3.483 \(\int \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2} \, dx\)

Optimal. Leaf size=154 \[ \frac{4 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{9/2}}{9 b^5 d}-\frac{8 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{7/2}}{7 b^5 d}+\frac{2 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{5/2}}{5 b^5 d}+\frac{2 (a+b \sin (c+d x))^{13/2}}{13 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{11/2}}{11 b^5 d} \]

[Out]

(2*(a^2 - b^2)^2*(a + b*Sin[c + d*x])^(5/2))/(5*b^5*d) - (8*a*(a^2 - b^2)*(a + b*Sin[c + d*x])^(7/2))/(7*b^5*d
) + (4*(3*a^2 - b^2)*(a + b*Sin[c + d*x])^(9/2))/(9*b^5*d) - (8*a*(a + b*Sin[c + d*x])^(11/2))/(11*b^5*d) + (2
*(a + b*Sin[c + d*x])^(13/2))/(13*b^5*d)

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Rubi [A]  time = 0.122571, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2668, 697} \[ \frac{4 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{9/2}}{9 b^5 d}-\frac{8 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{7/2}}{7 b^5 d}+\frac{2 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{5/2}}{5 b^5 d}+\frac{2 (a+b \sin (c+d x))^{13/2}}{13 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{11/2}}{11 b^5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a^2 - b^2)^2*(a + b*Sin[c + d*x])^(5/2))/(5*b^5*d) - (8*a*(a^2 - b^2)*(a + b*Sin[c + d*x])^(7/2))/(7*b^5*d
) + (4*(3*a^2 - b^2)*(a + b*Sin[c + d*x])^(9/2))/(9*b^5*d) - (8*a*(a + b*Sin[c + d*x])^(11/2))/(11*b^5*d) + (2
*(a + b*Sin[c + d*x])^(13/2))/(13*b^5*d)

Rule 2668

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*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^{3/2} \left (b^2-x^2\right )^2 \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\left (a^2-b^2\right )^2 (a+x)^{3/2}-4 \left (a^3-a b^2\right ) (a+x)^{5/2}+2 \left (3 a^2-b^2\right ) (a+x)^{7/2}-4 a (a+x)^{9/2}+(a+x)^{11/2}\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{2 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{5/2}}{5 b^5 d}-\frac{8 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{7/2}}{7 b^5 d}+\frac{4 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{9/2}}{9 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{11/2}}{11 b^5 d}+\frac{2 (a+b \sin (c+d x))^{13/2}}{13 b^5 d}\\ \end{align*}

Mathematica [A]  time = 0.70725, size = 131, normalized size = 0.85 \[ \frac{2 \left (\frac{2}{9} \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{9/2}+\frac{1}{5} \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{5/2}+\frac{1}{13} (a+b \sin (c+d x))^{13/2}-\frac{4}{11} a (a+b \sin (c+d x))^{11/2}-\frac{4}{7} a (a-b) (a+b) (a+b \sin (c+d x))^{7/2}\right )}{b^5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(((a^2 - b^2)^2*(a + b*Sin[c + d*x])^(5/2))/5 - (4*a*(a - b)*(a + b)*(a + b*Sin[c + d*x])^(7/2))/7 + (2*(3*
a^2 - b^2)*(a + b*Sin[c + d*x])^(9/2))/9 - (4*a*(a + b*Sin[c + d*x])^(11/2))/11 + (a + b*Sin[c + d*x])^(13/2)/
13))/(b^5*d)

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Maple [A]  time = 0.462, size = 126, normalized size = 0.8 \begin{align*}{\frac{6930\,{b}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{4}+5040\,a{b}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -3360\,{a}^{2}{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}+6160\,{b}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{2}-1920\,{a}^{3}b\sin \left ( dx+c \right ) +6400\,a{b}^{3}\sin \left ( dx+c \right ) +768\,{a}^{4}-1216\,{a}^{2}{b}^{2}+4928\,{b}^{4}}{45045\,{b}^{5}d} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*(a+b*sin(d*x+c))^(3/2),x)

[Out]

2/45045/b^5*(a+b*sin(d*x+c))^(5/2)*(3465*b^4*cos(d*x+c)^4+2520*a*b^3*cos(d*x+c)^2*sin(d*x+c)-1680*a^2*b^2*cos(
d*x+c)^2+3080*b^4*cos(d*x+c)^2-960*a^3*b*sin(d*x+c)+3200*a*b^3*sin(d*x+c)+384*a^4-608*a^2*b^2+2464*b^4)/d

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Maxima [A]  time = 0.950118, size = 157, normalized size = 1.02 \begin{align*} \frac{2 \,{\left (3465 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{13}{2}} - 16380 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{11}{2}} a + 10010 \,{\left (3 \, a^{2} - b^{2}\right )}{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{9}{2}} - 25740 \,{\left (a^{3} - a b^{2}\right )}{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} + 9009 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )}{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}}\right )}}{45045 \, b^{5} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+b*sin(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

2/45045*(3465*(b*sin(d*x + c) + a)^(13/2) - 16380*(b*sin(d*x + c) + a)^(11/2)*a + 10010*(3*a^2 - b^2)*(b*sin(d
*x + c) + a)^(9/2) - 25740*(a^3 - a*b^2)*(b*sin(d*x + c) + a)^(7/2) + 9009*(a^4 - 2*a^2*b^2 + b^4)*(b*sin(d*x
+ c) + a)^(5/2))/(b^5*d)

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Fricas [A]  time = 3.30025, size = 462, normalized size = 3. \begin{align*} -\frac{2 \,{\left (3465 \, b^{6} \cos \left (d x + c\right )^{6} - 384 \, a^{6} + 2144 \, a^{4} b^{2} - 8256 \, a^{2} b^{4} - 2464 \, b^{6} - 35 \,{\left (3 \, a^{2} b^{4} + 11 \, b^{6}\right )} \cos \left (d x + c\right )^{4} + 8 \,{\left (18 \, a^{4} b^{2} - 81 \, a^{2} b^{4} - 77 \, b^{6}\right )} \cos \left (d x + c\right )^{2} - 2 \,{\left (2205 \, a b^{5} \cos \left (d x + c\right )^{4} - 96 \, a^{5} b + 512 \, a^{3} b^{3} + 4064 \, a b^{5} + 20 \,{\left (3 \, a^{3} b^{3} + 137 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt{b \sin \left (d x + c\right ) + a}}{45045 \, b^{5} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+b*sin(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

-2/45045*(3465*b^6*cos(d*x + c)^6 - 384*a^6 + 2144*a^4*b^2 - 8256*a^2*b^4 - 2464*b^6 - 35*(3*a^2*b^4 + 11*b^6)
*cos(d*x + c)^4 + 8*(18*a^4*b^2 - 81*a^2*b^4 - 77*b^6)*cos(d*x + c)^2 - 2*(2205*a*b^5*cos(d*x + c)^4 - 96*a^5*
b + 512*a^3*b^3 + 4064*a*b^5 + 20*(3*a^3*b^3 + 137*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(b*sin(d*x + c) +
a)/(b^5*d)

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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(cos(d*x+c)**5*(a+b*sin(d*x+c))**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \cos \left (d x + c\right )^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+b*sin(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate((b*sin(d*x + c) + a)^(3/2)*cos(d*x + c)^5, x)