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

Optimal. Leaf size=97 \[ -\frac {3 \cosh (a+x (b-d)-c)}{8 (b-d)}+\frac {\cosh (3 a+x (3 b-d)-c)}{8 (3 b-d)}-\frac {3 \cosh (a+x (b+d)+c)}{8 (b+d)}+\frac {\cosh (3 a+x (3 b+d)+c)}{8 (3 b+d)} \]

[Out]

-3/8*cosh(a-c+(b-d)*x)/(b-d)+1/8*cosh(3*a-c+(3*b-d)*x)/(3*b-d)-3/8*cosh(a+c+(b+d)*x)/(b+d)+1/8*cosh(3*a+c+(3*b
+d)*x)/(3*b+d)

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Rubi [A]  time = 0.08, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {5618, 2638} \[ -\frac {3 \cosh (a+x (b-d)-c)}{8 (b-d)}+\frac {\cosh (3 a+x (3 b-d)-c)}{8 (3 b-d)}-\frac {3 \cosh (a+x (b+d)+c)}{8 (b+d)}+\frac {\cosh (3 a+x (3 b+d)+c)}{8 (3 b+d)} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]*Sinh[a + b*x]^3,x]

[Out]

(-3*Cosh[a - c + (b - d)*x])/(8*(b - d)) + Cosh[3*a - c + (3*b - d)*x]/(8*(3*b - d)) - (3*Cosh[a + c + (b + d)
*x])/(8*(b + d)) + Cosh[3*a + c + (3*b + d)*x]/(8*(3*b + d))

Rule 2638

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

Rule 5618

Int[Cosh[w_]^(q_.)*Sinh[v_]^(p_.), x_Symbol] :> Int[ExpandTrigReduce[Sinh[v]^p*Cosh[w]^q, x], x] /; IGtQ[p, 0]
&& IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/
w], x]))

Rubi steps

\begin {align*} \int \cosh (c+d x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac {3}{8} \sinh (a-c+(b-d) x)+\frac {1}{8} \sinh (3 a-c+(3 b-d) x)-\frac {3}{8} \sinh (a+c+(b+d) x)+\frac {1}{8} \sinh (3 a+c+(3 b+d) x)\right ) \, dx\\ &=\frac {1}{8} \int \sinh (3 a-c+(3 b-d) x) \, dx+\frac {1}{8} \int \sinh (3 a+c+(3 b+d) x) \, dx-\frac {3}{8} \int \sinh (a-c+(b-d) x) \, dx-\frac {3}{8} \int \sinh (a+c+(b+d) x) \, dx\\ &=-\frac {3 \cosh (a-c+(b-d) x)}{8 (b-d)}+\frac {\cosh (3 a-c+(3 b-d) x)}{8 (3 b-d)}-\frac {3 \cosh (a+c+(b+d) x)}{8 (b+d)}+\frac {\cosh (3 a+c+(3 b+d) x)}{8 (3 b+d)}\\ \end {align*}

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Mathematica [A]  time = 0.52, size = 90, normalized size = 0.93 \[ \frac {1}{8} \left (-\frac {3 \cosh (a+b x-c-d x)}{b-d}+\frac {\cosh (3 a+3 b x-c-d x)}{3 b-d}+\frac {\cosh (3 a+3 b x+c+d x)}{3 b+d}-\frac {3 \cosh (a+x (b+d)+c)}{b+d}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-3*Cosh[a - c + b*x - d*x])/(b - d) + Cosh[3*a - c + 3*b*x - d*x]/(3*b - d) + Cosh[3*a + c + 3*b*x + d*x]/(3
*b + d) - (3*Cosh[a + c + (b + d)*x])/(b + d))/8

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fricas [B]  time = 0.52, size = 243, normalized size = 2.51 \[ \frac {9 \, {\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} + 3 \, {\left ({\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{3} - {\left (9 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right ) - {\left ({\left (b^{2} d - d^{3}\right )} \sinh \left (b x + a\right )^{3} - 3 \, {\left (9 \, b^{2} d - d^{3} - {\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{4 \, {\left ({\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{4} - 2 \, {\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + {\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \sinh \left (b x + a\right )^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(9*(b^3 - b*d^2)*cosh(b*x + a)*cosh(d*x + c)*sinh(b*x + a)^2 + 3*((b^3 - b*d^2)*cosh(b*x + a)^3 - (9*b^3 -
b*d^2)*cosh(b*x + a))*cosh(d*x + c) - ((b^2*d - d^3)*sinh(b*x + a)^3 - 3*(9*b^2*d - d^3 - (b^2*d - d^3)*cosh(
b*x + a)^2)*sinh(b*x + a))*sinh(d*x + c))/((9*b^4 - 10*b^2*d^2 + d^4)*cosh(b*x + a)^4 - 2*(9*b^4 - 10*b^2*d^2
+ d^4)*cosh(b*x + a)^2*sinh(b*x + a)^2 + (9*b^4 - 10*b^2*d^2 + d^4)*sinh(b*x + a)^4)

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giac [B]  time = 0.14, size = 183, normalized size = 1.89 \[ \frac {e^{\left (3 \, b x + d x + 3 \, a + c\right )}}{16 \, {\left (3 \, b + d\right )}} + \frac {e^{\left (3 \, b x - d x + 3 \, a - c\right )}}{16 \, {\left (3 \, b - d\right )}} - \frac {3 \, e^{\left (b x + d x + a + c\right )}}{16 \, {\left (b + d\right )}} - \frac {3 \, e^{\left (b x - d x + a - c\right )}}{16 \, {\left (b - d\right )}} - \frac {3 \, e^{\left (-b x + d x - a + c\right )}}{16 \, {\left (b - d\right )}} - \frac {3 \, e^{\left (-b x - d x - a - c\right )}}{16 \, {\left (b + d\right )}} + \frac {e^{\left (-3 \, b x + d x - 3 \, a + c\right )}}{16 \, {\left (3 \, b - d\right )}} + \frac {e^{\left (-3 \, b x - d x - 3 \, a - c\right )}}{16 \, {\left (3 \, b + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*e^(3*b*x + d*x + 3*a + c)/(3*b + d) + 1/16*e^(3*b*x - d*x + 3*a - c)/(3*b - d) - 3/16*e^(b*x + d*x + a +
c)/(b + d) - 3/16*e^(b*x - d*x + a - c)/(b - d) - 3/16*e^(-b*x + d*x - a + c)/(b - d) - 3/16*e^(-b*x - d*x - a
- c)/(b + d) + 1/16*e^(-3*b*x + d*x - 3*a + c)/(3*b - d) + 1/16*e^(-3*b*x - d*x - 3*a - c)/(3*b + d)

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maple [A]  time = 0.10, size = 90, normalized size = 0.93 \[ -\frac {3 \cosh \left (a -c +\left (b -d \right ) x \right )}{8 \left (b -d \right )}+\frac {\cosh \left (3 a -c +\left (3 b -d \right ) x \right )}{24 b -8 d}-\frac {3 \cosh \left (a +c +\left (b +d \right ) x \right )}{8 \left (b +d \right )}+\frac {\cosh \left (3 a +c +\left (3 b +d \right ) x \right )}{24 b +8 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/8*cosh(a-c+(b-d)*x)/(b-d)+1/8*cosh(3*a-c+(3*b-d)*x)/(3*b-d)-3/8*cosh(a+c+(b+d)*x)/(b+d)+1/8*cosh(3*a+c+(3*b
+d)*x)/(3*b+d)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(-d/b>0)', see `assume?` for mo
re details)Is -d/b equal to -1?

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mupad [B]  time = 0.53, size = 183, normalized size = 1.89 \[ \frac {6\,b^2\,d\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )}{9\,b^4-10\,b^2\,d^2+d^4}-\frac {d\,{\mathrm {sinh}\left (a+b\,x\right )}^3\,\mathrm {sinh}\left (c+d\,x\right )\,\left (7\,b^2-d^2\right )}{9\,b^4-10\,b^2\,d^2+d^4}-\frac {3\,\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (b\,d^2-3\,b^3\right )}{9\,b^4-10\,b^2\,d^2+d^4}-\frac {6\,b^3\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,\mathrm {cosh}\left (c+d\,x\right )}{9\,b^4-10\,b^2\,d^2+d^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c + d*x)*sinh(a + b*x)^3,x)

[Out]

(6*b^2*d*cosh(a + b*x)^2*sinh(a + b*x)*sinh(c + d*x))/(9*b^4 + d^4 - 10*b^2*d^2) - (d*sinh(a + b*x)^3*sinh(c +
d*x)*(7*b^2 - d^2))/(9*b^4 + d^4 - 10*b^2*d^2) - (3*cosh(a + b*x)*cosh(c + d*x)*sinh(a + b*x)^2*(b*d^2 - 3*b^
3))/(9*b^4 + d^4 - 10*b^2*d^2) - (6*b^3*cosh(a + b*x)^3*cosh(c + d*x))/(9*b^4 + d^4 - 10*b^2*d^2)

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sympy [A]  time = 30.37, size = 935, normalized size = 9.64 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*sinh(a)**3*cosh(c), Eq(b, 0) & Eq(d, 0)), (3*x*sinh(a - d*x)**3*cosh(c + d*x)/8 + 3*x*sinh(a - d*
x)**2*sinh(c + d*x)*cosh(a - d*x)/8 - 3*x*sinh(a - d*x)*cosh(a - d*x)**2*cosh(c + d*x)/8 - 3*x*sinh(c + d*x)*c
osh(a - d*x)**3/8 - sinh(a - d*x)**3*sinh(c + d*x)/(8*d) - 3*sinh(a - d*x)**2*cosh(a - d*x)*cosh(c + d*x)/(4*d
) + 3*cosh(a - d*x)**3*cosh(c + d*x)/(8*d), Eq(b, -d)), (x*sinh(a - d*x/3)**3*cosh(c + d*x)/8 + 3*x*sinh(a - d
*x/3)**2*sinh(c + d*x)*cosh(a - d*x/3)/8 + 3*x*sinh(a - d*x/3)*cosh(a - d*x/3)**2*cosh(c + d*x)/8 + x*sinh(c +
d*x)*cosh(a - d*x/3)**3/8 + 7*sinh(a - d*x/3)**3*sinh(c + d*x)/(8*d) - 3*sinh(a - d*x/3)*sinh(c + d*x)*cosh(a
- d*x/3)**2/(4*d) - 3*cosh(a - d*x/3)**3*cosh(c + d*x)/(8*d), Eq(b, -d/3)), (x*sinh(a + d*x/3)**3*cosh(c + d*
x)/8 - 3*x*sinh(a + d*x/3)**2*sinh(c + d*x)*cosh(a + d*x/3)/8 + 3*x*sinh(a + d*x/3)*cosh(a + d*x/3)**2*cosh(c
+ d*x)/8 - x*sinh(c + d*x)*cosh(a + d*x/3)**3/8 + 9*sinh(a + d*x/3)**3*sinh(c + d*x)/(8*d) - 3*sinh(a + d*x/3)
**2*cosh(a + d*x/3)*cosh(c + d*x)/(4*d) + cosh(a + d*x/3)**3*cosh(c + d*x)/(8*d), Eq(b, d/3)), (3*x*sinh(a + d
*x)**3*cosh(c + d*x)/8 - 3*x*sinh(a + d*x)**2*sinh(c + d*x)*cosh(a + d*x)/8 - 3*x*sinh(a + d*x)*cosh(a + d*x)*
*2*cosh(c + d*x)/8 + 3*x*sinh(c + d*x)*cosh(a + d*x)**3/8 + 5*sinh(a + d*x)**3*sinh(c + d*x)/(8*d) - 3*sinh(a
+ d*x)*sinh(c + d*x)*cosh(a + d*x)**2/(4*d) + 3*cosh(a + d*x)**3*cosh(c + d*x)/(8*d), Eq(b, d)), (9*b**3*sinh(
a + b*x)**2*cosh(a + b*x)*cosh(c + d*x)/(9*b**4 - 10*b**2*d**2 + d**4) - 6*b**3*cosh(a + b*x)**3*cosh(c + d*x)
/(9*b**4 - 10*b**2*d**2 + d**4) - 7*b**2*d*sinh(a + b*x)**3*sinh(c + d*x)/(9*b**4 - 10*b**2*d**2 + d**4) + 6*b
**2*d*sinh(a + b*x)*sinh(c + d*x)*cosh(a + b*x)**2/(9*b**4 - 10*b**2*d**2 + d**4) - 3*b*d**2*sinh(a + b*x)**2*
cosh(a + b*x)*cosh(c + d*x)/(9*b**4 - 10*b**2*d**2 + d**4) + d**3*sinh(a + b*x)**3*sinh(c + d*x)/(9*b**4 - 10*
b**2*d**2 + d**4), True))

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