3.276 \(\int (d \cos (a+b x))^{9/2} (c \sin (a+b x))^{5/2} \, dx\)

Optimal. Leaf size=166 \[ \frac {3 c^2 d^4 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}{40 b \sqrt {\sin (2 a+2 b x)}}+\frac {c d^3 (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{20 b}-\frac {c (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{11/2}}{7 b d}+\frac {3 c d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{7/2}}{70 b} \]

[Out]

1/20*c*d^3*(d*cos(b*x+a))^(3/2)*(c*sin(b*x+a))^(3/2)/b+3/70*c*d*(d*cos(b*x+a))^(7/2)*(c*sin(b*x+a))^(3/2)/b-1/
7*c*(d*cos(b*x+a))^(11/2)*(c*sin(b*x+a))^(3/2)/b/d-3/40*c^2*d^4*(sin(a+1/4*Pi+b*x)^2)^(1/2)/sin(a+1/4*Pi+b*x)*
EllipticE(cos(a+1/4*Pi+b*x),2^(1/2))*(d*cos(b*x+a))^(1/2)*(c*sin(b*x+a))^(1/2)/b/sin(2*b*x+2*a)^(1/2)

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Rubi [A]  time = 0.24, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2568, 2569, 2572, 2639} \[ \frac {3 c^2 d^4 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}{40 b \sqrt {\sin (2 a+2 b x)}}+\frac {c d^3 (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{20 b}-\frac {c (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{11/2}}{7 b d}+\frac {3 c d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{7/2}}{70 b} \]

Antiderivative was successfully verified.

[In]

Int[(d*Cos[a + b*x])^(9/2)*(c*Sin[a + b*x])^(5/2),x]

[Out]

(c*d^3*(d*Cos[a + b*x])^(3/2)*(c*Sin[a + b*x])^(3/2))/(20*b) + (3*c*d*(d*Cos[a + b*x])^(7/2)*(c*Sin[a + b*x])^
(3/2))/(70*b) - (c*(d*Cos[a + b*x])^(11/2)*(c*Sin[a + b*x])^(3/2))/(7*b*d) + (3*c^2*d^4*Sqrt[d*Cos[a + b*x]]*E
llipticE[a - Pi/4 + b*x, 2]*Sqrt[c*Sin[a + b*x]])/(40*b*Sqrt[Sin[2*a + 2*b*x]])

Rule 2568

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(a*(b*Cos[e
+ f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Cos[e + f*x])
^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*
m, 2*n]

Rule 2569

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*(b*Sin[e +
f*x])^(n + 1)*(a*Cos[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Sin[e + f*x])^
n*(a*Cos[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m
, 2*n]

Rule 2572

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(Sqrt[a*Sin[e +
f*x]]*Sqrt[b*Cos[e + f*x]])/Sqrt[Sin[2*e + 2*f*x]], Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f},
x]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

\begin {align*} \int (d \cos (a+b x))^{9/2} (c \sin (a+b x))^{5/2} \, dx &=-\frac {c (d \cos (a+b x))^{11/2} (c \sin (a+b x))^{3/2}}{7 b d}+\frac {1}{14} \left (3 c^2\right ) \int (d \cos (a+b x))^{9/2} \sqrt {c \sin (a+b x)} \, dx\\ &=\frac {3 c d (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{70 b}-\frac {c (d \cos (a+b x))^{11/2} (c \sin (a+b x))^{3/2}}{7 b d}+\frac {1}{20} \left (3 c^2 d^2\right ) \int (d \cos (a+b x))^{5/2} \sqrt {c \sin (a+b x)} \, dx\\ &=\frac {c d^3 (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2}}{20 b}+\frac {3 c d (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{70 b}-\frac {c (d \cos (a+b x))^{11/2} (c \sin (a+b x))^{3/2}}{7 b d}+\frac {1}{40} \left (3 c^2 d^4\right ) \int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)} \, dx\\ &=\frac {c d^3 (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2}}{20 b}+\frac {3 c d (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{70 b}-\frac {c (d \cos (a+b x))^{11/2} (c \sin (a+b x))^{3/2}}{7 b d}+\frac {\left (3 c^2 d^4 \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}\right ) \int \sqrt {\sin (2 a+2 b x)} \, dx}{40 \sqrt {\sin (2 a+2 b x)}}\\ &=\frac {c d^3 (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2}}{20 b}+\frac {3 c d (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{70 b}-\frac {c (d \cos (a+b x))^{11/2} (c \sin (a+b x))^{3/2}}{7 b d}+\frac {3 c^2 d^4 \sqrt {d \cos (a+b x)} E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {c \sin (a+b x)}}{40 b \sqrt {\sin (2 a+2 b x)}}\\ \end {align*}

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Mathematica [C]  time = 0.17, size = 72, normalized size = 0.43 \[ \frac {2 \sqrt [4]{\cos ^2(a+b x)} \sec ^5(a+b x) (c \sin (a+b x))^{7/2} (d \cos (a+b x))^{9/2} \, _2F_1\left (-\frac {7}{4},\frac {7}{4};\frac {11}{4};\sin ^2(a+b x)\right )}{7 b c} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Cos[a + b*x])^(9/2)*(c*Sin[a + b*x])^(5/2),x]

[Out]

(2*(d*Cos[a + b*x])^(9/2)*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[-7/4, 7/4, 11/4, Sin[a + b*x]^2]*Sec[a + b*
x]^5*(c*Sin[a + b*x])^(7/2))/(7*b*c)

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fricas [F]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (c^{2} d^{4} \cos \left (b x + a\right )^{6} - c^{2} d^{4} \cos \left (b x + a\right )^{4}\right )} \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^(9/2)*(c*sin(b*x+a))^(5/2),x, algorithm="fricas")

[Out]

integral(-(c^2*d^4*cos(b*x + a)^6 - c^2*d^4*cos(b*x + a)^4)*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a)), x)

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giac [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((d*cos(b*x+a))^(9/2)*(c*sin(b*x+a))^(5/2),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.16, size = 545, normalized size = 3.28 \[ \frac {\left (40 \left (\cos ^{8}\left (b x +a \right )\right ) \sqrt {2}-52 \left (\cos ^{6}\left (b x +a \right )\right ) \sqrt {2}-2 \left (\cos ^{4}\left (b x +a \right )\right ) \sqrt {2}+21 \cos \left (b x +a \right ) \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )-42 \cos \left (b x +a \right ) \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticE \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )+21 \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )-42 \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticE \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )-7 \left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {2}+21 \cos \left (b x +a \right ) \sqrt {2}\right ) \left (d \cos \left (b x +a \right )\right )^{\frac {9}{2}} \left (c \sin \left (b x +a \right )\right )^{\frac {5}{2}} \sqrt {2}}{560 b \sin \left (b x +a \right )^{3} \cos \left (b x +a \right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cos(b*x+a))^(9/2)*(c*sin(b*x+a))^(5/2),x)

[Out]

1/560/b*(40*cos(b*x+a)^8*2^(1/2)-52*cos(b*x+a)^6*2^(1/2)-2*cos(b*x+a)^4*2^(1/2)+21*cos(b*x+a)*((1-cos(b*x+a)+s
in(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*
EllipticF(((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))-42*cos(b*x+a)*((1-cos(b*x+a)+sin(b*x+a))/s
in(b*x+a))^(1/2)*((-1+cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticE(((
1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))+21*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+c
os(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticF(((1-cos(b*x+a)+sin(b*x+a)
)/sin(b*x+a))^(1/2),1/2*2^(1/2))-42*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)+sin(b*x+a))/s
in(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticE(((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2),1/2
*2^(1/2))-7*cos(b*x+a)^2*2^(1/2)+21*cos(b*x+a)*2^(1/2))*(d*cos(b*x+a))^(9/2)*(c*sin(b*x+a))^(5/2)/sin(b*x+a)^3
/cos(b*x+a)^5*2^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}} \left (c \sin \left (b x + a\right )\right )^{\frac {5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^(9/2)*(c*sin(b*x+a))^(5/2),x, algorithm="maxima")

[Out]

integrate((d*cos(b*x + a))^(9/2)*(c*sin(b*x + a))^(5/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d\,\cos \left (a+b\,x\right )\right )}^{9/2}\,{\left (c\,\sin \left (a+b\,x\right )\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cos(a + b*x))^(9/2)*(c*sin(a + b*x))^(5/2),x)

[Out]

int((d*cos(a + b*x))^(9/2)*(c*sin(a + b*x))^(5/2), 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((d*cos(b*x+a))**(9/2)*(c*sin(b*x+a))**(5/2),x)

[Out]

Timed out

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