3.3.83 \(\int \cosh ^4(c+d x) (a+b \sinh ^2(c+d x)) \, dx\) [283]

Optimal. Leaf size=89 \[ \frac {1}{16} (6 a-b) x+\frac {(6 a-b) \cosh (c+d x) \sinh (c+d x)}{16 d}+\frac {(6 a-b) \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac {b \cosh ^5(c+d x) \sinh (c+d x)}{6 d} \]

[Out]

1/16*(6*a-b)*x+1/16*(6*a-b)*cosh(d*x+c)*sinh(d*x+c)/d+1/24*(6*a-b)*cosh(d*x+c)^3*sinh(d*x+c)/d+1/6*b*cosh(d*x+
c)^5*sinh(d*x+c)/d

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Rubi [A]
time = 0.04, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3270, 393, 205, 212} \begin {gather*} \frac {(6 a-b) \sinh (c+d x) \cosh ^3(c+d x)}{24 d}+\frac {(6 a-b) \sinh (c+d x) \cosh (c+d x)}{16 d}+\frac {1}{16} x (6 a-b)+\frac {b \sinh (c+d x) \cosh ^5(c+d x)}{6 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((6*a - b)*x)/16 + ((6*a - b)*Cosh[c + d*x]*Sinh[c + d*x])/(16*d) + ((6*a - b)*Cosh[c + d*x]^3*Sinh[c + d*x])/
(24*d) + (b*Cosh[c + d*x]^5*Sinh[c + d*x])/(6*d)

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 393

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 3270

Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, T
an[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rubi steps

\begin {align*} \int \cosh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx &=\frac {\text {Subst}\left (\int \frac {a-(a-b) x^2}{\left (1-x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b \cosh ^5(c+d x) \sinh (c+d x)}{6 d}+\frac {(6 a-b) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{6 d}\\ &=\frac {(6 a-b) \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac {b \cosh ^5(c+d x) \sinh (c+d x)}{6 d}+\frac {(6 a-b) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac {(6 a-b) \cosh (c+d x) \sinh (c+d x)}{16 d}+\frac {(6 a-b) \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac {b \cosh ^5(c+d x) \sinh (c+d x)}{6 d}+\frac {(6 a-b) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{16 d}\\ &=\frac {1}{16} (6 a-b) x+\frac {(6 a-b) \cosh (c+d x) \sinh (c+d x)}{16 d}+\frac {(6 a-b) \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac {b \cosh ^5(c+d x) \sinh (c+d x)}{6 d}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 63, normalized size = 0.71 \begin {gather*} \frac {72 a c+72 a d x-12 b d x+(48 a-3 b) \sinh (2 (c+d x))+3 (2 a+b) \sinh (4 (c+d x))+b \sinh (6 (c+d x))}{192 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(72*a*c + 72*a*d*x - 12*b*d*x + (48*a - 3*b)*Sinh[2*(c + d*x)] + 3*(2*a + b)*Sinh[4*(c + d*x)] + b*Sinh[6*(c +
d*x)])/(192*d)

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Maple [A]
time = 1.70, size = 67, normalized size = 0.75

method	result	size
default	\(\frac {\left (-\frac {b}{32}+\frac {a}{2}\right ) \sinh \left (2 d x +2 c \right )}{2 d}+\frac {\left (\frac {b}{16}+\frac {a}{8}\right ) \sinh \left (4 d x +4 c \right )}{4 d}+\frac {3 a x}{8}-\frac {b x}{16}+\frac {b \sinh \left (6 d x +6 c \right )}{192 d}\)	\(67\)
risch	\(-\frac {b x}{16}+\frac {3 a x}{8}+\frac {b \,{\mathrm e}^{6 d x +6 c}}{384 d}+\frac {{\mathrm e}^{4 d x +4 c} a}{64 d}+\frac {{\mathrm e}^{4 d x +4 c} b}{128 d}-\frac {{\mathrm e}^{2 d x +2 c} b}{128 d}+\frac {{\mathrm e}^{2 d x +2 c} a}{8 d}+\frac {{\mathrm e}^{-2 d x -2 c} b}{128 d}-\frac {{\mathrm e}^{-2 d x -2 c} a}{8 d}-\frac {{\mathrm e}^{-4 d x -4 c} a}{64 d}-\frac {{\mathrm e}^{-4 d x -4 c} b}{128 d}-\frac {b \,{\mathrm e}^{-6 d x -6 c}}{384 d}\)	\(160\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(-1/32*b+1/2*a)*sinh(2*d*x+2*c)/d+1/4*(1/16*b+1/8*a)*sinh(4*d*x+4*c)/d+3/8*a*x-1/16*b*x+1/192*b*sinh(6*d*x
+6*c)/d

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Maxima [A]
time = 0.27, size = 152, normalized size = 1.71 \begin {gather*} \frac {1}{64} \, a {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + \frac {1}{384} \, b {\left (\frac {{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} - \frac {24 \, {\left (d x + c\right )}}{d} + \frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} - e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*a*(24*x + e^(4*d*x + 4*c)/d + 8*e^(2*d*x + 2*c)/d - 8*e^(-2*d*x - 2*c)/d - e^(-4*d*x - 4*c)/d) + 1/384*b*
((3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + 1)*e^(6*d*x + 6*c)/d - 24*(d*x + c)/d + (3*e^(-2*d*x - 2*c) - 3*e^
(-4*d*x - 4*c) - e^(-6*d*x - 6*c))/d)

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Fricas [A]
time = 0.37, size = 117, normalized size = 1.31 \begin {gather*} \frac {3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 2 \, {\left (5 \, b \cosh \left (d x + c\right )^{3} + 3 \, {\left (2 \, a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 6 \, {\left (6 \, a - b\right )} d x + 3 \, {\left (b \cosh \left (d x + c\right )^{5} + 2 \, {\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{3} + {\left (16 \, a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{96 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(3*b*cosh(d*x + c)*sinh(d*x + c)^5 + 2*(5*b*cosh(d*x + c)^3 + 3*(2*a + b)*cosh(d*x + c))*sinh(d*x + c)^3
+ 6*(6*a - b)*d*x + 3*(b*cosh(d*x + c)^5 + 2*(2*a + b)*cosh(d*x + c)^3 + (16*a - b)*cosh(d*x + c))*sinh(d*x +
c))/d

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (76) = 152\).
time = 0.46, size = 250, normalized size = 2.81 \begin {gather*} \begin {cases} \frac {3 a x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac {3 a x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {3 a x \cosh ^{4}{\left (c + d x \right )}}{8} - \frac {3 a \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} + \frac {5 a \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac {b x \sinh ^{6}{\left (c + d x \right )}}{16} - \frac {3 b x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{16} + \frac {3 b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{16} - \frac {b x \cosh ^{6}{\left (c + d x \right )}}{16} - \frac {b \sinh ^{5}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{16 d} + \frac {b \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{6 d} + \frac {b \sinh {\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right ) \cosh ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((3*a*x*sinh(c + d*x)**4/8 - 3*a*x*sinh(c + d*x)**2*cosh(c + d*x)**2/4 + 3*a*x*cosh(c + d*x)**4/8 - 3
*a*sinh(c + d*x)**3*cosh(c + d*x)/(8*d) + 5*a*sinh(c + d*x)*cosh(c + d*x)**3/(8*d) + b*x*sinh(c + d*x)**6/16 -
3*b*x*sinh(c + d*x)**4*cosh(c + d*x)**2/16 + 3*b*x*sinh(c + d*x)**2*cosh(c + d*x)**4/16 - b*x*cosh(c + d*x)**
6/16 - b*sinh(c + d*x)**5*cosh(c + d*x)/(16*d) + b*sinh(c + d*x)**3*cosh(c + d*x)**3/(6*d) + b*sinh(c + d*x)*c
osh(c + d*x)**5/(16*d), Ne(d, 0)), (x*(a + b*sinh(c)**2)*cosh(c)**4, True))

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Giac [A]
time = 0.42, size = 121, normalized size = 1.36 \begin {gather*} \frac {1}{16} \, {\left (6 \, a - b\right )} x + \frac {b e^{\left (6 \, d x + 6 \, c\right )}}{384 \, d} + \frac {{\left (2 \, a + b\right )} e^{\left (4 \, d x + 4 \, c\right )}}{128 \, d} + \frac {{\left (16 \, a - b\right )} e^{\left (2 \, d x + 2 \, c\right )}}{128 \, d} - \frac {{\left (16 \, a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{128 \, d} - \frac {{\left (2 \, a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{128 \, d} - \frac {b e^{\left (-6 \, d x - 6 \, c\right )}}{384 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(6*a - b)*x + 1/384*b*e^(6*d*x + 6*c)/d + 1/128*(2*a + b)*e^(4*d*x + 4*c)/d + 1/128*(16*a - b)*e^(2*d*x +
2*c)/d - 1/128*(16*a - b)*e^(-2*d*x - 2*c)/d - 1/128*(2*a + b)*e^(-4*d*x - 4*c)/d - 1/384*b*e^(-6*d*x - 6*c)/
d

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Mupad [B]
time = 1.43, size = 76, normalized size = 0.85 \begin {gather*} \frac {12\,a\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+\frac {3\,a\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{2}-\frac {3\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{4}+\frac {3\,b\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{4}+\frac {b\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )}{4}+18\,a\,d\,x-3\,b\,d\,x}{48\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c + d*x)^4*(a + b*sinh(c + d*x)^2),x)

[Out]

(12*a*sinh(2*c + 2*d*x) + (3*a*sinh(4*c + 4*d*x))/2 - (3*b*sinh(2*c + 2*d*x))/4 + (3*b*sinh(4*c + 4*d*x))/4 +
(b*sinh(6*c + 6*d*x))/4 + 18*a*d*x - 3*b*d*x)/(48*d)

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