3.107 \(\int (a+b \cosh (x))^{5/2} (A+B \cosh (x)) \, dx\)

Optimal. Leaf size=233 \[ \frac{2 i \left (a^2-b^2\right ) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}} \text{EllipticF}\left (\frac{i x}{2},\frac{2 b}{a+b}\right )}{105 b \sqrt{a+b \cosh (x)}}+\frac{2}{105} \sinh (x) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sqrt{a+b \cosh (x)}-\frac{2 i \left (161 a^2 A b+15 a^3 B+145 a b^2 B+63 A b^3\right ) \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{105 b \sqrt{\frac{a+b \cosh (x)}{a+b}}}+\frac{2}{35} \sinh (x) (5 a B+7 A b) (a+b \cosh (x))^{3/2}+\frac{2}{7} B \sinh (x) (a+b \cosh (x))^{5/2} \]

[Out]

(((-2*I)/105)*(161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*Sqrt[a + b*Cosh[x]]*EllipticE[(I/2)*x, (2*b)/(
a + b)])/(b*Sqrt[(a + b*Cosh[x])/(a + b)]) + (((2*I)/105)*(a^2 - b^2)*(56*a*A*b + 15*a^2*B + 25*b^2*B)*Sqrt[(a
+ b*Cosh[x])/(a + b)]*EllipticF[(I/2)*x, (2*b)/(a + b)])/(b*Sqrt[a + b*Cosh[x]]) + (2*(56*a*A*b + 15*a^2*B +
25*b^2*B)*Sqrt[a + b*Cosh[x]]*Sinh[x])/105 + (2*(7*A*b + 5*a*B)*(a + b*Cosh[x])^(3/2)*Sinh[x])/35 + (2*B*(a +
b*Cosh[x])^(5/2)*Sinh[x])/7

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Rubi [A]  time = 0.454098, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {2753, 2752, 2663, 2661, 2655, 2653} \[ \frac{2}{105} \sinh (x) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sqrt{a+b \cosh (x)}+\frac{2 i \left (a^2-b^2\right ) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}} F\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{105 b \sqrt{a+b \cosh (x)}}-\frac{2 i \left (161 a^2 A b+15 a^3 B+145 a b^2 B+63 A b^3\right ) \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{105 b \sqrt{\frac{a+b \cosh (x)}{a+b}}}+\frac{2}{35} \sinh (x) (5 a B+7 A b) (a+b \cosh (x))^{3/2}+\frac{2}{7} B \sinh (x) (a+b \cosh (x))^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cosh[x])^(5/2)*(A + B*Cosh[x]),x]

[Out]

(((-2*I)/105)*(161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*Sqrt[a + b*Cosh[x]]*EllipticE[(I/2)*x, (2*b)/(
a + b)])/(b*Sqrt[(a + b*Cosh[x])/(a + b)]) + (((2*I)/105)*(a^2 - b^2)*(56*a*A*b + 15*a^2*B + 25*b^2*B)*Sqrt[(a
+ b*Cosh[x])/(a + b)]*EllipticF[(I/2)*x, (2*b)/(a + b)])/(b*Sqrt[a + b*Cosh[x]]) + (2*(56*a*A*b + 15*a^2*B +
25*b^2*B)*Sqrt[a + b*Cosh[x]]*Sinh[x])/105 + (2*(7*A*b + 5*a*B)*(a + b*Cosh[x])^(3/2)*Sinh[x])/35 + (2*B*(a +
b*Cosh[x])^(5/2)*Sinh[x])/7

Rule 2753

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(d
*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[
b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*
c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]

Rule 2752

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/b, Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]

Rule 2663

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2655

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rubi steps

\begin{align*} \int (a+b \cosh (x))^{5/2} (A+B \cosh (x)) \, dx &=\frac{2}{7} B (a+b \cosh (x))^{5/2} \sinh (x)+\frac{2}{7} \int (a+b \cosh (x))^{3/2} \left (\frac{1}{2} (7 a A+5 b B)+\frac{1}{2} (7 A b+5 a B) \cosh (x)\right ) \, dx\\ &=\frac{2}{35} (7 A b+5 a B) (a+b \cosh (x))^{3/2} \sinh (x)+\frac{2}{7} B (a+b \cosh (x))^{5/2} \sinh (x)+\frac{4}{35} \int \sqrt{a+b \cosh (x)} \left (\frac{1}{4} \left (35 a^2 A+21 A b^2+40 a b B\right )+\frac{1}{4} \left (56 a A b+15 a^2 B+25 b^2 B\right ) \cosh (x)\right ) \, dx\\ &=\frac{2}{105} \left (56 a A b+15 a^2 B+25 b^2 B\right ) \sqrt{a+b \cosh (x)} \sinh (x)+\frac{2}{35} (7 A b+5 a B) (a+b \cosh (x))^{3/2} \sinh (x)+\frac{2}{7} B (a+b \cosh (x))^{5/2} \sinh (x)+\frac{8}{105} \int \frac{\frac{1}{8} \left (105 a^3 A+119 a A b^2+135 a^2 b B+25 b^3 B\right )+\frac{1}{8} \left (161 a^2 A b+63 A b^3+15 a^3 B+145 a b^2 B\right ) \cosh (x)}{\sqrt{a+b \cosh (x)}} \, dx\\ &=\frac{2}{105} \left (56 a A b+15 a^2 B+25 b^2 B\right ) \sqrt{a+b \cosh (x)} \sinh (x)+\frac{2}{35} (7 A b+5 a B) (a+b \cosh (x))^{3/2} \sinh (x)+\frac{2}{7} B (a+b \cosh (x))^{5/2} \sinh (x)-\frac{\left (\left (a^2-b^2\right ) \left (56 a A b+15 a^2 B+25 b^2 B\right )\right ) \int \frac{1}{\sqrt{a+b \cosh (x)}} \, dx}{105 b}+\frac{\left (161 a^2 A b+63 A b^3+15 a^3 B+145 a b^2 B\right ) \int \sqrt{a+b \cosh (x)} \, dx}{105 b}\\ &=\frac{2}{105} \left (56 a A b+15 a^2 B+25 b^2 B\right ) \sqrt{a+b \cosh (x)} \sinh (x)+\frac{2}{35} (7 A b+5 a B) (a+b \cosh (x))^{3/2} \sinh (x)+\frac{2}{7} B (a+b \cosh (x))^{5/2} \sinh (x)+\frac{\left (\left (161 a^2 A b+63 A b^3+15 a^3 B+145 a b^2 B\right ) \sqrt{a+b \cosh (x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \cosh (x)}{a+b}} \, dx}{105 b \sqrt{\frac{a+b \cosh (x)}{a+b}}}-\frac{\left (\left (a^2-b^2\right ) \left (56 a A b+15 a^2 B+25 b^2 B\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}}\right ) \int \frac{1}{\sqrt{\frac{a}{a+b}+\frac{b \cosh (x)}{a+b}}} \, dx}{105 b \sqrt{a+b \cosh (x)}}\\ &=-\frac{2 i \left (161 a^2 A b+63 A b^3+15 a^3 B+145 a b^2 B\right ) \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{105 b \sqrt{\frac{a+b \cosh (x)}{a+b}}}+\frac{2 i \left (a^2-b^2\right ) \left (56 a A b+15 a^2 B+25 b^2 B\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}} F\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{105 b \sqrt{a+b \cosh (x)}}+\frac{2}{105} \left (56 a A b+15 a^2 B+25 b^2 B\right ) \sqrt{a+b \cosh (x)} \sinh (x)+\frac{2}{35} (7 A b+5 a B) (a+b \cosh (x))^{3/2} \sinh (x)+\frac{2}{7} B (a+b \cosh (x))^{5/2} \sinh (x)\\ \end{align*}

Mathematica [A]  time = 0.562536, size = 203, normalized size = 0.87 \[ \frac{\sinh (x) (a+b \cosh (x)) \left (90 a^2 B+6 b \cosh (x) (15 a B+7 A b)+154 a A b+15 b^2 B \cosh (2 x)+65 b^2 B\right )-\frac{2 i \sqrt{\frac{a+b \cosh (x)}{a+b}} \left (b \left (105 a^3 A+135 a^2 b B+119 a A b^2+25 b^3 B\right ) \text{EllipticF}\left (\frac{i x}{2},\frac{2 b}{a+b}\right )+\left (161 a^2 A b+15 a^3 B+145 a b^2 B+63 A b^3\right ) \left ((a+b) E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )-a \text{EllipticF}\left (\frac{i x}{2},\frac{2 b}{a+b}\right )\right )\right )}{b}}{105 \sqrt{a+b \cosh (x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cosh[x])^(5/2)*(A + B*Cosh[x]),x]

[Out]

(((-2*I)*Sqrt[(a + b*Cosh[x])/(a + b)]*(b*(105*a^3*A + 119*a*A*b^2 + 135*a^2*b*B + 25*b^3*B)*EllipticF[(I/2)*x
, (2*b)/(a + b)] + (161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*((a + b)*EllipticE[(I/2)*x, (2*b)/(a + b)
] - a*EllipticF[(I/2)*x, (2*b)/(a + b)])))/b + (a + b*Cosh[x])*(154*a*A*b + 90*a^2*B + 65*b^2*B + 6*b*(7*A*b +
15*a*B)*Cosh[x] + 15*b^2*B*Cosh[2*x])*Sinh[x])/(105*Sqrt[a + b*Cosh[x]])

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Maple [B]  time = 0.168, size = 1365, normalized size = 5.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cosh(x))^(5/2)*(A+B*cosh(x)),x)

[Out]

2/105*(240*B*(-2*b/(a-b))^(1/2)*b^3*cosh(1/2*x)*sinh(1/2*x)^8+(168*A*(-2*b/(a-b))^(1/2)*b^3+480*B*(-2*b/(a-b))
^(1/2)*a*b^2+360*B*(-2*b/(a-b))^(1/2)*b^3)*sinh(1/2*x)^6*cosh(1/2*x)+(392*A*(-2*b/(a-b))^(1/2)*a*b^2+168*A*(-2
*b/(a-b))^(1/2)*b^3+360*B*(-2*b/(a-b))^(1/2)*a^2*b+480*B*(-2*b/(a-b))^(1/2)*a*b^2+280*B*(-2*b/(a-b))^(1/2)*b^3
)*sinh(1/2*x)^4*cosh(1/2*x)+(154*A*(-2*b/(a-b))^(1/2)*a^2*b+196*A*(-2*b/(a-b))^(1/2)*a*b^2+42*A*(-2*b/(a-b))^(
1/2)*b^3+90*B*(-2*b/(a-b))^(1/2)*a^3+180*B*(-2*b/(a-b))^(1/2)*a^2*b+170*B*(-2*b/(a-b))^(1/2)*a*b^2+80*B*(-2*b/
(a-b))^(1/2)*b^3)*sinh(1/2*x)^2*cosh(1/2*x)+105*A*a^3*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x
)^2)^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))+161*A*a^2*b*(2*b/(a-b)*sinh(1/2*x)
^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))+
119*A*a*b^2*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/(a-
b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))+63*A*b^3*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*
EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))-322*A*(-sinh(1/2*x)^2)^(1/2)*EllipticE(cosh(1
/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*a^2*b-126*A*(-sin
h(1/2*x)^2)^(1/2)*EllipticE(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))*(2*b/(a-b)*sinh(1/2*x)^2+(a
+b)/(a-b))^(1/2)*b^3+15*a^3*B*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticF(cos
h(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))+135*B*a^2*b*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-
sinh(1/2*x)^2)^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))+145*B*a*b^2*(2*b/(a-b)*s
inh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/
b)^(1/2))+25*B*b^3*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticF(cosh(1/2*x)*(-
2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))-30*B*(-sinh(1/2*x)^2)^(1/2)*EllipticE(cosh(1/2*x)*(-2*b/(a-b))^(1/2),
1/2*(-2*(a-b)/b)^(1/2))*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*a^3-290*B*(-sinh(1/2*x)^2)^(1/2)*EllipticE
(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*a*b^2)*((2
*cosh(1/2*x)^2*b+a-b)*sinh(1/2*x)^2)^(1/2)/(-2*b/(a-b))^(1/2)/(2*b*sinh(1/2*x)^4+(a+b)*sinh(1/2*x)^2)^(1/2)/si
nh(1/2*x)/(2*sinh(1/2*x)^2*b+a+b)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \cosh \left (x\right ) + A\right )}{\left (b \cosh \left (x\right ) + a\right )}^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(x))^(5/2)*(A+B*cosh(x)),x, algorithm="maxima")

[Out]

integrate((B*cosh(x) + A)*(b*cosh(x) + a)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B b^{2} \cosh \left (x\right )^{3} + A a^{2} +{\left (2 \, B a b + A b^{2}\right )} \cosh \left (x\right )^{2} +{\left (B a^{2} + 2 \, A a b\right )} \cosh \left (x\right )\right )} \sqrt{b \cosh \left (x\right ) + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(x))^(5/2)*(A+B*cosh(x)),x, algorithm="fricas")

[Out]

integral((B*b^2*cosh(x)^3 + A*a^2 + (2*B*a*b + A*b^2)*cosh(x)^2 + (B*a^2 + 2*A*a*b)*cosh(x))*sqrt(b*cosh(x) +
a), x)

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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((a+b*cosh(x))**(5/2)*(A+B*cosh(x)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \cosh \left (x\right ) + A\right )}{\left (b \cosh \left (x\right ) + a\right )}^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(x))^(5/2)*(A+B*cosh(x)),x, algorithm="giac")

[Out]

integrate((B*cosh(x) + A)*(b*cosh(x) + a)^(5/2), x)