3.1.8 \(\int (c+d x)^4 \sinh ^2(a+b x) \, dx\) [8]

Optimal. Leaf size=162 \[ -\frac {3 d^4 x}{4 b^4}-\frac {d (c+d x)^3}{2 b^2}-\frac {(c+d x)^5}{10 d}+\frac {3 d^4 \cosh (a+b x) \sinh (a+b x)}{4 b^5}+\frac {3 d^2 (c+d x)^2 \cosh (a+b x) \sinh (a+b x)}{2 b^3}+\frac {(c+d x)^4 \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {3 d^3 (c+d x) \sinh ^2(a+b x)}{2 b^4}-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2} \]

[Out]

-3/4*d^4*x/b^4-1/2*d*(d*x+c)^3/b^2-1/10*(d*x+c)^5/d+3/4*d^4*cosh(b*x+a)*sinh(b*x+a)/b^5+3/2*d^2*(d*x+c)^2*cosh
(b*x+a)*sinh(b*x+a)/b^3+1/2*(d*x+c)^4*cosh(b*x+a)*sinh(b*x+a)/b-3/2*d^3*(d*x+c)*sinh(b*x+a)^2/b^4-d*(d*x+c)^3*
sinh(b*x+a)^2/b^2

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Rubi [A]
time = 0.08, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3392, 32, 2715, 8} \begin {gather*} \frac {3 d^4 \sinh (a+b x) \cosh (a+b x)}{4 b^5}-\frac {3 d^3 (c+d x) \sinh ^2(a+b x)}{2 b^4}+\frac {3 d^2 (c+d x)^2 \sinh (a+b x) \cosh (a+b x)}{2 b^3}-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac {3 d^4 x}{4 b^4}-\frac {d (c+d x)^3}{2 b^2}-\frac {(c+d x)^5}{10 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^4*Sinh[a + b*x]^2,x]

[Out]

(-3*d^4*x)/(4*b^4) - (d*(c + d*x)^3)/(2*b^2) - (c + d*x)^5/(10*d) + (3*d^4*Cosh[a + b*x]*Sinh[a + b*x])/(4*b^5
) + (3*d^2*(c + d*x)^2*Cosh[a + b*x]*Sinh[a + b*x])/(2*b^3) + ((c + d*x)^4*Cosh[a + b*x]*Sinh[a + b*x])/(2*b)
- (3*d^3*(c + d*x)*Sinh[a + b*x]^2)/(2*b^4) - (d*(c + d*x)^3*Sinh[a + b*x]^2)/b^2

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

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)^4 \sinh ^2(a+b x) \, dx &=\frac {(c+d x)^4 \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2}-\frac {1}{2} \int (c+d x)^4 \, dx+\frac {\left (3 d^2\right ) \int (c+d x)^2 \sinh ^2(a+b x) \, dx}{b^2}\\ &=-\frac {(c+d x)^5}{10 d}+\frac {3 d^2 (c+d x)^2 \cosh (a+b x) \sinh (a+b x)}{2 b^3}+\frac {(c+d x)^4 \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {3 d^3 (c+d x) \sinh ^2(a+b x)}{2 b^4}-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2}-\frac {\left (3 d^2\right ) \int (c+d x)^2 \, dx}{2 b^2}+\frac {\left (3 d^4\right ) \int \sinh ^2(a+b x) \, dx}{2 b^4}\\ &=-\frac {d (c+d x)^3}{2 b^2}-\frac {(c+d x)^5}{10 d}+\frac {3 d^4 \cosh (a+b x) \sinh (a+b x)}{4 b^5}+\frac {3 d^2 (c+d x)^2 \cosh (a+b x) \sinh (a+b x)}{2 b^3}+\frac {(c+d x)^4 \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {3 d^3 (c+d x) \sinh ^2(a+b x)}{2 b^4}-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2}-\frac {\left (3 d^4\right ) \int 1 \, dx}{4 b^4}\\ &=-\frac {3 d^4 x}{4 b^4}-\frac {d (c+d x)^3}{2 b^2}-\frac {(c+d x)^5}{10 d}+\frac {3 d^4 \cosh (a+b x) \sinh (a+b x)}{4 b^5}+\frac {3 d^2 (c+d x)^2 \cosh (a+b x) \sinh (a+b x)}{2 b^3}+\frac {(c+d x)^4 \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {3 d^3 (c+d x) \sinh ^2(a+b x)}{2 b^4}-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2}\\ \end {align*}

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Mathematica [A]
time = 0.36, size = 132, normalized size = 0.81 \begin {gather*} \frac {-8 b^5 x \left (5 c^4+10 c^3 d x+10 c^2 d^2 x^2+5 c d^3 x^3+d^4 x^4\right )-20 b d (c+d x) \left (3 d^2+2 b^2 (c+d x)^2\right ) \cosh (2 (a+b x))+10 \left (3 d^4+6 b^2 d^2 (c+d x)^2+2 b^4 (c+d x)^4\right ) \sinh (2 (a+b x))}{80 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^4*Sinh[a + b*x]^2,x]

[Out]

(-8*b^5*x*(5*c^4 + 10*c^3*d*x + 10*c^2*d^2*x^2 + 5*c*d^3*x^3 + d^4*x^4) - 20*b*d*(c + d*x)*(3*d^2 + 2*b^2*(c +
d*x)^2)*Cosh[2*(a + b*x)] + 10*(3*d^4 + 6*b^2*d^2*(c + d*x)^2 + 2*b^4*(c + d*x)^4)*Sinh[2*(a + b*x)])/(80*b^5
)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(909\) vs. \(2(148)=296\).
time = 0.44, size = 910, normalized size = 5.62

method	result	size
risch	\(-\frac {d^{4} x^{5}}{10}-\frac {d^{3} c \,x^{4}}{2}-d^{2} c^{2} x^{3}-d \,c^{3} x^{2}-\frac {c^{4} x}{2}-\frac {c^{5}}{10 d}+\frac {\left (2 d^{4} x^{4} b^{4}+8 b^{4} c \,d^{3} x^{3}+12 b^{4} c^{2} d^{2} x^{2}-4 b^{3} d^{4} x^{3}+8 b^{4} c^{3} d x -12 b^{3} c \,d^{3} x^{2}+2 b^{4} c^{4}-12 b^{3} c^{2} d^{2} x +6 b^{2} d^{4} x^{2}-4 b^{3} c^{3} d +12 b^{2} c \,d^{3} x +6 b^{2} c^{2} d^{2}-6 b \,d^{4} x -6 b c \,d^{3}+3 d^{4}\right ) {\mathrm e}^{2 b x +2 a}}{16 b^{5}}-\frac {\left (2 d^{4} x^{4} b^{4}+8 b^{4} c \,d^{3} x^{3}+12 b^{4} c^{2} d^{2} x^{2}+4 b^{3} d^{4} x^{3}+8 b^{4} c^{3} d x +12 b^{3} c \,d^{3} x^{2}+2 b^{4} c^{4}+12 b^{3} c^{2} d^{2} x +6 b^{2} d^{4} x^{2}+4 b^{3} c^{3} d +12 b^{2} c \,d^{3} x +6 b^{2} c^{2} d^{2}+6 b \,d^{4} x +6 b c \,d^{3}+3 d^{4}\right ) {\mathrm e}^{-2 b x -2 a}}{16 b^{5}}\)	\(383\)
derivativedivides	\(\text {Expression too large to display}\)	\(910\)
default	\(\text {Expression too large to display}\)	\(910\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^4*sinh(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/b*(d^4/b^4*(1/2*(b*x+a)^4*cosh(b*x+a)*sinh(b*x+a)-1/10*(b*x+a)^5-(b*x+a)^3*cosh(b*x+a)^2+3/2*(b*x+a)^2*cosh(
b*x+a)*sinh(b*x+a)+1/2*(b*x+a)^3-3/2*(b*x+a)*cosh(b*x+a)^2+3/4*cosh(b*x+a)*sinh(b*x+a)+3/4*b*x+3/4*a)+d^4/b^4*
a^4*(1/2*cosh(b*x+a)*sinh(b*x+a)-1/2*b*x-1/2*a)-12*d^3/b^3*a*c*(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)+12*d^3/b^3*a^2*c*(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)-12*d^2/b^2*a*c^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)+c^4*(1/2*cosh(b*x+a)*sinh(b*x+a)-1/2*b*x-1/2*a)-4*d^4/b^4*a*(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)*cosh(b*x+a)*sinh(b*x+a)+3/8*(b*x+a)^2-
3/8*cosh(b*x+a)^2)+4*d^3/b^3*c*(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)*cosh(b*x+a)*sinh(b*x+a)+3/8*(b*x+a)^2-3/8*cosh(b*x+a)^2)+6*d^4/b^4*a^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)*sinh(b*x+a)+1/4*b*x+1/4*a)+6*d^2/b^2*c^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)*sinh(b*x+a)+1/
4*b*x+1/4*a)-4*d^4/b^4*a^3*(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)+4*d/b*c^3*(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)-4*d^3/b^3*a^3*c*(1/2*cosh(b*x+a)*sinh(b*x+a
)-1/2*b*x-1/2*a)+6*d^2/b^2*a^2*c^2*(1/2*cosh(b*x+a)*sinh(b*x+a)-1/2*b*x-1/2*a)-4*d/b*a*c^3*(1/2*cosh(b*x+a)*si
nh(b*x+a)-1/2*b*x-1/2*a))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 382 vs. \(2 (148) = 296\).
time = 0.32, size = 382, normalized size = 2.36 \begin {gather*} -\frac {1}{4} \, {\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^{3} d - \frac {1}{8} \, {\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^{2} d^{2} - \frac {1}{8} \, {\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 )} c d^{3} - \frac {1}{80} \, {\left (8 \, x^{5} - \frac {5 \, {\left (2 \, b^{4} x^{4} e^{\left (2 \, a\right )} - 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^{5}} + \frac {5 \, {\left (2 \, b^{4} x^{4} + 4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{5}}\right )} d^{4} - \frac {1}{8} \, c^{4} {\left (4 \, x - \frac {e^{\left (2 \, b x + 2 \, a\right )}}{b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{b}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*sinh(b*x+a)^2,x, algorithm="maxima")

[Out]

-1/4*(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^3*d - 1/8*(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^2*d^2 - 1/8*(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)*c*d^3 - 1/80*(8*x^5 - 5*(2*b^4*x^4*e^(2*a) - 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^5 + 5*(2*b^4*x^4 + 4*b^3*x^3 + 6*b^2*x
^2 + 6*b*x + 3)*e^(-2*b*x - 2*a)/b^5)*d^4 - 1/8*c^4*(4*x - e^(2*b*x + 2*a)/b + e^(-2*b*x - 2*a)/b)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 312 vs. \(2 (148) = 296\).
time = 0.37, size = 312, normalized size = 1.93 \begin {gather*} -\frac {2 \, b^{5} d^{4} x^{5} + 10 \, b^{5} c d^{3} x^{4} + 20 \, b^{5} c^{2} d^{2} x^{3} + 20 \, b^{5} c^{3} d x^{2} + 10 \, b^{5} c^{4} x + 5 \, {\left (2 \, b^{3} d^{4} x^{3} + 6 \, b^{3} c d^{3} x^{2} + 2 \, b^{3} c^{3} d + 3 \, b c d^{3} + 3 \, {\left (2 \, b^{3} c^{2} d^{2} + b d^{4}\right )} x\right )} \cosh \left (b x + a\right )^{2} - 5 \, {\left (2 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} + 2 \, b^{4} c^{4} + 6 \, b^{2} c^{2} d^{2} + 3 \, d^{4} + 6 \, {\left (2 \, b^{4} c^{2} d^{2} + b^{2} d^{4}\right )} x^{2} + 4 \, {\left (2 \, b^{4} c^{3} d + 3 \, b^{2} c d^{3}\right )} x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + 5 \, {\left (2 \, b^{3} d^{4} x^{3} + 6 \, b^{3} c d^{3} x^{2} + 2 \, b^{3} c^{3} d + 3 \, b c d^{3} + 3 \, {\left (2 \, b^{3} c^{2} d^{2} + b d^{4}\right )} x\right )} \sinh \left (b x + a\right )^{2}}{20 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*sinh(b*x+a)^2,x, algorithm="fricas")

[Out]

-1/20*(2*b^5*d^4*x^5 + 10*b^5*c*d^3*x^4 + 20*b^5*c^2*d^2*x^3 + 20*b^5*c^3*d*x^2 + 10*b^5*c^4*x + 5*(2*b^3*d^4*
x^3 + 6*b^3*c*d^3*x^2 + 2*b^3*c^3*d + 3*b*c*d^3 + 3*(2*b^3*c^2*d^2 + b*d^4)*x)*cosh(b*x + a)^2 - 5*(2*b^4*d^4*
x^4 + 8*b^4*c*d^3*x^3 + 2*b^4*c^4 + 6*b^2*c^2*d^2 + 3*d^4 + 6*(2*b^4*c^2*d^2 + b^2*d^4)*x^2 + 4*(2*b^4*c^3*d +
3*b^2*c*d^3)*x)*cosh(b*x + a)*sinh(b*x + a) + 5*(2*b^3*d^4*x^3 + 6*b^3*c*d^3*x^2 + 2*b^3*c^3*d + 3*b*c*d^3 +
3*(2*b^3*c^2*d^2 + b*d^4)*x)*sinh(b*x + a)^2)/b^5

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 660 vs. \(2 (156) = 312\).
time = 0.70, size = 660, normalized size = 4.07 \begin {gather*} \begin {cases} \frac {c^{4} x \sinh ^{2}{\left (a + b x \right )}}{2} - \frac {c^{4} x \cosh ^{2}{\left (a + b x \right )}}{2} + c^{3} d x^{2} \sinh ^{2}{\left (a + b x \right )} - c^{3} d x^{2} \cosh ^{2}{\left (a + b x \right )} + c^{2} d^{2} x^{3} \sinh ^{2}{\left (a + b x \right )} - c^{2} d^{2} x^{3} \cosh ^{2}{\left (a + b x \right )} + \frac {c d^{3} x^{4} \sinh ^{2}{\left (a + b x \right )}}{2} - \frac {c d^{3} x^{4} \cosh ^{2}{\left (a + b x \right )}}{2} + \frac {d^{4} x^{5} \sinh ^{2}{\left (a + b x \right )}}{10} - \frac {d^{4} x^{5} \cosh ^{2}{\left (a + b x \right )}}{10} + \frac {c^{4} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} + \frac {2 c^{3} d x \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} + \frac {3 c^{2} d^{2} x^{2} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} + \frac {2 c d^{3} x^{3} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} + \frac {d^{4} x^{4} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} - \frac {c^{3} d \cosh ^{2}{\left (a + b x \right )}}{b^{2}} - \frac {3 c^{2} d^{2} x \sinh ^{2}{\left (a + b x \right )}}{2 b^{2}} - \frac {3 c^{2} d^{2} x \cosh ^{2}{\left (a + b x \right )}}{2 b^{2}} - \frac {3 c d^{3} x^{2} \sinh ^{2}{\left (a + b x \right )}}{2 b^{2}} - \frac {3 c d^{3} x^{2} \cosh ^{2}{\left (a + b x \right )}}{2 b^{2}} - \frac {d^{4} x^{3} \sinh ^{2}{\left (a + b x \right )}}{2 b^{2}} - \frac {d^{4} x^{3} \cosh ^{2}{\left (a + b x \right )}}{2 b^{2}} + \frac {3 c^{2} d^{2} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b^{3}} + \frac {3 c d^{3} x \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b^{3}} + \frac {3 d^{4} x^{2} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b^{3}} - \frac {3 c d^{3} \cosh ^{2}{\left (a + b x \right )}}{2 b^{4}} - \frac {3 d^{4} x \sinh ^{2}{\left (a + b x \right )}}{4 b^{4}} - \frac {3 d^{4} x \cosh ^{2}{\left (a + b x \right )}}{4 b^{4}} + \frac {3 d^{4} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{4 b^{5}} & \text {for}\: b \neq 0 \\\left (c^{4} x + 2 c^{3} d x^{2} + 2 c^{2} d^{2} x^{3} + c d^{3} x^{4} + \frac {d^{4} x^{5}}{5}\right ) \sinh ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**4*sinh(b*x+a)**2,x)

[Out]

Piecewise((c**4*x*sinh(a + b*x)**2/2 - c**4*x*cosh(a + b*x)**2/2 + c**3*d*x**2*sinh(a + b*x)**2 - c**3*d*x**2*
cosh(a + b*x)**2 + c**2*d**2*x**3*sinh(a + b*x)**2 - c**2*d**2*x**3*cosh(a + b*x)**2 + c*d**3*x**4*sinh(a + b*
x)**2/2 - c*d**3*x**4*cosh(a + b*x)**2/2 + d**4*x**5*sinh(a + b*x)**2/10 - d**4*x**5*cosh(a + b*x)**2/10 + c**
4*sinh(a + b*x)*cosh(a + b*x)/(2*b) + 2*c**3*d*x*sinh(a + b*x)*cosh(a + b*x)/b + 3*c**2*d**2*x**2*sinh(a + b*x
)*cosh(a + b*x)/b + 2*c*d**3*x**3*sinh(a + b*x)*cosh(a + b*x)/b + d**4*x**4*sinh(a + b*x)*cosh(a + b*x)/(2*b)
- c**3*d*cosh(a + b*x)**2/b**2 - 3*c**2*d**2*x*sinh(a + b*x)**2/(2*b**2) - 3*c**2*d**2*x*cosh(a + b*x)**2/(2*b
**2) - 3*c*d**3*x**2*sinh(a + b*x)**2/(2*b**2) - 3*c*d**3*x**2*cosh(a + b*x)**2/(2*b**2) - d**4*x**3*sinh(a +
b*x)**2/(2*b**2) - d**4*x**3*cosh(a + b*x)**2/(2*b**2) + 3*c**2*d**2*sinh(a + b*x)*cosh(a + b*x)/(2*b**3) + 3*
c*d**3*x*sinh(a + b*x)*cosh(a + b*x)/b**3 + 3*d**4*x**2*sinh(a + b*x)*cosh(a + b*x)/(2*b**3) - 3*c*d**3*cosh(a
+ b*x)**2/(2*b**4) - 3*d**4*x*sinh(a + b*x)**2/(4*b**4) - 3*d**4*x*cosh(a + b*x)**2/(4*b**4) + 3*d**4*sinh(a
+ b*x)*cosh(a + b*x)/(4*b**5), Ne(b, 0)), ((c**4*x + 2*c**3*d*x**2 + 2*c**2*d**2*x**3 + c*d**3*x**4 + d**4*x**
5/5)*sinh(a)**2, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (148) = 296\).
time = 0.45, size = 374, normalized size = 2.31 \begin {gather*} -\frac {1}{10} \, d^{4} x^{5} - \frac {1}{2} \, c d^{3} x^{4} - c^{2} d^{2} x^{3} - c^{3} d x^{2} - \frac {1}{2} \, c^{4} x + \frac {{\left (2 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} + 12 \, b^{4} c^{2} d^{2} x^{2} - 4 \, b^{3} d^{4} x^{3} + 8 \, b^{4} c^{3} d x - 12 \, b^{3} c d^{3} x^{2} + 2 \, b^{4} c^{4} - 12 \, b^{3} c^{2} d^{2} x + 6 \, b^{2} d^{4} x^{2} - 4 \, b^{3} c^{3} d + 12 \, b^{2} c d^{3} x + 6 \, b^{2} c^{2} d^{2} - 6 \, b d^{4} x - 6 \, b c d^{3} + 3 \, d^{4}\right )} e^{\left (2 \, b x + 2 \, a\right )}}{16 \, b^{5}} - \frac {{\left (2 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} + 12 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{3} d^{4} x^{3} + 8 \, b^{4} c^{3} d x + 12 \, b^{3} c d^{3} x^{2} + 2 \, b^{4} c^{4} + 12 \, b^{3} c^{2} d^{2} x + 6 \, b^{2} d^{4} x^{2} + 4 \, b^{3} c^{3} d + 12 \, b^{2} c d^{3} x + 6 \, b^{2} c^{2} d^{2} + 6 \, b d^{4} x + 6 \, b c d^{3} + 3 \, d^{4}\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*sinh(b*x+a)^2,x, algorithm="giac")

[Out]

-1/10*d^4*x^5 - 1/2*c*d^3*x^4 - c^2*d^2*x^3 - c^3*d*x^2 - 1/2*c^4*x + 1/16*(2*b^4*d^4*x^4 + 8*b^4*c*d^3*x^3 +
12*b^4*c^2*d^2*x^2 - 4*b^3*d^4*x^3 + 8*b^4*c^3*d*x - 12*b^3*c*d^3*x^2 + 2*b^4*c^4 - 12*b^3*c^2*d^2*x + 6*b^2*d
^4*x^2 - 4*b^3*c^3*d + 12*b^2*c*d^3*x + 6*b^2*c^2*d^2 - 6*b*d^4*x - 6*b*c*d^3 + 3*d^4)*e^(2*b*x + 2*a)/b^5 - 1
/16*(2*b^4*d^4*x^4 + 8*b^4*c*d^3*x^3 + 12*b^4*c^2*d^2*x^2 + 4*b^3*d^4*x^3 + 8*b^4*c^3*d*x + 12*b^3*c*d^3*x^2 +
2*b^4*c^4 + 12*b^3*c^2*d^2*x + 6*b^2*d^4*x^2 + 4*b^3*c^3*d + 12*b^2*c*d^3*x + 6*b^2*c^2*d^2 + 6*b*d^4*x + 6*b
*c*d^3 + 3*d^4)*e^(-2*b*x - 2*a)/b^5

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Mupad [B]
time = 0.69, size = 334, normalized size = 2.06 \begin {gather*} \frac {c^4\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4\,b}-\frac {d^4\,x^5}{10}-c^3\,d\,x^2-\frac {c\,d^3\,x^4}{2}-\frac {c^4\,x}{2}+\frac {3\,d^4\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{8\,b^5}-c^2\,d^2\,x^3-\frac {c^3\,d\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{2\,b^2}-\frac {3\,c\,d^3\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{4\,b^4}-\frac {3\,d^4\,x\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{4\,b^4}+\frac {3\,c^2\,d^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4\,b^3}-\frac {d^4\,x^3\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{2\,b^2}+\frac {d^4\,x^4\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4\,b}+\frac {3\,d^4\,x^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4\,b^3}+\frac {3\,c^2\,d^2\,x^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{2\,b}+\frac {c^3\,d\,x\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{b}+\frac {3\,c\,d^3\,x\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{2\,b^3}-\frac {3\,c^2\,d^2\,x\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{2\,b^2}-\frac {3\,c\,d^3\,x^2\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{2\,b^2}+\frac {c\,d^3\,x^3\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b*x)^2*(c + d*x)^4,x)

[Out]

(c^4*sinh(2*a + 2*b*x))/(4*b) - (d^4*x^5)/10 - c^3*d*x^2 - (c*d^3*x^4)/2 - (c^4*x)/2 + (3*d^4*sinh(2*a + 2*b*x
))/(8*b^5) - c^2*d^2*x^3 - (c^3*d*cosh(2*a + 2*b*x))/(2*b^2) - (3*c*d^3*cosh(2*a + 2*b*x))/(4*b^4) - (3*d^4*x*
cosh(2*a + 2*b*x))/(4*b^4) + (3*c^2*d^2*sinh(2*a + 2*b*x))/(4*b^3) - (d^4*x^3*cosh(2*a + 2*b*x))/(2*b^2) + (d^
4*x^4*sinh(2*a + 2*b*x))/(4*b) + (3*d^4*x^2*sinh(2*a + 2*b*x))/(4*b^3) + (3*c^2*d^2*x^2*sinh(2*a + 2*b*x))/(2*
b) + (c^3*d*x*sinh(2*a + 2*b*x))/b + (3*c*d^3*x*sinh(2*a + 2*b*x))/(2*b^3) - (3*c^2*d^2*x*cosh(2*a + 2*b*x))/(
2*b^2) - (3*c*d^3*x^2*cosh(2*a + 2*b*x))/(2*b^2) + (c*d^3*x^3*sinh(2*a + 2*b*x))/b

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