3.4.100 \(\int (d \cosh (e+f x))^m (a+b \sinh ^2(e+f x))^p \, dx\) [400]

Optimal. Leaf size=117 \[ \frac {d F_1\left (\frac {1}{2};\frac {1-m}{2},-p;\frac {3}{2};-\sinh ^2(e+f x),-\frac {b \sinh ^2(e+f x)}{a}\right ) (d \cosh (e+f x))^{-1+m} \cosh ^2(e+f x)^{\frac {1-m}{2}} \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \left (1+\frac {b \sinh ^2(e+f x)}{a}\right )^{-p}}{f} \]

[Out]

d*AppellF1(1/2,1/2-1/2*m,-p,3/2,-sinh(f*x+e)^2,-b*sinh(f*x+e)^2/a)*(d*cosh(f*x+e))^(-1+m)*(cosh(f*x+e)^2)^(1/2
-1/2*m)*sinh(f*x+e)*(a+b*sinh(f*x+e)^2)^p/f/((1+b*sinh(f*x+e)^2/a)^p)

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Rubi [A]
time = 0.08, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3272, 441, 440} \begin {gather*} \frac {d \sinh (e+f x) \cosh ^2(e+f x)^{\frac {1-m}{2}} (d \cosh (e+f x))^{m-1} \left (a+b \sinh ^2(e+f x)\right )^p \left (\frac {b \sinh ^2(e+f x)}{a}+1\right )^{-p} F_1\left (\frac {1}{2};\frac {1-m}{2},-p;\frac {3}{2};-\sinh ^2(e+f x),-\frac {b \sinh ^2(e+f x)}{a}\right )}{f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*Cosh[e + f*x])^m*(a + b*Sinh[e + f*x]^2)^p,x]

[Out]

(d*AppellF1[1/2, (1 - m)/2, -p, 3/2, -Sinh[e + f*x]^2, -((b*Sinh[e + f*x]^2)/a)]*(d*Cosh[e + f*x])^(-1 + m)*(C
osh[e + f*x]^2)^((1 - m)/2)*Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^p)/(f*(1 + (b*Sinh[e + f*x]^2)/a)^p)

Rule 440

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 441

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^F
racPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 3272

Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff
= FreeFactors[Sin[e + f*x], x]}, Dist[ff*d^(2*IntPart[(m - 1)/2] + 1)*((d*Cos[e + f*x])^(2*FracPart[(m - 1)/2
])/(f*(Cos[e + f*x]^2)^FracPart[(m - 1)/2])), Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x,
Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] &&  !IntegerQ[m]

Rubi steps

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

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Mathematica [F]
time = 6.71, size = 0, normalized size = 0.00 \begin {gather*} \int (d \cosh (e+f x))^m \left (a+b \sinh ^2(e+f x)\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(d*Cosh[e + f*x])^m*(a + b*Sinh[e + f*x]^2)^p,x]

[Out]

Integrate[(d*Cosh[e + f*x])^m*(a + b*Sinh[e + f*x]^2)^p, x]

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Maple [F]
time = 1.57, size = 0, normalized size = 0.00 \[\int \left (d \cosh \left (f x +e \right )\right )^{m} \left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cosh(f*x+e))^m*(a+b*sinh(f*x+e)^2)^p,x)

[Out]

int((d*cosh(f*x+e))^m*(a+b*sinh(f*x+e)^2)^p,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((d*cosh(f*x+e))^m*(a+b*sinh(f*x+e)^2)^p,x, algorithm="maxima")

[Out]

integrate((b*sinh(f*x + e)^2 + a)^p*(d*cosh(f*x + e))^m, x)

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Fricas [F]
time = 0.45, size = 27, normalized size = 0.23 \begin {gather*} {\rm integral}\left ({\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{p} \left (d \cosh \left (f x + e\right )\right )^{m}, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cosh(f*x+e))^m*(a+b*sinh(f*x+e)^2)^p,x, algorithm="fricas")

[Out]

integral((b*sinh(f*x + e)^2 + a)^p*(d*cosh(f*x + e))^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((d*cosh(f*x+e))**m*(a+b*sinh(f*x+e)**2)**p,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((d*cosh(f*x+e))^m*(a+b*sinh(f*x+e)^2)^p,x, algorithm="giac")

[Out]

integrate((b*sinh(f*x + e)^2 + a)^p*(d*cosh(f*x + e))^m, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (d\,\mathrm {cosh}\left (e+f\,x\right )\right )}^m\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cosh(e + f*x))^m*(a + b*sinh(e + f*x)^2)^p,x)

[Out]

int((d*cosh(e + f*x))^m*(a + b*sinh(e + f*x)^2)^p, x)

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