3.348 \(\int \sec ^3(c+d x) (a+a \sin (c+d x))^m \, dx\)

Optimal. Leaf size=47 \[ -\frac{a (a \sin (c+d x)+a)^{m-1} \, _2F_1\left (2,m-1;m;\frac{1}{2} (\sin (c+d x)+1)\right )}{4 d (1-m)} \]

[Out]

-(a*Hypergeometric2F1[2, -1 + m, m, (1 + Sin[c + d*x])/2]*(a + a*Sin[c + d*x])^(-1 + m))/(4*d*(1 - m))

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Rubi [A]  time = 0.0545228, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2667, 68} \[ -\frac{a (a \sin (c+d x)+a)^{m-1} \, _2F_1\left (2,m-1;m;\frac{1}{2} (\sin (c+d x)+1)\right )}{4 d (1-m)} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^m,x]

[Out]

-(a*Hypergeometric2F1[2, -1 + m, m, (1 + Sin[c + d*x])/2]*(a + a*Sin[c + d*x])^(-1 + m))/(4*d*(1 - m))

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rubi steps

\begin{align*} \int \sec ^3(c+d x) (a+a \sin (c+d x))^m \, dx &=\frac{a^3 \operatorname{Subst}\left (\int \frac{(a+x)^{-2+m}}{(a-x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{a \, _2F_1\left (2,-1+m;m;\frac{1}{2} (1+\sin (c+d x))\right ) (a+a \sin (c+d x))^{-1+m}}{4 d (1-m)}\\ \end{align*}

Mathematica [B]  time = 0.337913, size = 111, normalized size = 2.36 \[ \frac{(a (\sin (c+d x)+1))^m \left (\frac{2 (\sin (c+d x)+1) \, _2F_1\left (1,m+1;m+2;\frac{1}{2} (\sin (c+d x)+1)\right )}{m+1}+\frac{(\sin (c+d x)+1) \, _2F_1\left (2,m+1;m+2;\frac{1}{2} (\sin (c+d x)+1)\right )}{m+1}+4 \left (\frac{1}{(m-1) (\sin (c+d x)+1)}+\frac{1}{m}\right )\right )}{16 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^m,x]

[Out]

((a*(1 + Sin[c + d*x]))^m*((2*Hypergeometric2F1[1, 1 + m, 2 + m, (1 + Sin[c + d*x])/2]*(1 + Sin[c + d*x]))/(1
+ m) + (Hypergeometric2F1[2, 1 + m, 2 + m, (1 + Sin[c + d*x])/2]*(1 + Sin[c + d*x]))/(1 + m) + 4*(m^(-1) + 1/(
(-1 + m)*(1 + Sin[c + d*x])))))/(16*d)

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( dx+c \right ) \right ) ^{3} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^3*(a+a*sin(d*x+c))^m,x)

[Out]

int(sec(d*x+c)^3*(a+a*sin(d*x+c))^m,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3*(a+a*sin(d*x+c))^m,x, algorithm="maxima")

[Out]

integrate((a*sin(d*x + c) + a)^m*sec(d*x + c)^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3*(a+a*sin(d*x+c))^m,x, algorithm="fricas")

[Out]

integral((a*sin(d*x + c) + a)^m*sec(d*x + c)^3, 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(sec(d*x+c)**3*(a+a*sin(d*x+c))**m,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3*(a+a*sin(d*x+c))^m,x, algorithm="giac")

[Out]

integrate((a*sin(d*x + c) + a)^m*sec(d*x + c)^3, x)