3.131 \(\int \frac{A+B \sinh (x)}{(a+b \sinh (x))^3} \, dx\)

Optimal. Leaf size=128 \[ -\frac{\left (2 a^2 A+3 a b B-A b^2\right ) \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac{\cosh (x) \left (a^2 (-B)+3 a A b+2 b^2 B\right )}{2 \left (a^2+b^2\right )^2 (a+b \sinh (x))}-\frac{\cosh (x) (A b-a B)}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2} \]

[Out]

-(((2*a^2*A - A*b^2 + 3*a*b*B)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2)) - ((A*b - a*B)*C
osh[x])/(2*(a^2 + b^2)*(a + b*Sinh[x])^2) - ((3*a*A*b - a^2*B + 2*b^2*B)*Cosh[x])/(2*(a^2 + b^2)^2*(a + b*Sinh
[x]))

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Rubi [A]  time = 0.17506, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2754, 12, 2660, 618, 206} \[ -\frac{\left (2 a^2 A+3 a b B-A b^2\right ) \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac{\cosh (x) \left (a^2 (-B)+3 a A b+2 b^2 B\right )}{2 \left (a^2+b^2\right )^2 (a+b \sinh (x))}-\frac{\cosh (x) (A b-a B)}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Sinh[x])/(a + b*Sinh[x])^3,x]

[Out]

-(((2*a^2*A - A*b^2 + 3*a*b*B)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2)) - ((A*b - a*B)*C
osh[x])/(2*(a^2 + b^2)*(a + b*Sinh[x])^2) - ((3*a*A*b - a^2*B + 2*b^2*B)*Cosh[x])/(2*(a^2 + b^2)^2*(a + b*Sinh
[x]))

Rule 2754

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{A+B \sinh (x)}{(a+b \sinh (x))^3} \, dx &=-\frac{(A b-a B) \cosh (x)}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2}-\frac{\int \frac{-2 (a A+b B)+(A b-a B) \sinh (x)}{(a+b \sinh (x))^2} \, dx}{2 \left (a^2+b^2\right )}\\ &=-\frac{(A b-a B) \cosh (x)}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2}-\frac{\left (3 a A b-a^2 B+2 b^2 B\right ) \cosh (x)}{2 \left (a^2+b^2\right )^2 (a+b \sinh (x))}+\frac{\int \frac{2 a^2 A-A b^2+3 a b B}{a+b \sinh (x)} \, dx}{2 \left (a^2+b^2\right )^2}\\ &=-\frac{(A b-a B) \cosh (x)}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2}-\frac{\left (3 a A b-a^2 B+2 b^2 B\right ) \cosh (x)}{2 \left (a^2+b^2\right )^2 (a+b \sinh (x))}+\frac{\left (2 a^2 A-A b^2+3 a b B\right ) \int \frac{1}{a+b \sinh (x)} \, dx}{2 \left (a^2+b^2\right )^2}\\ &=-\frac{(A b-a B) \cosh (x)}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2}-\frac{\left (3 a A b-a^2 B+2 b^2 B\right ) \cosh (x)}{2 \left (a^2+b^2\right )^2 (a+b \sinh (x))}+\frac{\left (2 a^2 A-A b^2+3 a b B\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{\left (a^2+b^2\right )^2}\\ &=-\frac{(A b-a B) \cosh (x)}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2}-\frac{\left (3 a A b-a^2 B+2 b^2 B\right ) \cosh (x)}{2 \left (a^2+b^2\right )^2 (a+b \sinh (x))}-\frac{\left (2 \left (2 a^2 A-A b^2+3 a b B\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac{x}{2}\right )\right )}{\left (a^2+b^2\right )^2}\\ &=-\frac{\left (2 a^2 A-A b^2+3 a b B\right ) \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac{(A b-a B) \cosh (x)}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2}-\frac{\left (3 a A b-a^2 B+2 b^2 B\right ) \cosh (x)}{2 \left (a^2+b^2\right )^2 (a+b \sinh (x))}\\ \end{align*}

Mathematica [A]  time = 0.270176, size = 131, normalized size = 1.02 \[ \frac{\frac{2 \left (2 a^2 A+3 a b B-A b^2\right ) \tan ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{-a^2-b^2}}\right )}{\sqrt{-a^2-b^2}}+\frac{\cosh (x) \left (a^2 B-3 a A b-2 b^2 B\right )}{a+b \sinh (x)}+\frac{\left (a^2+b^2\right ) \cosh (x) (a B-A b)}{(a+b \sinh (x))^2}}{2 \left (a^2+b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Sinh[x])/(a + b*Sinh[x])^3,x]

[Out]

((2*(2*a^2*A - A*b^2 + 3*a*b*B)*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] + ((a^2 + b^2)*(-
(A*b) + a*B)*Cosh[x])/(a + b*Sinh[x])^2 + ((-3*a*A*b + a^2*B - 2*b^2*B)*Cosh[x])/(a + b*Sinh[x]))/(2*(a^2 + b^
2)^2)

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Maple [B]  time = 0.036, size = 314, normalized size = 2.5 \begin{align*} -2\,{\frac{1}{ \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) b-a \right ) ^{2}} \left ( -1/2\,{\frac{b \left ( 5\,A{a}^{2}b+2\,A{b}^{3}-3\,{a}^{3}B \right ) \left ( \tanh \left ( x/2 \right ) \right ) ^{3}}{a \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) }}-1/2\,{\frac{ \left ( 4\,A{a}^{4}b-7\,A{a}^{2}{b}^{3}-2\,A{b}^{5}-2\,B{a}^{5}+5\,B{a}^{3}{b}^{2}-2\,Ba{b}^{4} \right ) \left ( \tanh \left ( x/2 \right ) \right ) ^{2}}{ \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ){a}^{2}}}+1/2\,{\frac{b \left ( 11\,A{a}^{2}b+2\,A{b}^{3}-5\,{a}^{3}B+4\,Ba{b}^{2} \right ) \tanh \left ( x/2 \right ) }{a \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) }}+1/2\,{\frac{4\,A{a}^{2}b+A{b}^{3}-2\,{a}^{3}B+Ba{b}^{2}}{{a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4}}} \right ) }+{\frac{2\,{a}^{2}A-A{b}^{2}+3\,bBa}{{a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4}}{\it Artanh} \left ({\frac{1}{2} \left ( 2\,a\tanh \left ( x/2 \right ) -2\,b \right ){\frac{1}{\sqrt{{a}^{2}+{b}^{2}}}}} \right ){\frac{1}{\sqrt{{a}^{2}+{b}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*sinh(x))/(a+b*sinh(x))^3,x)

[Out]

-2*(-1/2*b*(5*A*a^2*b+2*A*b^3-3*B*a^3)/a/(a^4+2*a^2*b^2+b^4)*tanh(1/2*x)^3-1/2*(4*A*a^4*b-7*A*a^2*b^3-2*A*b^5-
2*B*a^5+5*B*a^3*b^2-2*B*a*b^4)/(a^4+2*a^2*b^2+b^4)/a^2*tanh(1/2*x)^2+1/2*b*(11*A*a^2*b+2*A*b^3-5*B*a^3+4*B*a*b
^2)/(a^4+2*a^2*b^2+b^4)/a*tanh(1/2*x)+1/2*(4*A*a^2*b+A*b^3-2*B*a^3+B*a*b^2)/(a^4+2*a^2*b^2+b^4))/(a*tanh(1/2*x
)^2-2*tanh(1/2*x)*b-a)^2+(2*A*a^2-A*b^2+3*B*a*b)/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tanh(1/2
*x)-2*b)/(a^2+b^2)^(1/2))

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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((A+B*sinh(x))/(a+b*sinh(x))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.03506, size = 3617, normalized size = 28.26 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sinh(x))/(a+b*sinh(x))^3,x, algorithm="fricas")

[Out]

-1/2*(2*B*a^4*b^2 - 6*A*a^3*b^3 - 2*B*a^2*b^4 - 6*A*a*b^5 - 4*B*b^6 - 2*(2*A*a^4*b^2 + 3*B*a^3*b^3 + A*a^2*b^4
+ 3*B*a*b^5 - A*b^6)*cosh(x)^3 - 2*(2*A*a^4*b^2 + 3*B*a^3*b^3 + A*a^2*b^4 + 3*B*a*b^5 - A*b^6)*sinh(x)^3 + 2*
(2*B*a^6 - 6*A*a^5*b - 3*B*a^4*b^2 - 3*A*a^3*b^3 - 3*B*a^2*b^4 + 3*A*a*b^5 + 2*B*b^6)*cosh(x)^2 + 2*(2*B*a^6 -
6*A*a^5*b - 3*B*a^4*b^2 - 3*A*a^3*b^3 - 3*B*a^2*b^4 + 3*A*a*b^5 + 2*B*b^6 - 3*(2*A*a^4*b^2 + 3*B*a^3*b^3 + A*
a^2*b^4 + 3*B*a*b^5 - A*b^6)*cosh(x))*sinh(x)^2 + (2*A*a^2*b^3 + 3*B*a*b^4 - A*b^5 + (2*A*a^2*b^3 + 3*B*a*b^4
- A*b^5)*cosh(x)^4 + (2*A*a^2*b^3 + 3*B*a*b^4 - A*b^5)*sinh(x)^4 + 4*(2*A*a^3*b^2 + 3*B*a^2*b^3 - A*a*b^4)*cos
h(x)^3 + 4*(2*A*a^3*b^2 + 3*B*a^2*b^3 - A*a*b^4 + (2*A*a^2*b^3 + 3*B*a*b^4 - A*b^5)*cosh(x))*sinh(x)^3 + 2*(4*
A*a^4*b + 6*B*a^3*b^2 - 4*A*a^2*b^3 - 3*B*a*b^4 + A*b^5)*cosh(x)^2 + 2*(4*A*a^4*b + 6*B*a^3*b^2 - 4*A*a^2*b^3
- 3*B*a*b^4 + A*b^5 + 3*(2*A*a^2*b^3 + 3*B*a*b^4 - A*b^5)*cosh(x)^2 + 6*(2*A*a^3*b^2 + 3*B*a^2*b^3 - A*a*b^4)*
cosh(x))*sinh(x)^2 - 4*(2*A*a^3*b^2 + 3*B*a^2*b^3 - A*a*b^4)*cosh(x) - 4*(2*A*a^3*b^2 + 3*B*a^2*b^3 - A*a*b^4
- (2*A*a^2*b^3 + 3*B*a*b^4 - A*b^5)*cosh(x)^3 - 3*(2*A*a^3*b^2 + 3*B*a^2*b^3 - A*a*b^4)*cosh(x)^2 - (4*A*a^4*b
+ 6*B*a^3*b^2 - 4*A*a^2*b^3 - 3*B*a*b^4 + A*b^5)*cosh(x))*sinh(x))*sqrt(a^2 + b^2)*log((b^2*cosh(x)^2 + b^2*s
inh(x)^2 + 2*a*b*cosh(x) + 2*a^2 + b^2 + 2*(b^2*cosh(x) + a*b)*sinh(x) + 2*sqrt(a^2 + b^2)*(b*cosh(x) + b*sinh
(x) + a))/(b*cosh(x)^2 + b*sinh(x)^2 + 2*a*cosh(x) + 2*(b*cosh(x) + a)*sinh(x) - b)) - 2*(4*B*a^5*b - 10*A*a^4
*b^2 - B*a^3*b^3 - 11*A*a^2*b^4 - 5*B*a*b^5 - A*b^6)*cosh(x) - 2*(4*B*a^5*b - 10*A*a^4*b^2 - B*a^3*b^3 - 11*A*
a^2*b^4 - 5*B*a*b^5 - A*b^6 + 3*(2*A*a^4*b^2 + 3*B*a^3*b^3 + A*a^2*b^4 + 3*B*a*b^5 - A*b^6)*cosh(x)^2 - 2*(2*B
*a^6 - 6*A*a^5*b - 3*B*a^4*b^2 - 3*A*a^3*b^3 - 3*B*a^2*b^4 + 3*A*a*b^5 + 2*B*b^6)*cosh(x))*sinh(x))/(a^6*b^3 +
3*a^4*b^5 + 3*a^2*b^7 + b^9 + (a^6*b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9)*cosh(x)^4 + (a^6*b^3 + 3*a^4*b^5 + 3*a^
2*b^7 + b^9)*sinh(x)^4 + 4*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8)*cosh(x)^3 + 4*(a^7*b^2 + 3*a^5*b^4 + 3*a^
3*b^6 + a*b^8 + (a^6*b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9)*cosh(x))*sinh(x)^3 + 2*(2*a^8*b + 5*a^6*b^3 + 3*a^4*b^
5 - a^2*b^7 - b^9)*cosh(x)^2 + 2*(2*a^8*b + 5*a^6*b^3 + 3*a^4*b^5 - a^2*b^7 - b^9 + 3*(a^6*b^3 + 3*a^4*b^5 + 3
*a^2*b^7 + b^9)*cosh(x)^2 + 6*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8)*cosh(x))*sinh(x)^2 - 4*(a^7*b^2 + 3*a^
5*b^4 + 3*a^3*b^6 + a*b^8)*cosh(x) - 4*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8 - (a^6*b^3 + 3*a^4*b^5 + 3*a^2
*b^7 + b^9)*cosh(x)^3 - 3*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8)*cosh(x)^2 - (2*a^8*b + 5*a^6*b^3 + 3*a^4*b
^5 - a^2*b^7 - b^9)*cosh(x))*sinh(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*sinh(x))/(a+b*sinh(x))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.27497, size = 377, normalized size = 2.95 \begin{align*} -\frac{{\left (2 \, A a^{2} + 3 \, B a b - A b^{2}\right )} \log \left (\frac{{\left | -2 \, b e^{x} - 2 \, a - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | -2 \, b e^{x} - 2 \, a + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{2 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt{a^{2} + b^{2}}} + \frac{2 \, A a^{2} b^{2} e^{\left (3 \, x\right )} + 3 \, B a b^{3} e^{\left (3 \, x\right )} - A b^{4} e^{\left (3 \, x\right )} - 2 \, B a^{4} e^{\left (2 \, x\right )} + 6 \, A a^{3} b e^{\left (2 \, x\right )} + 5 \, B a^{2} b^{2} e^{\left (2 \, x\right )} - 3 \, A a b^{3} e^{\left (2 \, x\right )} - 2 \, B b^{4} e^{\left (2 \, x\right )} + 4 \, B a^{3} b e^{x} - 10 \, A a^{2} b^{2} e^{x} - 5 \, B a b^{3} e^{x} - A b^{4} e^{x} - B a^{2} b^{2} + 3 \, A a b^{3} + 2 \, B b^{4}}{{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )}{\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sinh(x))/(a+b*sinh(x))^3,x, algorithm="giac")

[Out]

-1/2*(2*A*a^2 + 3*B*a*b - A*b^2)*log(abs(-2*b*e^x - 2*a - 2*sqrt(a^2 + b^2))/abs(-2*b*e^x - 2*a + 2*sqrt(a^2 +
b^2)))/((a^4 + 2*a^2*b^2 + b^4)*sqrt(a^2 + b^2)) + (2*A*a^2*b^2*e^(3*x) + 3*B*a*b^3*e^(3*x) - A*b^4*e^(3*x) -
2*B*a^4*e^(2*x) + 6*A*a^3*b*e^(2*x) + 5*B*a^2*b^2*e^(2*x) - 3*A*a*b^3*e^(2*x) - 2*B*b^4*e^(2*x) + 4*B*a^3*b*e
^x - 10*A*a^2*b^2*e^x - 5*B*a*b^3*e^x - A*b^4*e^x - B*a^2*b^2 + 3*A*a*b^3 + 2*B*b^4)/((a^4*b + 2*a^2*b^3 + b^5
)*(b*e^(2*x) + 2*a*e^x - b)^2)