3.258 \(\int \frac {(b \cos (c+d x))^n}{\cos ^{\frac {7}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=80 \[ \frac {2 \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2 n-5);\frac {1}{4} (2 n-1);\cos ^2(c+d x)\right )}{d (5-2 n) \sqrt {\sin ^2(c+d x)} \cos ^{\frac {5}{2}}(c+d x)} \]

[Out]

2*(b*cos(d*x+c))^n*hypergeom([1/2, -5/4+1/2*n],[-1/4+1/2*n],cos(d*x+c)^2)*sin(d*x+c)/d/(5-2*n)/cos(d*x+c)^(5/2
)/(sin(d*x+c)^2)^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 80, normalized size of antiderivative = 1.00, 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 = {20, 2643} \[ \frac {2 \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2 n-5);\frac {1}{4} (2 n-1);\cos ^2(c+d x)\right )}{d (5-2 n) \sqrt {\sin ^2(c+d x)} \cos ^{\frac {5}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[(b*Cos[c + d*x])^n/Cos[c + d*x]^(7/2),x]

[Out]

(2*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (-5 + 2*n)/4, (-1 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 -
2*n)*Cos[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2])

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

\begin {align*} \int \frac {(b \cos (c+d x))^n}{\cos ^{\frac {7}{2}}(c+d x)} \, dx &=\left (\cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{-\frac {7}{2}+n}(c+d x) \, dx\\ &=\frac {2 (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (-5+2 n);\frac {1}{4} (-1+2 n);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (5-2 n) \cos ^{\frac {5}{2}}(c+d x) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 80, normalized size = 1.00 \[ -\frac {\sqrt {\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} \left (n-\frac {5}{2}\right );\frac {1}{2} \left (n-\frac {1}{2}\right );\cos ^2(c+d x)\right )}{d \left (n-\frac {5}{2}\right ) \cos ^{\frac {5}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Cos[c + d*x])^n/Cos[c + d*x]^(7/2),x]

[Out]

-(((b*Cos[c + d*x])^n*Csc[c + d*x]*Hypergeometric2F1[1/2, (-5/2 + n)/2, (-1/2 + n)/2, Cos[c + d*x]^2]*Sqrt[Sin
[c + d*x]^2])/(d*(-5/2 + n)*Cos[c + d*x]^(5/2)))

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fricas [F]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (b \cos \left (d x + c\right )\right )^{n}}{\cos \left (d x + c\right )^{\frac {7}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*cos(d*x+c))^n/cos(d*x+c)^(7/2),x, algorithm="fricas")

[Out]

integral((b*cos(d*x + c))^n/cos(d*x + c)^(7/2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \cos \left (d x + c\right )\right )^{n}}{\cos \left (d x + c\right )^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*cos(d*x+c))^n/cos(d*x+c)^(7/2),x, algorithm="giac")

[Out]

integrate((b*cos(d*x + c))^n/cos(d*x + c)^(7/2), x)

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maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \cos \left (d x +c \right )\right )^{n}}{\cos \left (d x +c \right )^{\frac {7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*cos(d*x+c))^n/cos(d*x+c)^(7/2),x)

[Out]

int((b*cos(d*x+c))^n/cos(d*x+c)^(7/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \cos \left (d x + c\right )\right )^{n}}{\cos \left (d x + c\right )^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*cos(d*x+c))^n/cos(d*x+c)^(7/2),x, algorithm="maxima")

[Out]

integrate((b*cos(d*x + c))^n/cos(d*x + c)^(7/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,\cos \left (c+d\,x\right )\right )}^n}{{\cos \left (c+d\,x\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*cos(c + d*x))^n/cos(c + d*x)^(7/2),x)

[Out]

int((b*cos(c + d*x))^n/cos(c + d*x)^(7/2), 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((b*cos(d*x+c))**n/cos(d*x+c)**(7/2),x)

[Out]

Timed out

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