3.49 \(\int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{7/2} \, dx\)

Optimal. Leaf size=242 \[ -\frac{10 a e^3 \left (11 a^2+6 b^2\right ) \cos (c+d x) \sqrt{e \sin (c+d x)}}{231 d}+\frac{10 a e^4 \left (11 a^2+6 b^2\right ) \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{231 d \sqrt{e \sin (c+d x)}}+\frac{2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{1287 d e}-\frac{2 a e \left (11 a^2+6 b^2\right ) \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac{2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))^2}{13 d e}+\frac{34 a b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{143 d e} \]

[Out]

(10*a*(11*a^2 + 6*b^2)*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(231*d*Sqrt[e*Sin[c + d*x]]) -
(10*a*(11*a^2 + 6*b^2)*e^3*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(231*d) - (2*a*(11*a^2 + 6*b^2)*e*Cos[c + d*x]*
(e*Sin[c + d*x])^(5/2))/(77*d) + (2*b*(177*a^2 + 44*b^2)*(e*Sin[c + d*x])^(9/2))/(1287*d*e) + (34*a*b*(a + b*C
os[c + d*x])*(e*Sin[c + d*x])^(9/2))/(143*d*e) + (2*b*(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(9/2))/(13*d*e)

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Rubi [A]  time = 0.320394, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {2692, 2862, 2669, 2635, 2642, 2641} \[ -\frac{10 a e^3 \left (11 a^2+6 b^2\right ) \cos (c+d x) \sqrt{e \sin (c+d x)}}{231 d}+\frac{10 a e^4 \left (11 a^2+6 b^2\right ) \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{231 d \sqrt{e \sin (c+d x)}}+\frac{2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{1287 d e}-\frac{2 a e \left (11 a^2+6 b^2\right ) \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac{2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))^2}{13 d e}+\frac{34 a b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{143 d e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*a*(11*a^2 + 6*b^2)*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(231*d*Sqrt[e*Sin[c + d*x]]) -
(10*a*(11*a^2 + 6*b^2)*e^3*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(231*d) - (2*a*(11*a^2 + 6*b^2)*e*Cos[c + d*x]*
(e*Sin[c + d*x])^(5/2))/(77*d) + (2*b*(177*a^2 + 44*b^2)*(e*Sin[c + d*x])^(9/2))/(1287*d*e) + (34*a*b*(a + b*C
os[c + d*x])*(e*Sin[c + d*x])^(9/2))/(143*d*e) + (2*b*(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(9/2))/(13*d*e)

Rule 2692

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

Rule 2862

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> -Simp[(d*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(f*g*(m + p + 1)), x]
+ Dist[1/(m + p + 1), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*Simp[a*c*(m + p + 1) + b*d*m + (a*d*
m + b*c*(m + p + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && Gt
Q[m, 0] &&  !LtQ[p, -1] && IntegerQ[2*m] &&  !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && SimplerQ[c + d*x, a + b*x])

Rule 2669

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b*(g*Cos[
e + f*x])^(p + 1))/(f*g*(p + 1)), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x]
&& (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

Rule 2635

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

Rule 2642

Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rule 2641

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

Rubi steps

\begin{align*} \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{7/2} \, dx &=\frac{2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{9/2}}{13 d e}+\frac{2}{13} \int (a+b \cos (c+d x)) \left (\frac{13 a^2}{2}+2 b^2+\frac{17}{2} a b \cos (c+d x)\right ) (e \sin (c+d x))^{7/2} \, dx\\ &=\frac{34 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{143 d e}+\frac{2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{9/2}}{13 d e}+\frac{4}{143} \int \left (\frac{13}{4} a \left (11 a^2+6 b^2\right )+\frac{1}{4} b \left (177 a^2+44 b^2\right ) \cos (c+d x)\right ) (e \sin (c+d x))^{7/2} \, dx\\ &=\frac{2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{1287 d e}+\frac{34 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{143 d e}+\frac{2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{9/2}}{13 d e}+\frac{1}{11} \left (a \left (11 a^2+6 b^2\right )\right ) \int (e \sin (c+d x))^{7/2} \, dx\\ &=-\frac{2 a \left (11 a^2+6 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac{2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{1287 d e}+\frac{34 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{143 d e}+\frac{2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{9/2}}{13 d e}+\frac{1}{77} \left (5 a \left (11 a^2+6 b^2\right ) e^2\right ) \int (e \sin (c+d x))^{3/2} \, dx\\ &=-\frac{10 a \left (11 a^2+6 b^2\right ) e^3 \cos (c+d x) \sqrt{e \sin (c+d x)}}{231 d}-\frac{2 a \left (11 a^2+6 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac{2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{1287 d e}+\frac{34 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{143 d e}+\frac{2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{9/2}}{13 d e}+\frac{1}{231} \left (5 a \left (11 a^2+6 b^2\right ) e^4\right ) \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx\\ &=-\frac{10 a \left (11 a^2+6 b^2\right ) e^3 \cos (c+d x) \sqrt{e \sin (c+d x)}}{231 d}-\frac{2 a \left (11 a^2+6 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac{2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{1287 d e}+\frac{34 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{143 d e}+\frac{2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{9/2}}{13 d e}+\frac{\left (5 a \left (11 a^2+6 b^2\right ) e^4 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{231 \sqrt{e \sin (c+d x)}}\\ &=\frac{10 a \left (11 a^2+6 b^2\right ) e^4 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{231 d \sqrt{e \sin (c+d x)}}-\frac{10 a \left (11 a^2+6 b^2\right ) e^3 \cos (c+d x) \sqrt{e \sin (c+d x)}}{231 d}-\frac{2 a \left (11 a^2+6 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac{2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{1287 d e}+\frac{34 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{143 d e}+\frac{2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{9/2}}{13 d e}\\ \end{align*}

Mathematica [A]  time = 2.50023, size = 205, normalized size = 0.85 \[ \frac{(e \sin (c+d x))^{7/2} \left (154 b \left (78 a^2+11 b^2\right ) \csc ^3(c+d x)+\frac{1}{3} \csc ^3(c+d x) \left (-77 b \left (624 a^2+73 b^2\right ) \cos (2 (c+d x))-154 b \left (b^2-78 a^2\right ) \cos (4 (c+d x))-156 a \left (506 a^2+213 b^2\right ) \cos (c+d x)+234 a \left (44 a^2-39 b^2\right ) \cos (3 (c+d x))+4914 a b^2 \cos (5 (c+d x))+693 b^3 \cos (6 (c+d x))\right )-\frac{2080 a \left (11 a^2+6 b^2\right ) F\left (\left .\frac{1}{4} (-2 c-2 d x+\pi )\right |2\right )}{\sin ^{\frac{7}{2}}(c+d x)}\right )}{48048 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((154*b*(78*a^2 + 11*b^2)*Csc[c + d*x]^3 + ((-156*a*(506*a^2 + 213*b^2)*Cos[c + d*x] - 77*b*(624*a^2 + 73*b^2)
*Cos[2*(c + d*x)] + 234*a*(44*a^2 - 39*b^2)*Cos[3*(c + d*x)] - 154*b*(-78*a^2 + b^2)*Cos[4*(c + d*x)] + 4914*a
*b^2*Cos[5*(c + d*x)] + 693*b^3*Cos[6*(c + d*x)])*Csc[c + d*x]^3)/3 - (2080*a*(11*a^2 + 6*b^2)*EllipticF[(-2*c
+ Pi - 2*d*x)/4, 2])/Sin[c + d*x]^(7/2))*(e*Sin[c + d*x])^(7/2))/(48048*d)

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Maple [A]  time = 2.711, size = 252, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({\frac{2\,b \left ( 9\,{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}+39\,{a}^{2}+4\,{b}^{2} \right ) }{117\,e} \left ( e\sin \left ( dx+c \right ) \right ) ^{{\frac{9}{2}}}}-{\frac{{e}^{4}a}{231\,\cos \left ( dx+c \right ) } \left ( -126\,{b}^{2}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{6}+ \left ( -66\,{a}^{2}+216\,{b}^{2} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}\sin \left ( dx+c \right ) + \left ( 176\,{a}^{2}-30\,{b}^{2} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) +55\,\sqrt{1-\sin \left ( dx+c \right ) }\sqrt{2+2\,\sin \left ( dx+c \right ) }\sqrt{\sin \left ( dx+c \right ) }{\it EllipticF} \left ( \sqrt{1-\sin \left ( dx+c \right ) },1/2\,\sqrt{2} \right ){a}^{2}+30\,\sqrt{1-\sin \left ( dx+c \right ) }\sqrt{2+2\,\sin \left ( dx+c \right ) }\sqrt{\sin \left ( dx+c \right ) }{\it EllipticF} \left ( \sqrt{1-\sin \left ( dx+c \right ) },1/2\,\sqrt{2} \right ){b}^{2} \right ){\frac{1}{\sqrt{e\sin \left ( dx+c \right ) }}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2/117/e*b*(e*sin(d*x+c))^(9/2)*(9*b^2*cos(d*x+c)^2+39*a^2+4*b^2)-1/231*e^4*a*(-126*b^2*sin(d*x+c)*cos(d*x+c)^
6+(-66*a^2+216*b^2)*cos(d*x+c)^4*sin(d*x+c)+(176*a^2-30*b^2)*cos(d*x+c)^2*sin(d*x+c)+55*(1-sin(d*x+c))^(1/2)*(
2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)*EllipticF((1-sin(d*x+c))^(1/2),1/2*2^(1/2))*a^2+30*(1-sin(d*x+c))^(1/2)
*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)*EllipticF((1-sin(d*x+c))^(1/2),1/2*2^(1/2))*b^2)/cos(d*x+c)/(e*sin(d*
x+c))^(1/2))/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \left (e \sin \left (d x + c\right )\right )^{\frac{7}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (b^{3} e^{3} \cos \left (d x + c\right )^{5} + 3 \, a b^{2} e^{3} \cos \left (d x + c\right )^{4} - 3 \, a^{2} b e^{3} \cos \left (d x + c\right ) +{\left (3 \, a^{2} b - b^{3}\right )} e^{3} \cos \left (d x + c\right )^{3} - a^{3} e^{3} +{\left (a^{3} - 3 \, a b^{2}\right )} e^{3} \cos \left (d x + c\right )^{2}\right )} \sqrt{e \sin \left (d x + c\right )} \sin \left (d x + c\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(b^3*e^3*cos(d*x + c)^5 + 3*a*b^2*e^3*cos(d*x + c)^4 - 3*a^2*b*e^3*cos(d*x + c) + (3*a^2*b - b^3)*e^
3*cos(d*x + c)^3 - a^3*e^3 + (a^3 - 3*a*b^2)*e^3*cos(d*x + c)^2)*sqrt(e*sin(d*x + c))*sin(d*x + c), x)

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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((a+b*cos(d*x+c))**3*(e*sin(d*x+c))**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \left (e \sin \left (d x + c\right )\right )^{\frac{7}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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