### 3.961 $$\int e^{c+d x} \cosh ^3(a+b x) \sinh (a+b x) \, dx$$

Optimal. Leaf size=137 $-\frac{d e^{c+d x} \sinh (2 a+2 b x)}{4 \left (4 b^2-d^2\right )}-\frac{d e^{c+d x} \sinh (4 a+4 b x)}{8 \left (16 b^2-d^2\right )}+\frac{b e^{c+d x} \cosh (2 a+2 b x)}{2 \left (4 b^2-d^2\right )}+\frac{b e^{c+d x} \cosh (4 a+4 b x)}{2 \left (16 b^2-d^2\right )}$

[Out]

(b*E^(c + d*x)*Cosh[2*a + 2*b*x])/(2*(4*b^2 - d^2)) + (b*E^(c + d*x)*Cosh[4*a + 4*b*x])/(2*(16*b^2 - d^2)) - (
d*E^(c + d*x)*Sinh[2*a + 2*b*x])/(4*(4*b^2 - d^2)) - (d*E^(c + d*x)*Sinh[4*a + 4*b*x])/(8*(16*b^2 - d^2))

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Rubi [A]  time = 0.0936001, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.091, Rules used = {5509, 5474} $-\frac{d e^{c+d x} \sinh (2 a+2 b x)}{4 \left (4 b^2-d^2\right )}-\frac{d e^{c+d x} \sinh (4 a+4 b x)}{8 \left (16 b^2-d^2\right )}+\frac{b e^{c+d x} \cosh (2 a+2 b x)}{2 \left (4 b^2-d^2\right )}+\frac{b e^{c+d x} \cosh (4 a+4 b x)}{2 \left (16 b^2-d^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(b*E^(c + d*x)*Cosh[2*a + 2*b*x])/(2*(4*b^2 - d^2)) + (b*E^(c + d*x)*Cosh[4*a + 4*b*x])/(2*(16*b^2 - d^2)) - (
d*E^(c + d*x)*Sinh[2*a + 2*b*x])/(4*(4*b^2 - d^2)) - (d*E^(c + d*x)*Sinh[4*a + 4*b*x])/(8*(16*b^2 - d^2))

Rule 5509

Int[Cosh[(f_.) + (g_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sinh[(d_.) + (e_.)*(x_)]^(m_.), x_Symbol]
:> Int[ExpandTrigReduce[F^(c*(a + b*x)), Sinh[d + e*x]^m*Cosh[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e,
f, g}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 5474

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sinh[(d_.) + (e_.)*(x_)], x_Symbol] :> -Simp[(b*c*Log[F]*F^(c*(a + b*x))
*Sinh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2), x] + Simp[(e*F^(c*(a + b*x))*Cosh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2)
, x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 - b^2*c^2*Log[F]^2, 0]

Rubi steps

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

Mathematica [A]  time = 0.830326, size = 86, normalized size = 0.63 $\frac{1}{8} e^{c+d x} \left (\frac{4 b \cosh (2 (a+b x))-2 d \sinh (2 (a+b x))}{4 b^2-d^2}+\frac{4 b \cosh (4 (a+b x))-d \sinh (4 (a+b x))}{16 b^2-d^2}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(E^(c + d*x)*((4*b*Cosh[2*(a + b*x)] - 2*d*Sinh[2*(a + b*x)])/(4*b^2 - d^2) + (4*b*Cosh[4*(a + b*x)] - d*Sinh[
4*(a + b*x)])/(16*b^2 - d^2)))/8

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Maple [A]  time = 0.009, size = 202, normalized size = 1.5 \begin{align*} -{\frac{\sinh \left ( 2\,a-c+ \left ( 2\,b-d \right ) x \right ) }{16\,b-8\,d}}+{\frac{\sinh \left ( 2\,a+c+ \left ( 2\,b+d \right ) x \right ) }{16\,b+8\,d}}-{\frac{\sinh \left ( \left ( 4\,b-d \right ) x+4\,a-c \right ) }{64\,b-16\,d}}+{\frac{\sinh \left ( \left ( 4\,b+d \right ) x+4\,a+c \right ) }{64\,b+16\,d}}+{\frac{\cosh \left ( 2\,a-c+ \left ( 2\,b-d \right ) x \right ) }{16\,b-8\,d}}+{\frac{\cosh \left ( 2\,a+c+ \left ( 2\,b+d \right ) x \right ) }{16\,b+8\,d}}+{\frac{\cosh \left ( \left ( 4\,b-d \right ) x+4\,a-c \right ) }{64\,b-16\,d}}+{\frac{\cosh \left ( \left ( 4\,b+d \right ) x+4\,a+c \right ) }{64\,b+16\,d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sinh(2*a-c+(2*b-d)*x)/(2*b-d)+1/8*sinh(2*a+c+(2*b+d)*x)/(2*b+d)-1/16/(4*b-d)*sinh((4*b-d)*x+4*a-c)+1/16/(
4*b+d)*sinh((4*b+d)*x+4*a+c)+1/8*cosh(2*a-c+(2*b-d)*x)/(2*b-d)+1/8*cosh(2*a+c+(2*b+d)*x)/(2*b+d)+1/16*cosh((4*
b-d)*x+4*a-c)/(4*b-d)+1/16*cosh((4*b+d)*x+4*a+c)/(4*b+d)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.07682, size = 1161, normalized size = 8.47 \begin{align*} -\frac{{\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{3} -{\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{4} -{\left (16 \, b^{3} - b d^{2} + 6 \,{\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} +{\left ({\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{3} +{\left (16 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) -{\left ({\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{4} +{\left (16 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right ) -{\left ({\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{4} -{\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} +{\left (4 \, b^{3} - b d^{2}\right )} \sinh \left (b x + a\right )^{4} +{\left (16 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2} +{\left (16 \, b^{3} - b d^{2} + 6 \,{\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{2} -{\left ({\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{3} +{\left (16 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{2 \,{\left ({\left (64 \, b^{4} - 20 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{4} - 2 \,{\left (64 \, b^{4} - 20 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} +{\left (64 \, b^{4} - 20 \, b^{2} d^{2} + d^{4}\right )} \sinh \left (b x + a\right )^{4}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((4*b^2*d - d^3)*cosh(b*x + a)*cosh(d*x + c)*sinh(b*x + a)^3 - (4*b^3 - b*d^2)*cosh(d*x + c)*sinh(b*x + a
)^4 - (16*b^3 - b*d^2 + 6*(4*b^3 - b*d^2)*cosh(b*x + a)^2)*cosh(d*x + c)*sinh(b*x + a)^2 + ((4*b^2*d - d^3)*co
sh(b*x + a)^3 + (16*b^2*d - d^3)*cosh(b*x + a))*cosh(d*x + c)*sinh(b*x + a) - ((4*b^3 - b*d^2)*cosh(b*x + a)^4
+ (16*b^3 - b*d^2)*cosh(b*x + a)^2)*cosh(d*x + c) - ((4*b^3 - b*d^2)*cosh(b*x + a)^4 - (4*b^2*d - d^3)*cosh(b
*x + a)*sinh(b*x + a)^3 + (4*b^3 - b*d^2)*sinh(b*x + a)^4 + (16*b^3 - b*d^2)*cosh(b*x + a)^2 + (16*b^3 - b*d^2
+ 6*(4*b^3 - b*d^2)*cosh(b*x + a)^2)*sinh(b*x + a)^2 - ((4*b^2*d - d^3)*cosh(b*x + a)^3 + (16*b^2*d - d^3)*co
sh(b*x + a))*sinh(b*x + a))*sinh(d*x + c))/((64*b^4 - 20*b^2*d^2 + d^4)*cosh(b*x + a)^4 - 2*(64*b^4 - 20*b^2*d
^2 + d^4)*cosh(b*x + a)^2*sinh(b*x + a)^2 + (64*b^4 - 20*b^2*d^2 + d^4)*sinh(b*x + a)^4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.15977, size = 126, normalized size = 0.92 \begin{align*} \frac{e^{\left (4 \, b x + d x + 4 \, a + c\right )}}{16 \,{\left (4 \, b + d\right )}} + \frac{e^{\left (2 \, b x + d x + 2 \, a + c\right )}}{8 \,{\left (2 \, b + d\right )}} + \frac{e^{\left (-2 \, b x + d x - 2 \, a + c\right )}}{8 \,{\left (2 \, b - d\right )}} + \frac{e^{\left (-4 \, b x + d x - 4 \, a + c\right )}}{16 \,{\left (4 \, b - d\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*e^(4*b*x + d*x + 4*a + c)/(4*b + d) + 1/8*e^(2*b*x + d*x + 2*a + c)/(2*b + d) + 1/8*e^(-2*b*x + d*x - 2*a
+ c)/(2*b - d) + 1/16*e^(-4*b*x + d*x - 4*a + c)/(4*b - d)