3.130 \(\int (d \sinh (e+f x))^m (a+b \sinh ^2(e+f x))^p \, dx\)

Optimal. Leaf size=128 \[ \frac {d \cosh (e+f x) \left (-\sinh ^2(e+f x)\right )^{\frac {1-m}{2}} (d \sinh (e+f x))^{m-1} \left (a+b \cosh ^2(e+f x)-b\right )^p \left (\frac {b \cosh ^2(e+f x)}{a-b}+1\right )^{-p} F_1\left (\frac {1}{2};\frac {1-m}{2},-p;\frac {3}{2};\cosh ^2(e+f x),-\frac {b \cosh ^2(e+f x)}{a-b}\right )}{f} \]

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 128, 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 = {3189, 430, 429} \[ \frac {d \cosh (e+f x) \left (-\sinh ^2(e+f x)\right )^{\frac {1-m}{2}} (d \sinh (e+f x))^{m-1} \left (a+b \cosh ^2(e+f x)-b\right )^p \left (\frac {b \cosh ^2(e+f x)}{a-b}+1\right )^{-p} F_1\left (\frac {1}{2};\frac {1-m}{2},-p;\frac {3}{2};\cosh ^2(e+f x),-\frac {b \cosh ^2(e+f x)}{a-b}\right )}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 429

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 430

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 3189

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

Rubi steps

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

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Mathematica [F]  time = 9.65, size = 0, normalized size = 0.00 \[ \int (d \sinh (e+f x))^m \left (a+b \sinh ^2(e+f x)\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F]  time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{p} \left (d \sinh \left (f x + e\right )\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sinh(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*sinh(f*x + e))^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sinh(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*sinh(f*x + e))^m, x)

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maple [F]  time = 0.90, size = 0, normalized size = 0.00 \[ \int \left (d \sinh \left (f x +e \right )\right )^{m} \left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((d*sinh(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 \[ \int {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{p} \left (d \sinh \left (f x + e\right )\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sinh(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*sinh(f*x + e))^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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