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

Optimal. Leaf size=124 \[ \frac{10 a e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 d}+\frac{10 a e^4 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d \sqrt{e \cos (c+d x)}}+\frac{2 a e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}-\frac{2 b (e \cos (c+d x))^{9/2}}{9 d e} \]

[Out]

(-2*b*(e*Cos[c + d*x])^(9/2))/(9*d*e) + (10*a*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*d*Sqrt[e*C
os[c + d*x]]) + (10*a*e^3*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*e*(e*Cos[c + d*x])^(5/2)*Sin[c + d*
x])/(7*d)

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Rubi [A]  time = 0.0921037, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2669, 2635, 2642, 2641} \[ \frac{10 a e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 d}+\frac{10 a e^4 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d \sqrt{e \cos (c+d x)}}+\frac{2 a e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}-\frac{2 b (e \cos (c+d x))^{9/2}}{9 d e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*(e*Cos[c + d*x])^(9/2))/(9*d*e) + (10*a*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*d*Sqrt[e*C
os[c + d*x]]) + (10*a*e^3*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*e*(e*Cos[c + d*x])^(5/2)*Sin[c + d*
x])/(7*d)

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)) \, dx &=-\frac{2 b (e \cos (c+d x))^{9/2}}{9 d e}+a \int (e \cos (c+d x))^{7/2} \, dx\\ &=-\frac{2 b (e \cos (c+d x))^{9/2}}{9 d e}+\frac{2 a e (e \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{1}{7} \left (5 a e^2\right ) \int (e \cos (c+d x))^{3/2} \, dx\\ &=-\frac{2 b (e \cos (c+d x))^{9/2}}{9 d e}+\frac{10 a e^3 \sqrt{e \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a e (e \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{1}{21} \left (5 a e^4\right ) \int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx\\ &=-\frac{2 b (e \cos (c+d x))^{9/2}}{9 d e}+\frac{10 a e^3 \sqrt{e \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a e (e \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{\left (5 a e^4 \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{21 \sqrt{e \cos (c+d x)}}\\ &=-\frac{2 b (e \cos (c+d x))^{9/2}}{9 d e}+\frac{10 a e^4 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d \sqrt{e \cos (c+d x)}}+\frac{10 a e^3 \sqrt{e \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a e (e \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}\\ \end{align*}

Mathematica [A]  time = 0.824756, size = 104, normalized size = 0.84 \[ \frac{e^3 \sqrt{e \cos (c+d x)} \left (\sqrt{\cos (c+d x)} (138 a \sin (c+d x)+18 a \sin (3 (c+d x))-28 b \cos (2 (c+d x))-7 b \cos (4 (c+d x))-21 b)+120 a F\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{252 d \sqrt{\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e^3*Sqrt[e*Cos[c + d*x]]*(120*a*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(-21*b - 28*b*Cos[2*(c + d*x)]
- 7*b*Cos[4*(c + d*x)] + 138*a*Sin[c + d*x] + 18*a*Sin[3*(c + d*x)])))/(252*d*Sqrt[Cos[c + d*x]])

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Maple [A]  time = 0.99, size = 259, normalized size = 2.1 \begin{align*} -{\frac{2\,{e}^{4}}{63\,d} \left ( -224\,b \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{11}+144\,a\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}+560\,b \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{9}-216\,a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) -560\,b \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{7}+168\,a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) +280\,b \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}+15\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) a-48\,a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) -70\,b \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}+7\,b\sin \left ( 1/2\,dx+c/2 \right ) \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}e+e}}}} \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)),x)

[Out]

-2/63/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^4*(-224*b*sin(1/2*d*x+1/2*c)^11+144*a*cos(1/2*d
*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+560*b*sin(1/2*d*x+1/2*c)^9-216*a*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-560*b*
sin(1/2*d*x+1/2*c)^7+168*a*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+280*b*sin(1/2*d*x+1/2*c)^5+15*(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-48*a*sin(1/2*d*x+1/2*
c)^2*cos(1/2*d*x+1/2*c)-70*b*sin(1/2*d*x+1/2*c)^3+7*b*sin(1/2*d*x+1/2*c))/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 )}\,{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)),x, algorithm="maxima")

[Out]

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

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

[Out]

integral((b*e^3*cos(d*x + c)^3*sin(d*x + c) + a*e^3*cos(d*x + c)^3)*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)),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 )}\,{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)),x, algorithm="giac")

[Out]

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