3.9.54 \(\int (a+b \cosh (c+d x) \sinh (c+d x))^m \, dx\) [854]

Optimal. Leaf size=147 \[ \frac {i F_1\left (\frac {1}{2};\frac {1}{2},-m;\frac {3}{2};\frac {1}{2} (1-i \sinh (2 c+2 d x)),\frac {b (1-i \sinh (2 c+2 d x))}{2 i a+b}\right ) \cosh (2 c+2 d x) \left (a+\frac {1}{2} b \sinh (2 c+2 d x)\right )^m \left (\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}\right )^{-m}}{\sqrt {2} d \sqrt {1+i \sinh (2 c+2 d x)}} \]

[Out]

1/2*I*AppellF1(1/2,-m,1/2,3/2,b*(1-I*sinh(2*d*x+2*c))/(2*I*a+b),1/2-1/2*I*sinh(2*d*x+2*c))*cosh(2*d*x+2*c)*(a+
1/2*b*sinh(2*d*x+2*c))^m/d/(((2*a+b*sinh(2*d*x+2*c))/(2*a-I*b))^m)*2^(1/2)/(1+I*sinh(2*d*x+2*c))^(1/2)

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Rubi [A]
time = 0.09, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2745, 2744, 144, 143} \begin {gather*} \frac {i \cosh (2 c+2 d x) \left (a+\frac {1}{2} b \sinh (2 c+2 d x)\right )^m \left (\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}\right )^{-m} F_1\left (\frac {1}{2};\frac {1}{2},-m;\frac {3}{2};\frac {1}{2} (1-i \sinh (2 c+2 d x)),\frac {b (1-i \sinh (2 c+2 d x))}{2 i a+b}\right )}{\sqrt {2} d \sqrt {1+i \sinh (2 c+2 d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(I*AppellF1[1/2, 1/2, -m, 3/2, (1 - I*Sinh[2*c + 2*d*x])/2, (b*(1 - I*Sinh[2*c + 2*d*x]))/((2*I)*a + b)]*Cosh[
2*c + 2*d*x]*(a + (b*Sinh[2*c + 2*d*x])/2)^m)/(Sqrt[2]*d*Sqrt[1 + I*Sinh[2*c + 2*d*x]]*((2*a + b*Sinh[2*c + 2*
d*x])/(2*a - I*b))^m)

Rule 143

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x)
^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n*(b/(b*e - a*f))^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(b*c
- a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !Inte
gerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !(GtQ[d/(d*a - c*b), 0] && GtQ[
d/(d*e - c*f), 0] && SimplerQ[c + d*x, a + b*x]) &&  !(GtQ[f/(f*a - e*b), 0] && GtQ[f/(f*c - e*d), 0] && Simpl
erQ[e + f*x, a + b*x])

Rule 144

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(e + f*x)^
FracPart[p]/((b/(b*e - a*f))^IntPart[p]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p]), Int[(a + b*x)^m*(c + d*x)^n*
(b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !GtQ[b/(b*e - a*f), 0]

Rule 2744

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

Rule 2745

Int[((a_) + cos[(c_.) + (d_.)*(x_)]*(b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Int[(a + b*(Sin[2*c + 2*
d*x]/2))^n, x] /; FreeQ[{a, b, c, d, n}, x]

Rubi steps

\begin {align*} \int (a+b \cosh (c+d x) \sinh (c+d x))^m \, dx &=\int \left (a+\frac {1}{2} b \sinh (2 c+2 d x)\right )^m \, dx\\ &=-\frac {(i \cosh (2 c+2 d x)) \text {Subst}\left (\int \frac {\left (a-\frac {i b x}{2}\right )^m}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,i \sinh (2 c+2 d x)\right )}{2 d \sqrt {1-i \sinh (2 c+2 d x)} \sqrt {1+i \sinh (2 c+2 d x)}}\\ &=-\frac {\left (i \cosh (2 c+2 d x) \left (a+\frac {1}{2} b \sinh (2 c+2 d x)\right )^m \left (-\frac {a+\frac {1}{2} b \sinh (2 c+2 d x)}{-a+\frac {i b}{2}}\right )^{-m}\right ) \text {Subst}\left (\int \frac {\left (-\frac {a}{-a+\frac {i b}{2}}+\frac {i b x}{2 \left (-a+\frac {i b}{2}\right )}\right )^m}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,i \sinh (2 c+2 d x)\right )}{2 d \sqrt {1-i \sinh (2 c+2 d x)} \sqrt {1+i \sinh (2 c+2 d x)}}\\ &=\frac {i F_1\left (\frac {1}{2};\frac {1}{2},-m;\frac {3}{2};\frac {1}{2} (1-i \sinh (2 c+2 d x)),\frac {b (1-i \sinh (2 c+2 d x))}{2 i a+b}\right ) \cosh (2 c+2 d x) \left (a+\frac {1}{2} b \sinh (2 c+2 d x)\right )^m \left (\frac {2 a+b \sinh (2 c+2 d x)}{2 a-i b}\right )^{-m}}{\sqrt {2} d \sqrt {1+i \sinh (2 c+2 d x)}}\\ \end {align*}

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Mathematica [A]
time = 0.47, size = 162, normalized size = 1.10 \begin {gather*} \frac {F_1\left (1+m;\frac {1}{2},\frac {1}{2};2+m;\frac {2 a+b \sinh (2 (c+d x))}{2 a+i b},\frac {2 a+b \sinh (2 (c+d x))}{2 a-i b}\right ) \text {sech}(2 (c+d x)) \sqrt {\frac {b (1-i \sinh (2 (c+d x)))}{2 i a+b}} \sqrt {\frac {b (1+i \sinh (2 (c+d x)))}{-2 i a+b}} \left (a+\frac {1}{2} b \sinh (2 (c+d x))\right )^{1+m}}{b d (1+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(AppellF1[1 + m, 1/2, 1/2, 2 + m, (2*a + b*Sinh[2*(c + d*x)])/(2*a + I*b), (2*a + b*Sinh[2*(c + d*x)])/(2*a -
I*b)]*Sech[2*(c + d*x)]*Sqrt[(b*(1 - I*Sinh[2*(c + d*x)]))/((2*I)*a + b)]*Sqrt[(b*(1 + I*Sinh[2*(c + d*x)]))/(
(-2*I)*a + b)]*(a + (b*Sinh[2*(c + d*x)])/2)^(1 + m))/(b*d*(1 + m))

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Maple [F]
time = 4.88, size = 0, normalized size = 0.00 \[\int \left (a +b \cosh \left (d x +c \right ) \sinh \left (d x +c \right )\right )^{m}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*cosh(d*x + c)*sinh(d*x + c) + a)^m, x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+b\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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