3.156 \(\int \frac {(b \cos (c+d x))^{3/2}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=33 \[ \frac {b \sin (c+d x) \sqrt {b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)} \]

[Out]

b*sin(d*x+c)*(b*cos(d*x+c))^(1/2)/d/cos(d*x+c)^(3/2)

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Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {17, 3767, 8} \[ \frac {b \sin (c+d x) \sqrt {b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[(b*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(7/2),x]

[Out]

(b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 17

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(a^(m + 1/2)*b^(n - 1/2)*Sqrt[b*v])/Sqrt[a*v]
, Int[u*v^(m + n), x], x] /; FreeQ[{a, b, m}, x] &&  !IntegerQ[m] && IGtQ[n + 1/2, 0] && IntegerQ[m + n]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

\begin {align*} \int \frac {(b \cos (c+d x))^{3/2}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx &=\frac {\left (b \sqrt {b \cos (c+d x)}\right ) \int \sec ^2(c+d x) \, dx}{\sqrt {\cos (c+d x)}}\\ &=-\frac {\left (b \sqrt {b \cos (c+d x)}\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d \sqrt {\cos (c+d x)}}\\ &=\frac {b \sqrt {b \cos (c+d x)} \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 32, normalized size = 0.97 \[ \frac {\sin (c+d x) (b \cos (c+d x))^{3/2}}{d \cos ^{\frac {5}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(7/2),x]

[Out]

((b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(d*Cos[c + d*x]^(5/2))

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fricas [A]  time = 0.79, size = 29, normalized size = 0.88 \[ \frac {\sqrt {b \cos \left (d x + c\right )} b \sin \left (d x + c\right )}{d \cos \left (d x + c\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*cos(d*x+c))^(3/2)/cos(d*x+c)^(7/2),x, algorithm="fricas")

[Out]

sqrt(b*cos(d*x + c))*b*sin(d*x + c)/(d*cos(d*x + c)^(3/2))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \cos \left (d x + c\right )\right )^{\frac {3}{2}}}{\cos \left (d x + c\right )^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*cos(d*x+c))^(3/2)/cos(d*x+c)^(7/2),x, algorithm="giac")

[Out]

integrate((b*cos(d*x + c))^(3/2)/cos(d*x + c)^(7/2), x)

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maple [A]  time = 0.10, size = 29, normalized size = 0.88 \[ \frac {\left (b \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{d \cos \left (d x +c \right )^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*cos(d*x+c))^(3/2)/cos(d*x+c)^(7/2),x)

[Out]

1/d*(b*cos(d*x+c))^(3/2)*sin(d*x+c)/cos(d*x+c)^(5/2)

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maxima [A]  time = 0.83, size = 54, normalized size = 1.64 \[ \frac {2 \, b^{\frac {3}{2}} \sin \left (2 \, d x + 2 \, c\right )}{{\left (\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*cos(d*x+c))^(3/2)/cos(d*x+c)^(7/2),x, algorithm="maxima")

[Out]

2*b^(3/2)*sin(2*d*x + 2*c)/((cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*d)

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mupad [B]  time = 0.50, size = 60, normalized size = 1.82 \[ \frac {b\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (\cos \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}+\sin \left (2\,c+2\,d\,x\right )+1{}\mathrm {i}\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*cos(c + d*x))^(3/2)/cos(c + d*x)^(7/2),x)

[Out]

(b*(b*cos(c + d*x))^(1/2)*(cos(2*c + 2*d*x)*1i + sin(2*c + 2*d*x) + 1i))/(d*cos(c + d*x)^(1/2)*(cos(2*c + 2*d*
x) + 1))

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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))**(3/2)/cos(d*x+c)**(7/2),x)

[Out]

Timed out

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