### 3.966 $$\int (-\frac{3 d^2 e^{a+b x}}{4 (b^2-\frac{9 d^2}{4}) \sqrt{\sinh (c+d x)}}+e^{a+b x} \sinh ^{\frac{3}{2}}(c+d x)) \, dx$$

Optimal. Leaf size=73 $\frac{4 b e^{a+b x} \sinh ^{\frac{3}{2}}(c+d x)}{4 b^2-9 d^2}-\frac{6 d e^{a+b x} \sqrt{\sinh (c+d x)} \cosh (c+d x)}{4 b^2-9 d^2}$

[Out]

(-6*d*E^(a + b*x)*Cosh[c + d*x]*Sqrt[Sinh[c + d*x]])/(4*b^2 - 9*d^2) + (4*b*E^(a + b*x)*Sinh[c + d*x]^(3/2))/(
4*b^2 - 9*d^2)

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Rubi [A]  time = 0.613906, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 56, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.089, Rules used = {5482, 2253, 2252, 2251, 5476} $\frac{4 b e^{a+b x} \sinh ^{\frac{3}{2}}(c+d x)}{4 b^2-9 d^2}-\frac{6 d e^{a+b x} \sqrt{\sinh (c+d x)} \cosh (c+d x)}{4 b^2-9 d^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(-3*d^2*E^(a + b*x))/(4*(b^2 - (9*d^2)/4)*Sqrt[Sinh[c + d*x]]) + E^(a + b*x)*Sinh[c + d*x]^(3/2),x]

[Out]

(-6*d*E^(a + b*x)*Cosh[c + d*x]*Sqrt[Sinh[c + d*x]])/(4*b^2 - 9*d^2) + (4*b*E^(a + b*x)*Sinh[c + d*x]^(3/2))/(
4*b^2 - 9*d^2)

Rule 5482

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sinh[(d_.) + (e_.)*(x_)]^(n_), x_Symbol] :> Dist[(E^(n*(d + e*x))*Sinh[d
+ e*x]^n)/(-1 + E^(2*(d + e*x)))^n, Int[(F^(c*(a + b*x))*(-1 + E^(2*(d + e*x)))^n)/E^(n*(d + e*x)), x], x] /;
FreeQ[{F, a, b, c, d, e, n}, x] &&  !IntegerQ[n]

Rule 2253

Int[((a_) + (b_.)*(F_)^((e_.)*(v_)))^(p_)*(G_)^((h_.)*(u_)), x_Symbol] :> Int[G^(h*ExpandToSum[u, x])*(a + b*F
^(e*ExpandToSum[v, x]))^p, x] /; FreeQ[{F, G, a, b, e, h, p}, x] && LinearQ[{u, v}, x] &&  !LinearMatchQ[{u, v
}, x]

Rule 2252

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Dist
[(a + b*F^(e*(c + d*x)))^p/(1 + (b/a)*F^(e*(c + d*x)))^p, Int[G^(h*(f + g*x))*(1 + (b*F^(e*(c + d*x)))/a)^p, x
], x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 2251

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Simp
[(a^p*G^(h*(f + g*x))*Hypergeometric2F1[-p, (g*h*Log[G])/(d*e*Log[F]), (g*h*Log[G])/(d*e*Log[F]) + 1, Simplify
[-((b*F^(e*(c + d*x)))/a)]])/(g*h*Log[G]), x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && (ILtQ[p, 0] ||
GtQ[a, 0])

Rule 5476

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sinh[(d_.) + (e_.)*(x_)]^(n_), x_Symbol] :> -Simp[(b*c*Log[F]*F^(c*(a +
b*x))*Sinh[d + e*x]^n)/(e^2*n^2 - b^2*c^2*Log[F]^2), x] + (-Dist[(n*(n - 1)*e^2)/(e^2*n^2 - b^2*c^2*Log[F]^2),
Int[F^(c*(a + b*x))*Sinh[d + e*x]^(n - 2), x], x] + Simp[(e*n*F^(c*(a + b*x))*Cosh[d + e*x]*Sinh[d + e*x]^(n
- 1))/(e^2*n^2 - b^2*c^2*Log[F]^2), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 - b^2*c^2*Log[F]^2, 0]
&& GtQ[n, 1]

Rubi steps

\begin{align*} \int \left (-\frac{3 d^2 e^{a+b x}}{4 \left (b^2-\frac{9 d^2}{4}\right ) \sqrt{\sinh (c+d x)}}+e^{a+b x} \sinh ^{\frac{3}{2}}(c+d x)\right ) \, dx &=-\frac{\left (3 d^2\right ) \int \frac{e^{a+b x}}{\sqrt{\sinh (c+d x)}} \, dx}{4 b^2-9 d^2}+\int e^{a+b x} \sinh ^{\frac{3}{2}}(c+d x) \, dx\\ &=-\frac{6 d e^{a+b x} \cosh (c+d x) \sqrt{\sinh (c+d x)}}{4 b^2-9 d^2}+\frac{4 b e^{a+b x} \sinh ^{\frac{3}{2}}(c+d x)}{4 b^2-9 d^2}+\frac{\left (3 d^2\right ) \int \frac{e^{a+b x}}{\sqrt{\sinh (c+d x)}} \, dx}{4 b^2-9 d^2}-\frac{\left (3 d^2 e^{\frac{1}{2} (-c-d x)} \sqrt{-1+e^{2 (c+d x)}}\right ) \int \frac{e^{a+b x+\frac{1}{2} (c+d x)}}{\sqrt{-1+e^{2 (c+d x)}}} \, dx}{\left (4 b^2-9 d^2\right ) \sqrt{\sinh (c+d x)}}\\ &=-\frac{6 d e^{a+b x} \cosh (c+d x) \sqrt{\sinh (c+d x)}}{4 b^2-9 d^2}+\frac{4 b e^{a+b x} \sinh ^{\frac{3}{2}}(c+d x)}{4 b^2-9 d^2}-\frac{\left (3 d^2 e^{\frac{1}{2} (-c-d x)} \sqrt{-1+e^{2 (c+d x)}}\right ) \int \frac{e^{\frac{1}{2} (2 a+c)+\frac{1}{2} (2 b+d) x}}{\sqrt{-1+e^{2 (c+d x)}}} \, dx}{\left (4 b^2-9 d^2\right ) \sqrt{\sinh (c+d x)}}+\frac{\left (3 d^2 e^{\frac{1}{2} (-c-d x)} \sqrt{-1+e^{2 (c+d x)}}\right ) \int \frac{e^{a+b x+\frac{1}{2} (c+d x)}}{\sqrt{-1+e^{2 (c+d x)}}} \, dx}{\left (4 b^2-9 d^2\right ) \sqrt{\sinh (c+d x)}}\\ &=-\frac{6 d e^{a+b x} \cosh (c+d x) \sqrt{\sinh (c+d x)}}{4 b^2-9 d^2}+\frac{4 b e^{a+b x} \sinh ^{\frac{3}{2}}(c+d x)}{4 b^2-9 d^2}-\frac{\left (3 d^2 e^{\frac{1}{2} (-c-d x)} \sqrt{1-e^{2 (c+d x)}}\right ) \int \frac{e^{\frac{1}{2} (2 a+c)+\frac{1}{2} (2 b+d) x}}{\sqrt{1-e^{2 (c+d x)}}} \, dx}{\left (4 b^2-9 d^2\right ) \sqrt{\sinh (c+d x)}}+\frac{\left (3 d^2 e^{\frac{1}{2} (-c-d x)} \sqrt{-1+e^{2 (c+d x)}}\right ) \int \frac{e^{\frac{1}{2} (2 a+c)+\frac{1}{2} (2 b+d) x}}{\sqrt{-1+e^{2 (c+d x)}}} \, dx}{\left (4 b^2-9 d^2\right ) \sqrt{\sinh (c+d x)}}\\ &=-\frac{6 d^2 \exp \left (\frac{1}{2} (2 a+c)+\frac{1}{2} (2 b+d) x+\frac{1}{2} (-c-d x)\right ) \sqrt{1-e^{2 (c+d x)}} \, _2F_1\left (\frac{1}{2},\frac{2 b+d}{4 d};\frac{1}{4} \left (5+\frac{2 b}{d}\right );e^{2 (c+d x)}\right )}{(2 b+d) \left (4 b^2-9 d^2\right ) \sqrt{\sinh (c+d x)}}-\frac{6 d e^{a+b x} \cosh (c+d x) \sqrt{\sinh (c+d x)}}{4 b^2-9 d^2}+\frac{4 b e^{a+b x} \sinh ^{\frac{3}{2}}(c+d x)}{4 b^2-9 d^2}+\frac{\left (3 d^2 e^{\frac{1}{2} (-c-d x)} \sqrt{1-e^{2 (c+d x)}}\right ) \int \frac{e^{\frac{1}{2} (2 a+c)+\frac{1}{2} (2 b+d) x}}{\sqrt{1-e^{2 (c+d x)}}} \, dx}{\left (4 b^2-9 d^2\right ) \sqrt{\sinh (c+d x)}}\\ &=-\frac{6 d e^{a+b x} \cosh (c+d x) \sqrt{\sinh (c+d x)}}{4 b^2-9 d^2}+\frac{4 b e^{a+b x} \sinh ^{\frac{3}{2}}(c+d x)}{4 b^2-9 d^2}\\ \end{align*}

Mathematica [C]  time = 1.32062, size = 155, normalized size = 2.12 $\frac{2 e^{a+b x} \left (e^{2 (c+d x)}-1\right ) \left (\left (4 b^2+8 b d+3 d^2\right ) \sinh ^2(c+d x) \, _2F_1\left (1,\frac{1}{4} \left (\frac{2 b}{d}+7\right );\frac{2 b+d}{4 d};e^{2 (c+d x)}\right )-3 d^2 \, _2F_1\left (1,\frac{1}{4} \left (\frac{2 b}{d}+3\right );\frac{1}{4} \left (\frac{2 b}{d}+5\right );e^{2 (c+d x)}\right )\right )}{(2 b+d) (3 d-2 b) (2 b+3 d) \sqrt{\sinh (c+d x)}}$

Warning: Unable to verify antiderivative.

[In]

Integrate[(-3*d^2*E^(a + b*x))/(4*(b^2 - (9*d^2)/4)*Sqrt[Sinh[c + d*x]]) + E^(a + b*x)*Sinh[c + d*x]^(3/2),x]

[Out]

(2*E^(a + b*x)*(-1 + E^(2*(c + d*x)))*(-3*d^2*Hypergeometric2F1[1, (3 + (2*b)/d)/4, (5 + (2*b)/d)/4, E^(2*(c +
d*x))] + (4*b^2 + 8*b*d + 3*d^2)*Hypergeometric2F1[1, (7 + (2*b)/d)/4, (2*b + d)/(4*d), E^(2*(c + d*x))]*Sinh
[c + d*x]^2))/((2*b + d)*(-2*b + 3*d)*(2*b + 3*d)*Sqrt[Sinh[c + d*x]])

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Maple [F]  time = 0.214, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{bx+a}} \left ( \sinh \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}-{\frac{3\,{d}^{2}{{\rm e}^{bx+a}}}{4} \left ({b}^{2}-{\frac{9\,{d}^{2}}{4}} \right ) ^{-1}{\frac{1}{\sqrt{\sinh \left ( dx+c \right ) }}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(exp(b*x+a)*sinh(d*x+c)^(3/2)-3/4*d^2*exp(b*x+a)/(b^2-9/4*d^2)/sinh(d*x+c)^(1/2),x)

[Out]

int(exp(b*x+a)*sinh(d*x+c)^(3/2)-3/4*d^2*exp(b*x+a)/(b^2-9/4*d^2)/sinh(d*x+c)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\left (b x + a\right )} \sinh \left (d x + c\right )^{\frac{3}{2}} - \frac{3 \, d^{2} e^{\left (b x + a\right )}}{{\left (4 \, b^{2} - 9 \, d^{2}\right )} \sqrt{\sinh \left (d x + c\right )}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)*sinh(d*x+c)^(3/2)-3/4*d^2*exp(b*x+a)/(b^2-9/4*d^2)/sinh(d*x+c)^(1/2),x, algorithm="maxima
")

[Out]

integrate(e^(b*x + a)*sinh(d*x + c)^(3/2) - 3*d^2*e^(b*x + a)/((4*b^2 - 9*d^2)*sqrt(sinh(d*x + c))), x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)*sinh(d*x+c)^(3/2)-3/4*d^2*exp(b*x+a)/(b^2-9/4*d^2)/sinh(d*x+c)^(1/2),x, algorithm="fricas
")

[Out]

Exception raised: UnboundLocalError

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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(b*x+a)*sinh(d*x+c)**(3/2)-3/4*d**2*exp(b*x+a)/(b**2-9/4*d**2)/sinh(d*x+c)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\left (b x + a\right )} \sinh \left (d x + c\right )^{\frac{3}{2}} - \frac{3 \, d^{2} e^{\left (b x + a\right )}}{{\left (4 \, b^{2} - 9 \, d^{2}\right )} \sqrt{\sinh \left (d x + c\right )}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)*sinh(d*x+c)^(3/2)-3/4*d^2*exp(b*x+a)/(b^2-9/4*d^2)/sinh(d*x+c)^(1/2),x, algorithm="giac")

[Out]

integrate(e^(b*x + a)*sinh(d*x + c)^(3/2) - 3*d^2*e^(b*x + a)/((4*b^2 - 9*d^2)*sqrt(sinh(d*x + c))), x)