 162*d^3 - (9*b^2*d^3*x^2 + 18*b^2*c*d^2*x + 9*b^2*c^2*d + 2*d^3)*cosh(b*x + a)^2)*sinh(b*x
+ a))/b^4

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Sympy [A]  time = 6.79592, size = 495, normalized size = 2.83 \begin{align*} \begin{cases} \frac{c^{3} \sinh ^{2}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{b} - \frac{2 c^{3} \cosh ^{3}{\left (a + b x \right )}}{3 b} + \frac{3 c^{2} d x \sinh ^{2}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{b} - \frac{2 c^{2} d x \cosh ^{3}{\left (a + b x \right )}}{b} + \frac{3 c d^{2} x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{b} - \frac{2 c d^{2} x^{2} \cosh ^{3}{\left (a + b x \right )}}{b} + \frac{d^{3} x^{3} \sinh ^{2}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{b} - \frac{2 d^{3} x^{3} \cosh ^{3}{\left (a + b x \right )}}{3 b} - \frac{7 c^{2} d \sinh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac{2 c^{2} d \sinh{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b^{2}} - \frac{14 c d^{2} x \sinh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac{4 c d^{2} x \sinh{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b^{2}} - \frac{7 d^{3} x^{2} \sinh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac{2 d^{3} x^{2} \sinh{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b^{2}} + \frac{14 c d^{2} \sinh ^{2}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{3 b^{3}} - \frac{40 c d^{2} \cosh ^{3}{\left (a + b x \right )}}{9 b^{3}} + \frac{14 d^{3} x \sinh ^{2}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{3 b^{3}} - \frac{40 d^{3} x \cosh ^{3}{\left (a + b x \right )}}{9 b^{3}} - \frac{122 d^{3} \sinh ^{3}{\left (a + b x \right )}}{27 b^{4}} + \frac{40 d^{3} \sinh{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{9 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 ^{3}{\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)**3,x)

[Out]

Piecewise((c**3*sinh(a + b*x)**2*cosh(a + b*x)/b - 2*c**3*cosh(a + b*x)**3/(3*b) + 3*c**2*d*x*sinh(a + b*x)**2
*cosh(a + b*x)/b - 2*c**2*d*x*cosh(a + b*x)**3/b + 3*c*d**2*x**2*sinh(a + b*x)**2*cosh(a + b*x)/b - 2*c*d**2*x
**2*cosh(a + b*x)**3/b + d**3*x**3*sinh(a + b*x)**2*cosh(a + b*x)/b - 2*d**3*x**3*cosh(a + b*x)**3/(3*b) - 7*c
**2*d*sinh(a + b*x)**3/(3*b**2) + 2*c**2*d*sinh(a + b*x)*cosh(a + b*x)**2/b**2 - 14*c*d**2*x*sinh(a + b*x)**3/
(3*b**2) + 4*c*d**2*x*sinh(a + b*x)*cosh(a + b*x)**2/b**2 - 7*d**3*x**2*sinh(a + b*x)**3/(3*b**2) + 2*d**3*x**
2*sinh(a + b*x)*cosh(a + b*x)**2/b**2 + 14*c*d**2*sinh(a + b*x)**2*cosh(a + b*x)/(3*b**3) - 40*c*d**2*cosh(a +
b*x)**3/(9*b**3) + 14*d**3*x*sinh(a + b*x)**2*cosh(a + b*x)/(3*b**3) - 40*d**3*x*cosh(a + b*x)**3/(9*b**3) -
122*d**3*sinh(a + b*x)**3/(27*b**4) + 40*d**3*sinh(a + b*x)*cosh(a + b*x)**2/(9*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)**3, True))

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Giac [B]  time = 1.23016, size = 559, normalized size = 3.19 \begin{align*} \frac{{\left (9 \, b^{3} d^{3} x^{3} + 27 \, b^{3} c d^{2} x^{2} + 27 \, b^{3} c^{2} d x - 9 \, b^{2} d^{3} x^{2} + 9 \, b^{3} c^{3} - 18 \, b^{2} c d^{2} x - 9 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} - 2 \, d^{3}\right )} e^{\left (3 \, b x + 3 \, a\right )}}{216 \, b^{4}} - \frac{3 \,{\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 )}}{8 \, b^{4}} - \frac{3 \,{\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 )}}{8 \, b^{4}} + \frac{{\left (9 \, b^{3} d^{3} x^{3} + 27 \, b^{3} c d^{2} x^{2} + 27 \, b^{3} c^{2} d x + 9 \, b^{2} d^{3} x^{2} + 9 \, b^{3} c^{3} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} + 2 \, d^{3}\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{216 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/216*(9*b^3*d^3*x^3 + 27*b^3*c*d^2*x^2 + 27*b^3*c^2*d*x - 9*b^2*d^3*x^2 + 9*b^3*c^3 - 18*b^2*c*d^2*x - 9*b^2*
c^2*d + 6*b*d^3*x + 6*b*c*d^2 - 2*d^3)*e^(3*b*x + 3*a)/b^4 - 3/8*(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 - 3
/8*(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 + 1/216*(9*b^3*d^3*x^3 + 27*b^3*c*d^2*x^2 + 27*b^3*c^2*d*x + 9*b
^2*d^3*x^2 + 9*b^3*c^3 + 18*b^2*c*d^2*x + 9*b^2*c^2*d + 6*b*d^3*x + 6*b*c*d^2 + 2*d^3)*e^(-3*b*x - 3*a)/b^4