3.159 \(\int \frac{(b \cos (c+d x))^{3/2}}{\cos ^{\frac{13}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=110 \[ \frac{3 b \sin (c+d x) \sqrt{b \cos (c+d x)}}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{b \sin (c+d x) \sqrt{b \cos (c+d x)}}{4 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{3 b \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{8 d \sqrt{\cos (c+d x)}} \]

[Out]

(3*b*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(8*d*Sqrt[Cos[c + d*x]]) + (b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*
x])/(4*d*Cos[c + d*x]^(9/2)) + (3*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d*Cos[c + d*x]^(5/2))

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Rubi [A]  time = 0.0392478, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {17, 3768, 3770} \[ \frac{3 b \sin (c+d x) \sqrt{b \cos (c+d x)}}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{b \sin (c+d x) \sqrt{b \cos (c+d x)}}{4 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{3 b \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{8 d \sqrt{\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[(b*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(13/2),x]

[Out]

(3*b*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(8*d*Sqrt[Cos[c + d*x]]) + (b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*
x])/(4*d*Cos[c + d*x]^(9/2)) + (3*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d*Cos[c + d*x]^(5/2))

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 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{(b \cos (c+d x))^{3/2}}{\cos ^{\frac{13}{2}}(c+d x)} \, dx &=\frac{\left (b \sqrt{b \cos (c+d x)}\right ) \int \sec ^5(c+d x) \, dx}{\sqrt{\cos (c+d x)}}\\ &=\frac{b \sqrt{b \cos (c+d x)} \sin (c+d x)}{4 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{\left (3 b \sqrt{b \cos (c+d x)}\right ) \int \sec ^3(c+d x) \, dx}{4 \sqrt{\cos (c+d x)}}\\ &=\frac{b \sqrt{b \cos (c+d x)} \sin (c+d x)}{4 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{3 b \sqrt{b \cos (c+d x)} \sin (c+d x)}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{\left (3 b \sqrt{b \cos (c+d x)}\right ) \int \sec (c+d x) \, dx}{8 \sqrt{\cos (c+d x)}}\\ &=\frac{3 b \tanh ^{-1}(\sin (c+d x)) \sqrt{b \cos (c+d x)}}{8 d \sqrt{\cos (c+d x)}}+\frac{b \sqrt{b \cos (c+d x)} \sin (c+d x)}{4 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{3 b \sqrt{b \cos (c+d x)} \sin (c+d x)}{8 d \cos ^{\frac{5}{2}}(c+d x)}\\ \end{align*}

Mathematica [A]  time = 0.0887455, size = 67, normalized size = 0.61 \[ \frac{b \sqrt{b \cos (c+d x)} \left (\sin (c+d x) \left (3 \cos ^2(c+d x)+2\right )+3 \cos ^4(c+d x) \tanh ^{-1}(\sin (c+d x))\right )}{8 d \cos ^{\frac{9}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(13/2),x]

[Out]

(b*Sqrt[b*Cos[c + d*x]]*(3*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^4 + (2 + 3*Cos[c + d*x]^2)*Sin[c + d*x]))/(8*d*C
os[c + d*x]^(9/2))

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Maple [A]  time = 0.195, size = 121, normalized size = 1.1 \begin{align*} -{\frac{1}{8\,d} \left ( 3\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}\ln \left ( -{\frac{-1+\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) -3\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}\ln \left ( -{\frac{-1+\cos \left ( dx+c \right ) -\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) -3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -2\,\sin \left ( dx+c \right ) \right ) \left ( b\cos \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}} \left ( \cos \left ( dx+c \right ) \right ) ^{-{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*cos(d*x+c))^(3/2)/cos(d*x+c)^(13/2),x)

[Out]

-1/8/d*(3*cos(d*x+c)^4*ln(-(-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))-3*cos(d*x+c)^4*ln(-(-1+cos(d*x+c)-sin(d*x+c)
)/sin(d*x+c))-3*cos(d*x+c)^2*sin(d*x+c)-2*sin(d*x+c))*(b*cos(d*x+c))^(3/2)/cos(d*x+c)^(11/2)

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Maxima [B]  time = 2.19676, size = 2352, normalized size = 21.38 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*cos(d*x+c))^(3/2)/cos(d*x+c)^(13/2),x, algorithm="maxima")

[Out]

-1/16*(12*(b*sin(8*d*x + 8*c) + 4*b*sin(6*d*x + 6*c) + 6*b*sin(4*d*x + 4*c) + 4*b*sin(2*d*x + 2*c))*cos(7/2*ar
ctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 44*(b*sin(8*d*x + 8*c) + 4*b*sin(6*d*x + 6*c) + 6*b*sin(4*d*x + 4
*c) + 4*b*sin(2*d*x + 2*c))*cos(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 44*(b*sin(8*d*x + 8*c) + 4*
b*sin(6*d*x + 6*c) + 6*b*sin(4*d*x + 4*c) + 4*b*sin(2*d*x + 2*c))*cos(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x
+ 2*c))) - 12*(b*sin(8*d*x + 8*c) + 4*b*sin(6*d*x + 6*c) + 6*b*sin(4*d*x + 4*c) + 4*b*sin(2*d*x + 2*c))*cos(1/
2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 3*(b*cos(8*d*x + 8*c)^2 + 16*b*cos(6*d*x + 6*c)^2 + 36*b*cos(
4*d*x + 4*c)^2 + 16*b*cos(2*d*x + 2*c)^2 + b*sin(8*d*x + 8*c)^2 + 16*b*sin(6*d*x + 6*c)^2 + 36*b*sin(4*d*x + 4
*c)^2 + 48*b*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*b*sin(2*d*x + 2*c)^2 + 2*(4*b*cos(6*d*x + 6*c) + 6*b*cos(4
*d*x + 4*c) + 4*b*cos(2*d*x + 2*c) + b)*cos(8*d*x + 8*c) + 8*(6*b*cos(4*d*x + 4*c) + 4*b*cos(2*d*x + 2*c) + b)
*cos(6*d*x + 6*c) + 12*(4*b*cos(2*d*x + 2*c) + b)*cos(4*d*x + 4*c) + 8*b*cos(2*d*x + 2*c) + 4*(2*b*sin(6*d*x +
6*c) + 3*b*sin(4*d*x + 4*c) + 2*b*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*(3*b*sin(4*d*x + 4*c) + 2*b*sin(2*d
*x + 2*c))*sin(6*d*x + 6*c) + b)*log(cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + sin(1/2*arctan2(
sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) + 3*(b*co
s(8*d*x + 8*c)^2 + 16*b*cos(6*d*x + 6*c)^2 + 36*b*cos(4*d*x + 4*c)^2 + 16*b*cos(2*d*x + 2*c)^2 + b*sin(8*d*x +
8*c)^2 + 16*b*sin(6*d*x + 6*c)^2 + 36*b*sin(4*d*x + 4*c)^2 + 48*b*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*b*si
n(2*d*x + 2*c)^2 + 2*(4*b*cos(6*d*x + 6*c) + 6*b*cos(4*d*x + 4*c) + 4*b*cos(2*d*x + 2*c) + b)*cos(8*d*x + 8*c)
+ 8*(6*b*cos(4*d*x + 4*c) + 4*b*cos(2*d*x + 2*c) + b)*cos(6*d*x + 6*c) + 12*(4*b*cos(2*d*x + 2*c) + b)*cos(4*
d*x + 4*c) + 8*b*cos(2*d*x + 2*c) + 4*(2*b*sin(6*d*x + 6*c) + 3*b*sin(4*d*x + 4*c) + 2*b*sin(2*d*x + 2*c))*sin
(8*d*x + 8*c) + 16*(3*b*sin(4*d*x + 4*c) + 2*b*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + b)*log(cos(1/2*arctan2(sin
(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 - 2*sin(1/2*arcta
n2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) - 12*(b*cos(8*d*x + 8*c) + 4*b*cos(6*d*x + 6*c) + 6*b*cos(4*d*x +
4*c) + 4*b*cos(2*d*x + 2*c) + b)*sin(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 44*(b*cos(8*d*x + 8*c
) + 4*b*cos(6*d*x + 6*c) + 6*b*cos(4*d*x + 4*c) + 4*b*cos(2*d*x + 2*c) + b)*sin(5/2*arctan2(sin(2*d*x + 2*c),
cos(2*d*x + 2*c))) + 44*(b*cos(8*d*x + 8*c) + 4*b*cos(6*d*x + 6*c) + 6*b*cos(4*d*x + 4*c) + 4*b*cos(2*d*x + 2*
c) + b)*sin(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 12*(b*cos(8*d*x + 8*c) + 4*b*cos(6*d*x + 6*c) +
6*b*cos(4*d*x + 4*c) + 4*b*cos(2*d*x + 2*c) + b)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*sqrt(b
)/((2*(4*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + 4*cos(2*d*x + 2*c) + 1)*cos(8*d*x + 8*c) + cos(8*d*x + 8*c)^2
+ 8*(6*cos(4*d*x + 4*c) + 4*cos(2*d*x + 2*c) + 1)*cos(6*d*x + 6*c) + 16*cos(6*d*x + 6*c)^2 + 12*(4*cos(2*d*x
+ 2*c) + 1)*cos(4*d*x + 4*c) + 36*cos(4*d*x + 4*c)^2 + 16*cos(2*d*x + 2*c)^2 + 4*(2*sin(6*d*x + 6*c) + 3*sin(4
*d*x + 4*c) + 2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + sin(8*d*x + 8*c)^2 + 16*(3*sin(4*d*x + 4*c) + 2*sin(2*d*x
+ 2*c))*sin(6*d*x + 6*c) + 16*sin(6*d*x + 6*c)^2 + 36*sin(4*d*x + 4*c)^2 + 48*sin(4*d*x + 4*c)*sin(2*d*x + 2*
c) + 16*sin(2*d*x + 2*c)^2 + 8*cos(2*d*x + 2*c) + 1)*d)

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Fricas [A]  time = 2.15479, size = 641, normalized size = 5.83 \begin{align*} \left [\frac{3 \, b^{\frac{3}{2}} \cos \left (d x + c\right )^{5} \log \left (-\frac{b \cos \left (d x + c\right )^{3} - 2 \, \sqrt{b \cos \left (d x + c\right )} \sqrt{b} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 2 \, b \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{3}}\right ) + 2 \,{\left (3 \, b \cos \left (d x + c\right )^{2} + 2 \, b\right )} \sqrt{b \cos \left (d x + c\right )} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{16 \, d \cos \left (d x + c\right )^{5}}, -\frac{3 \, \sqrt{-b} b \arctan \left (\frac{\sqrt{b \cos \left (d x + c\right )} \sqrt{-b} \sin \left (d x + c\right )}{b \sqrt{\cos \left (d x + c\right )}}\right ) \cos \left (d x + c\right )^{5} -{\left (3 \, b \cos \left (d x + c\right )^{2} + 2 \, b\right )} \sqrt{b \cos \left (d x + c\right )} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{8 \, d \cos \left (d x + c\right )^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*cos(d*x+c))^(3/2)/cos(d*x+c)^(13/2),x, algorithm="fricas")

[Out]

[1/16*(3*b^(3/2)*cos(d*x + c)^5*log(-(b*cos(d*x + c)^3 - 2*sqrt(b*cos(d*x + c))*sqrt(b)*sqrt(cos(d*x + c))*sin
(d*x + c) - 2*b*cos(d*x + c))/cos(d*x + c)^3) + 2*(3*b*cos(d*x + c)^2 + 2*b)*sqrt(b*cos(d*x + c))*sqrt(cos(d*x
+ c))*sin(d*x + c))/(d*cos(d*x + c)^5), -1/8*(3*sqrt(-b)*b*arctan(sqrt(b*cos(d*x + c))*sqrt(-b)*sin(d*x + c)/
(b*sqrt(cos(d*x + c))))*cos(d*x + c)^5 - (3*b*cos(d*x + c)^2 + 2*b)*sqrt(b*cos(d*x + c))*sqrt(cos(d*x + c))*si
n(d*x + c))/(d*cos(d*x + c)^5)]

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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((b*cos(d*x+c))**(3/2)/cos(d*x+c)**(13/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b \cos \left (d x + c\right )\right )^{\frac{3}{2}}}{\cos \left (d x + c\right )^{\frac{13}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*cos(d*x+c))^(3/2)/cos(d*x+c)^(13/2),x, algorithm="giac")

[Out]

integrate((b*cos(d*x + c))^(3/2)/cos(d*x + c)^(13/2), x)