3.2.85 \(\int \sinh ^3(c+d x) (a+b \sinh ^4(c+d x)) \, dx\) [185]

Optimal. Leaf size=67 \[ -\frac {(a+b) \cosh (c+d x)}{d}+\frac {(a+3 b) \cosh ^3(c+d x)}{3 d}-\frac {3 b \cosh ^5(c+d x)}{5 d}+\frac {b \cosh ^7(c+d x)}{7 d} \]

[Out]

-(a+b)*cosh(d*x+c)/d+1/3*(a+3*b)*cosh(d*x+c)^3/d-3/5*b*cosh(d*x+c)^5/d+1/7*b*cosh(d*x+c)^7/d

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Rubi [A]
time = 0.05, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3294, 1167} \begin {gather*} \frac {(a+3 b) \cosh ^3(c+d x)}{3 d}-\frac {(a+b) \cosh (c+d x)}{d}+\frac {b \cosh ^7(c+d x)}{7 d}-\frac {3 b \cosh ^5(c+d x)}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^4),x]

[Out]

-(((a + b)*Cosh[c + d*x])/d) + ((a + 3*b)*Cosh[c + d*x]^3)/(3*d) - (3*b*Cosh[c + d*x]^5)/(5*d) + (b*Cosh[c + d
*x]^7)/(7*d)

Rule 1167

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 3294

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \sinh ^3(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx &=-\frac {\text {Subst}\left (\int \left (1-x^2\right ) \left (a+b-2 b x^2+b x^4\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \left (a \left (1+\frac {b}{a}\right )-(a+3 b) x^2+3 b x^4-b x^6\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {(a+b) \cosh (c+d x)}{d}+\frac {(a+3 b) \cosh ^3(c+d x)}{3 d}-\frac {3 b \cosh ^5(c+d x)}{5 d}+\frac {b \cosh ^7(c+d x)}{7 d}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 93, normalized size = 1.39 \begin {gather*} -\frac {3 a \cosh (c+d x)}{4 d}-\frac {35 b \cosh (c+d x)}{64 d}+\frac {a \cosh (3 (c+d x))}{12 d}+\frac {7 b \cosh (3 (c+d x))}{64 d}-\frac {7 b \cosh (5 (c+d x))}{320 d}+\frac {b \cosh (7 (c+d x))}{448 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^4),x]

[Out]

(-3*a*Cosh[c + d*x])/(4*d) - (35*b*Cosh[c + d*x])/(64*d) + (a*Cosh[3*(c + d*x)])/(12*d) + (7*b*Cosh[3*(c + d*x
)])/(64*d) - (7*b*Cosh[5*(c + d*x)])/(320*d) + (b*Cosh[7*(c + d*x)])/(448*d)

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Maple [A]
time = 0.70, size = 70, normalized size = 1.04

method	result	size
default	\(\frac {\left (-\frac {35 b}{64}-\frac {3 a}{4}\right ) \cosh \left (d x +c \right )}{d}+\frac {\left (\frac {21 b}{64}+\frac {a}{4}\right ) \cosh \left (3 d x +3 c \right )}{3 d}-\frac {7 b \cosh \left (5 d x +5 c \right )}{320 d}+\frac {b \cosh \left (7 d x +7 c \right )}{448 d}\)	\(70\)
risch	\(\frac {b \,{\mathrm e}^{7 d x +7 c}}{896 d}-\frac {7 b \,{\mathrm e}^{5 d x +5 c}}{640 d}+\frac {7 b \,{\mathrm e}^{3 d x +3 c}}{128 d}+\frac {{\mathrm e}^{3 d x +3 c} a}{24 d}-\frac {3 a \,{\mathrm e}^{d x +c}}{8 d}-\frac {35 b \,{\mathrm e}^{d x +c}}{128 d}-\frac {3 \,{\mathrm e}^{-d x -c} a}{8 d}-\frac {35 \,{\mathrm e}^{-d x -c} b}{128 d}+\frac {7 \,{\mathrm e}^{-3 d x -3 c} b}{128 d}+\frac {{\mathrm e}^{-3 d x -3 c} a}{24 d}-\frac {7 b \,{\mathrm e}^{-5 d x -5 c}}{640 d}+\frac {b \,{\mathrm e}^{-7 d x -7 c}}{896 d}\)	\(176\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^3*(a+b*sinh(d*x+c)^4),x,method=_RETURNVERBOSE)

[Out]

(-35/64*b-3/4*a)/d*cosh(d*x+c)+1/3*(21/64*b+1/4*a)/d*cosh(3*d*x+3*c)-7/320*b/d*cosh(5*d*x+5*c)+1/448*b/d*cosh(
7*d*x+7*c)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (61) = 122\).
time = 0.27, size = 157, normalized size = 2.34 \begin {gather*} -\frac {1}{4480} \, b {\left (\frac {{\left (49 \, e^{\left (-2 \, d x - 2 \, c\right )} - 245 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1225 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac {1225 \, e^{\left (-d x - c\right )} - 245 \, e^{\left (-3 \, d x - 3 \, c\right )} + 49 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac {1}{24} \, a {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^3*(a+b*sinh(d*x+c)^4),x, algorithm="maxima")

[Out]

-1/4480*b*((49*e^(-2*d*x - 2*c) - 245*e^(-4*d*x - 4*c) + 1225*e^(-6*d*x - 6*c) - 5)*e^(7*d*x + 7*c)/d + (1225*
e^(-d*x - c) - 245*e^(-3*d*x - 3*c) + 49*e^(-5*d*x - 5*c) - 5*e^(-7*d*x - 7*c))/d) + 1/24*a*(e^(3*d*x + 3*c)/d
- 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d + e^(-3*d*x - 3*c)/d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (61) = 122\).
time = 0.41, size = 155, normalized size = 2.31 \begin {gather*} \frac {15 \, b \cosh \left (d x + c\right )^{7} + 105 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} - 147 \, b \cosh \left (d x + c\right )^{5} + 105 \, {\left (5 \, b \cosh \left (d x + c\right )^{3} - 7 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 35 \, {\left (16 \, a + 21 \, b\right )} \cosh \left (d x + c\right )^{3} + 105 \, {\left (3 \, b \cosh \left (d x + c\right )^{5} - 14 \, b \cosh \left (d x + c\right )^{3} + {\left (16 \, a + 21 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 105 \, {\left (48 \, a + 35 \, b\right )} \cosh \left (d x + c\right )}{6720 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^3*(a+b*sinh(d*x+c)^4),x, algorithm="fricas")

[Out]

1/6720*(15*b*cosh(d*x + c)^7 + 105*b*cosh(d*x + c)*sinh(d*x + c)^6 - 147*b*cosh(d*x + c)^5 + 105*(5*b*cosh(d*x
+ c)^3 - 7*b*cosh(d*x + c))*sinh(d*x + c)^4 + 35*(16*a + 21*b)*cosh(d*x + c)^3 + 105*(3*b*cosh(d*x + c)^5 - 1
4*b*cosh(d*x + c)^3 + (16*a + 21*b)*cosh(d*x + c))*sinh(d*x + c)^2 - 105*(48*a + 35*b)*cosh(d*x + c))/d

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (56) = 112\).
time = 0.63, size = 128, normalized size = 1.91 \begin {gather*} \begin {cases} \frac {a \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {2 a \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {b \sinh ^{6}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {2 b \sinh ^{4}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {8 b \sinh ^{2}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac {16 b \cosh ^{7}{\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{4}{\left (c \right )}\right ) \sinh ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)**3*(a+b*sinh(d*x+c)**4),x)

[Out]

Piecewise((a*sinh(c + d*x)**2*cosh(c + d*x)/d - 2*a*cosh(c + d*x)**3/(3*d) + b*sinh(c + d*x)**6*cosh(c + d*x)/
d - 2*b*sinh(c + d*x)**4*cosh(c + d*x)**3/d + 8*b*sinh(c + d*x)**2*cosh(c + d*x)**5/(5*d) - 16*b*cosh(c + d*x)
**7/(35*d), Ne(d, 0)), (x*(a + b*sinh(c)**4)*sinh(c)**3, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (61) = 122\).
time = 0.43, size = 142, normalized size = 2.12 \begin {gather*} \frac {b e^{\left (7 \, d x + 7 \, c\right )}}{896 \, d} - \frac {7 \, b e^{\left (5 \, d x + 5 \, c\right )}}{640 \, d} + \frac {{\left (16 \, a + 21 \, b\right )} e^{\left (3 \, d x + 3 \, c\right )}}{384 \, d} - \frac {{\left (48 \, a + 35 \, b\right )} e^{\left (d x + c\right )}}{128 \, d} - \frac {{\left (48 \, a + 35 \, b\right )} e^{\left (-d x - c\right )}}{128 \, d} + \frac {{\left (16 \, a + 21 \, b\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{384 \, d} - \frac {7 \, b e^{\left (-5 \, d x - 5 \, c\right )}}{640 \, d} + \frac {b e^{\left (-7 \, d x - 7 \, c\right )}}{896 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^3*(a+b*sinh(d*x+c)^4),x, algorithm="giac")

[Out]

1/896*b*e^(7*d*x + 7*c)/d - 7/640*b*e^(5*d*x + 5*c)/d + 1/384*(16*a + 21*b)*e^(3*d*x + 3*c)/d - 1/128*(48*a +
35*b)*e^(d*x + c)/d - 1/128*(48*a + 35*b)*e^(-d*x - c)/d + 1/384*(16*a + 21*b)*e^(-3*d*x - 3*c)/d - 7/640*b*e^
(-5*d*x - 5*c)/d + 1/896*b*e^(-7*d*x - 7*c)/d

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Mupad [B]
time = 0.79, size = 66, normalized size = 0.99 \begin {gather*} -\frac {a\,\mathrm {cosh}\left (c+d\,x\right )+b\,\mathrm {cosh}\left (c+d\,x\right )-\frac {a\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}-b\,{\mathrm {cosh}\left (c+d\,x\right )}^3+\frac {3\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-\frac {b\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^3*(a + b*sinh(c + d*x)^4),x)

[Out]

-(a*cosh(c + d*x) + b*cosh(c + d*x) - (a*cosh(c + d*x)^3)/3 - b*cosh(c + d*x)^3 + (3*b*cosh(c + d*x)^5)/5 - (b
*cosh(c + d*x)^7)/7)/d

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