3.1.17 \(\int (c+d x)^3 \sinh ^3(a+b x) \, dx\) [17]

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

[Out]

-40/9*d^2*(d*x+c)*cosh(b*x+a)/b^3-2/3*(d*x+c)^3*cosh(b*x+a)/b+40/9*d^3*sinh(b*x+a)/b^4+2*d*(d*x+c)^2*sinh(b*x+
a)/b^2+2/9*d^2*(d*x+c)*cosh(b*x+a)*sinh(b*x+a)^2/b^3+1/3*(d*x+c)^3*cosh(b*x+a)*sinh(b*x+a)^2/b-2/27*d^3*sinh(b
*x+a)^3/b^4-1/3*d*(d*x+c)^2*sinh(b*x+a)^3/b^2

________________________________________________________________________________________

Rubi [A]
time = 0.16, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3392, 3377, 2717, 3391} \begin {gather*} -\frac {2 d^3 \sinh ^3(a+b x)}{27 b^4}+\frac {40 d^3 \sinh (a+b x)}{9 b^4}-\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 (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} \end {gather*}

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 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

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 3391

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 3392

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 \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 = 0.62, size = 127, normalized size = 0.73 \begin {gather*} \frac {-162 b (c+d x) \left (6 d^2+b^2 (c+d x)^2\right ) \cosh (a+b x)+6 b (c+d x) \left (2 d^2+3 b^2 (c+d x)^2\right ) \cosh (3 (a+b x))-4 d \left (-242 d^2-117 b^2 (c+d x)^2+\left (2 d^2+9 b^2 (c+d x)^2\right ) \cosh (2 (a+b x))\right ) \sinh (a+b x)}{216 b^4} \end {gather*}

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)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(709\) vs. \(2(161)=322\).
time = 0.43, size = 710, normalized size = 4.06

method	result	size
risch	\(\frac {\left (9 d^{3} x^{3} b^{3}+27 b^{3} c \,d^{2} x^{2}+27 b^{3} c^{2} d x -9 b^{2} d^{3} x^{2}+9 b^{3} c^{3}-18 b^{2} c \,d^{2} x -9 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}-2 d^{3}\right ) {\mathrm e}^{3 b x +3 a}}{216 b^{4}}-\frac {3 \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x -3 b^{2} d^{3} x^{2}+b^{3} c^{3}-6 b^{2} c \,d^{2} x -3 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}-6 d^{3}\right ) {\mathrm e}^{b x +a}}{8 b^{4}}-\frac {3 \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +3 b^{2} d^{3} x^{2}+b^{3} c^{3}+6 b^{2} c \,d^{2} x +3 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}+6 d^{3}\right ) {\mathrm e}^{-b x -a}}{8 b^{4}}+\frac {\left (9 d^{3} x^{3} b^{3}+27 b^{3} c \,d^{2} x^{2}+27 b^{3} c^{2} d x +9 b^{2} d^{3} x^{2}+9 b^{3} c^{3}+18 b^{2} c \,d^{2} x +9 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}+2 d^{3}\right ) {\mathrm e}^{-3 b x -3 a}}{216 b^{4}}\)	\(415\)
default	\(\frac {-\frac {3 d^{3} \left (\left (b x +a \right )^{3} \cosh \left (b x +a \right )-3 \left (b x +a \right )^{2} \sinh \left (b x +a \right )+6 \left (b x +a \right ) \cosh \left (b x +a \right )-6 \sinh \left (b x +a \right )\right )}{4 b^{3}}+\frac {9 d^{3} a \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )}{4 b^{3}}-\frac {9 d^{3} a^{2} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{4 b^{3}}+\frac {3 d^{3} a^{3} \cosh \left (b x +a \right )}{4 b^{3}}-\frac {9 c \,d^{2} \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )}{4 b^{2}}+\frac {9 c \,d^{2} a \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{2 b^{2}}-\frac {9 c \,d^{2} a^{2} \cosh \left (b x +a \right )}{4 b^{2}}-\frac {9 c^{2} d \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{4 b}+\frac {9 c^{2} d a \cosh \left (b x +a \right )}{4 b}-\frac {3 c^{3} \cosh \left (b x +a \right )}{4}}{b}+\frac {\frac {d^{3} \left (\left (3 b x +3 a \right )^{3} \cosh \left (3 b x +3 a \right )-3 \left (3 b x +3 a \right )^{2} \sinh \left (3 b x +3 a \right )+6 \left (3 b x +3 a \right ) \cosh \left (3 b x +3 a \right )-6 \sinh \left (3 b x +3 a \right )\right )}{b^{3}}-\frac {9 d^{3} a \left (\left (3 b x +3 a \right )^{2} \cosh \left (3 b x +3 a \right )-2 \left (3 b x +3 a \right ) \sinh \left (3 b x +3 a \right )+2 \cosh \left (3 b x +3 a \right )\right )}{b^{3}}+\frac {27 d^{3} a^{2} \left (\left (3 b x +3 a \right ) \cosh \left (3 b x +3 a \right )-\sinh \left (3 b x +3 a \right )\right )}{b^{3}}-\frac {27 d^{3} a^{3} \cosh \left (3 b x +3 a \right )}{b^{3}}+\frac {9 c \,d^{2} \left (\left (3 b x +3 a \right )^{2} \cosh \left (3 b x +3 a \right )-2 \left (3 b x +3 a \right ) \sinh \left (3 b x +3 a \right )+2 \cosh \left (3 b x +3 a \right )\right )}{b^{2}}-\frac {54 c \,d^{2} a \left (\left (3 b x +3 a \right ) \cosh \left (3 b x +3 a \right )-\sinh \left (3 b x +3 a \right )\right )}{b^{2}}+\frac {81 c \,d^{2} a^{2} \cosh \left (3 b x +3 a \right )}{b^{2}}+\frac {27 c^{2} d \left (\left (3 b x +3 a \right ) \cosh \left (3 b x +3 a \right )-\sinh \left (3 b x +3 a \right )\right )}{b}-\frac {81 c^{2} d a \cosh \left (3 b x +3 a \right )}{b}+27 c^{3} \cosh \left (3 b x +3 a \right )}{324 b}\)	\(710\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3*sinh(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

3/4/b*(-1/b^3*d^3*((b*x+a)^3*cosh(b*x+a)-3*(b*x+a)^2*sinh(b*x+a)+6*(b*x+a)*cosh(b*x+a)-6*sinh(b*x+a))+3/b^3*d^
3*a*((b*x+a)^2*cosh(b*x+a)-2*(b*x+a)*sinh(b*x+a)+2*cosh(b*x+a))-3/b^3*d^3*a^2*((b*x+a)*cosh(b*x+a)-sinh(b*x+a)
)+1/b^3*d^3*a^3*cosh(b*x+a)-3/b^2*c*d^2*((b*x+a)^2*cosh(b*x+a)-2*(b*x+a)*sinh(b*x+a)+2*cosh(b*x+a))+6/b^2*c*d^
2*a*((b*x+a)*cosh(b*x+a)-sinh(b*x+a))-3/b^2*c*d^2*a^2*cosh(b*x+a)-3/b*c^2*d*((b*x+a)*cosh(b*x+a)-sinh(b*x+a))+
3/b*c^2*d*a*cosh(b*x+a)-c^3*cosh(b*x+a))+1/324/b*(1/b^3*d^3*((3*b*x+3*a)^3*cosh(3*b*x+3*a)-3*(3*b*x+3*a)^2*sin
h(3*b*x+3*a)+6*(3*b*x+3*a)*cosh(3*b*x+3*a)-6*sinh(3*b*x+3*a))-9/b^3*d^3*a*((3*b*x+3*a)^2*cosh(3*b*x+3*a)-2*(3*
b*x+3*a)*sinh(3*b*x+3*a)+2*cosh(3*b*x+3*a))+27/b^3*d^3*a^2*((3*b*x+3*a)*cosh(3*b*x+3*a)-sinh(3*b*x+3*a))-27/b^
3*d^3*a^3*cosh(3*b*x+3*a)+9/b^2*c*d^2*((3*b*x+3*a)^2*cosh(3*b*x+3*a)-2*(3*b*x+3*a)*sinh(3*b*x+3*a)+2*cosh(3*b*
x+3*a))-54/b^2*c*d^2*a*((3*b*x+3*a)*cosh(3*b*x+3*a)-sinh(3*b*x+3*a))+81/b^2*c*d^2*a^2*cosh(3*b*x+3*a)+27/b*c^2
*d*((3*b*x+3*a)*cosh(3*b*x+3*a)-sinh(3*b*x+3*a))-81/b*c^2*d*a*cosh(3*b*x+3*a)+27*c^3*cosh(3*b*x+3*a))

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (161) = 322\).
time = 0.30, size = 435, normalized size = 2.49 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (161) = 322\).
time = 0.35, size = 345, normalized size = 1.97 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (173) = 346\).
time = 0.53, size = 495, normalized size = 2.83 \begin {gather*} \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 {gather*}

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))

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 414 vs. \(2 (161) = 322\).
time = 0.41, size = 414, normalized size = 2.37 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

Mupad [B]
time = 0.35, size = 364, normalized size = 2.08 \begin {gather*} \frac {\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (3\,b^2\,c^3+14\,c\,d^2\right )}{3\,b^3}-\frac {{\mathrm {sinh}\left (a+b\,x\right )}^3\,\left (63\,b^2\,c^2\,d+122\,d^3\right )}{27\,b^4}-\frac {2\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,\left (3\,b^2\,c^3+20\,c\,d^2\right )}{9\,b^3}+\frac {2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )\,\left (9\,b^2\,c^2\,d+20\,d^3\right )}{9\,b^4}-\frac {2\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,\left (9\,b^2\,c^2\,d+20\,d^3\right )}{9\,b^3}-\frac {2\,d^3\,x^3\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{3\,b}-\frac {7\,d^3\,x^2\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{3\,b^2}-\frac {14\,c\,d^2\,x\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{3\,b^2}+\frac {x\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (9\,b^2\,c^2\,d+14\,d^3\right )}{3\,b^3}-\frac {2\,c\,d^2\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{b}+\frac {d^3\,x^3\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{b}+\frac {2\,d^3\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{b^2}+\frac {3\,c\,d^2\,x^2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{b}+\frac {4\,c\,d^2\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b*x)^3*(c + d*x)^3,x)

[Out]

(cosh(a + b*x)*sinh(a + b*x)^2*(14*c*d^2 + 3*b^2*c^3))/(3*b^3) - (sinh(a + b*x)^3*(122*d^3 + 63*b^2*c^2*d))/(2
7*b^4) - (2*cosh(a + b*x)^3*(20*c*d^2 + 3*b^2*c^3))/(9*b^3) + (2*cosh(a + b*x)^2*sinh(a + b*x)*(20*d^3 + 9*b^2
*c^2*d))/(9*b^4) - (2*x*cosh(a + b*x)^3*(20*d^3 + 9*b^2*c^2*d))/(9*b^3) - (2*d^3*x^3*cosh(a + b*x)^3)/(3*b) -
(7*d^3*x^2*sinh(a + b*x)^3)/(3*b^2) - (14*c*d^2*x*sinh(a + b*x)^3)/(3*b^2) + (x*cosh(a + b*x)*sinh(a + b*x)^2*
(14*d^3 + 9*b^2*c^2*d))/(3*b^3) - (2*c*d^2*x^2*cosh(a + b*x)^3)/b + (d^3*x^3*cosh(a + b*x)*sinh(a + b*x)^2)/b
+ (2*d^3*x^2*cosh(a + b*x)^2*sinh(a + b*x))/b^2 + (3*c*d^2*x^2*cosh(a + b*x)*sinh(a + b*x)^2)/b + (4*c*d^2*x*c
osh(a + b*x)^2*sinh(a + b*x))/b^2

________________________________________________________________________________________