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

Optimal. Leaf size=237 \[ \frac{10 a e^3 \left (11 a^2+6 b^2\right ) \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}+\frac{10 a e^4 \left (11 a^2+6 b^2\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d \sqrt{e \cos (c+d x)}}-\frac{2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac{2 a e \left (11 a^2+6 b^2\right ) \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}-\frac{2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}-\frac{34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e} \]

[Out]

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

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Rubi [A]  time = 0.307995, antiderivative size = 237, 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} \[ \frac{10 a e^3 \left (11 a^2+6 b^2\right ) \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}+\frac{10 a e^4 \left (11 a^2+6 b^2\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d \sqrt{e \cos (c+d x)}}-\frac{2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac{2 a e \left (11 a^2+6 b^2\right ) \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}-\frac{2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}-\frac{34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 2.01581, size = 205, normalized size = 0.86 \[ \frac{(e \cos (c+d x))^{7/2} \left (-154 b \left (78 a^2+11 b^2\right ) \sqrt{\cos (c+d x)}+2080 \left (11 a^3+6 a b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\frac{1}{3} \sqrt{\cos (c+d x)} \left (156 a \left (506 a^2+213 b^2\right ) \sin (c+d x)+234 a \left (44 a^2-39 b^2\right ) \sin (3 (c+d x))-77 b \left (624 a^2+73 b^2\right ) \cos (2 (c+d x))+154 b \left (b^2-78 a^2\right ) \cos (4 (c+d x))-4914 a b^2 \sin (5 (c+d x))+693 b^3 \cos (6 (c+d x))\right )\right )}{48048 d \cos ^{\frac{7}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 2.566, size = 618, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/9009/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^4*(1170*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1
/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a*b^2+20592*a^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+
1/2*c)^8-308*sin(1/2*d*x+1/2*c)^3*b^3-18172*b^3*sin(1/2*d*x+1/2*c)^5-310464*b^3*sin(1/2*d*x+1/2*c)^13+433664*b
^3*sin(1/2*d*x+1/2*c)^11+88704*b^3*sin(1/2*d*x+1/2*c)^15-308000*b^3*sin(1/2*d*x+1/2*c)^9+113960*b^3*sin(1/2*d*
x+1/2*c)^7+308*b^3*sin(1/2*d*x+1/2*c)-30888*a^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6+24024*a^3*cos(1/2*d*x+
1/2*c)*sin(1/2*d*x+1/2*c)^4-6864*a^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+240240*a^2*b*sin(1/2*d*x+1/2*c)^9
-240240*a^2*b*sin(1/2*d*x+1/2*c)^7+120120*a^2*b*sin(1/2*d*x+1/2*c)^5-30030*a^2*b*sin(1/2*d*x+1/2*c)^3-96096*a^
2*b*sin(1/2*d*x+1/2*c)^11+3003*a^2*b*sin(1/2*d*x+1/2*c)-157248*a*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12+
393120*a*b^2*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)-381888*a*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+179
712*a*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-36036*a*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4+1170*a*b
^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+2145*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*
EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a^3)/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(3*a*b^2*e^3*cos(d*x + c)^5 - (a^3 + 3*a*b^2)*e^3*cos(d*x + c)^3 + (b^3*e^3*cos(d*x + c)^5 - (3*a^2*
b + b^3)*e^3*cos(d*x + c)^3)*sin(d*x + c))*sqrt(e*cos(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((e*cos(d*x+c))**(7/2)*(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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