\(\int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \, dx\) [548]

Optimal result

Integrand size = 25, antiderivative size = 149 \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \, dx=-\frac {22 a b (e \cos (c+d x))^{7/2}}{63 d e}+\frac {2 \left (9 a^2+2 b^2\right ) e^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)}}+\frac {2 \left (9 a^2+2 b^2\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e} \]

[Out]

-22/63*a*b*(e*cos(d*x+c))^(7/2)/d/e+2/45*(9*a^2+2*b^2)*e*(e*cos(d*x+c))^(3/2)*sin(d*x+c)/d-2/9*b*(e*cos(d*x+c)
)^(7/2)*(a+b*sin(d*x+c))/d/e+2/15*(9*a^2+2*b^2)*e^2*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(
sin(1/2*d*x+1/2*c),2^(1/2))*(e*cos(d*x+c))^(1/2)/d/cos(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2771, 2748, 2715, 2721, 2719} \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \, dx=\frac {2 e^2 \left (9 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{15 d \sqrt {\cos (c+d x)}}+\frac {2 e \left (9 a^2+2 b^2\right ) \sin (c+d x) (e \cos (c+d x))^{3/2}}{45 d}-\frac {22 a b (e \cos (c+d x))^{7/2}}{63 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e} \]

[In]

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

[Out]

(-22*a*b*(e*Cos[c + d*x])^(7/2))/(63*d*e) + (2*(9*a^2 + 2*b^2)*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2,
2])/(15*d*Sqrt[Cos[c + d*x]]) + (2*(9*a^2 + 2*b^2)*e*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45*d) - (2*b*(e*Co
s[c + d*x])^(7/2)*(a + b*Sin[c + d*x]))/(9*d*e)

Rule 2715

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] && Integ
erQ[2*n]

Rule 2719

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

Rule 2721

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

Rule 2748

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b)*((g*Co
s[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 2771

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])

Rubi steps \begin{align*} \text {integral}& = -\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e}+\frac {2}{9} \int (e \cos (c+d x))^{5/2} \left (\frac {9 a^2}{2}+b^2+\frac {11}{2} a b \sin (c+d x)\right ) \, dx \\ & = -\frac {22 a b (e \cos (c+d x))^{7/2}}{63 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e}+\frac {1}{9} \left (9 a^2+2 b^2\right ) \int (e \cos (c+d x))^{5/2} \, dx \\ & = -\frac {22 a b (e \cos (c+d x))^{7/2}}{63 d e}+\frac {2 \left (9 a^2+2 b^2\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e}+\frac {1}{15} \left (\left (9 a^2+2 b^2\right ) e^2\right ) \int \sqrt {e \cos (c+d x)} \, dx \\ & = -\frac {22 a b (e \cos (c+d x))^{7/2}}{63 d e}+\frac {2 \left (9 a^2+2 b^2\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e}+\frac {\left (\left (9 a^2+2 b^2\right ) e^2 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{15 \sqrt {\cos (c+d x)}} \\ & = -\frac {22 a b (e \cos (c+d x))^{7/2}}{63 d e}+\frac {2 \left (9 a^2+2 b^2\right ) e^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)}}+\frac {2 \left (9 a^2+2 b^2\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.97 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.76 \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \, dx=\frac {(e \cos (c+d x))^{5/2} \left (84 \left (9 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\cos ^{\frac {3}{2}}(c+d x) \left (-180 a b \cos (2 (c+d x))+21 \left (12 a^2+b^2\right ) \sin (c+d x)-5 b (36 a+7 b \sin (3 (c+d x)))\right )\right )}{630 d \cos ^{\frac {5}{2}}(c+d x)} \]

[In]

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

[Out]

((e*Cos[c + d*x])^(5/2)*(84*(9*a^2 + 2*b^2)*EllipticE[(c + d*x)/2, 2] + Cos[c + d*x]^(3/2)*(-180*a*b*Cos[2*(c
+ d*x)] + 21*(12*a^2 + b^2)*Sin[c + d*x] - 5*b*(36*a + 7*b*Sin[3*(c + d*x)]))))/(630*d*Cos[c + d*x]^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(407\) vs. \(2(157)=314\).

Time = 13.06 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.74

method	result	size
default	\(-\frac {2 e^{3} \left (1120 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}-2240 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}+1440 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -504 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}+1568 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}-2880 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b +504 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}-448 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}+2160 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -126 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}+42 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}-189 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2}-42 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{2}-720 a b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+90 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a b \right )}{315 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\)	\(408\)
parts	\(-\frac {2 a^{2} \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, e^{3} \left (-8 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{5 \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}+\frac {4 b^{2} \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, e^{3} \left (80 \left (\cos ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-240 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+272 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-144 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+35 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{45 \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}-\frac {4 a b \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}{7 d e}\)	\(462\)

[In]

int((e*cos(d*x+c))^(5/2)*(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

-2/315/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^3*(1120*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^
10*b^2-2240*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8*b^2+1440*sin(1/2*d*x+1/2*c)^9*a*b-504*cos(1/2*d*x+1/2*c)*s
in(1/2*d*x+1/2*c)^6*a^2+1568*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6*b^2-2880*sin(1/2*d*x+1/2*c)^7*a*b+504*cos
(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4*a^2-448*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4*b^2+2160*sin(1/2*d*x+1/2*
c)^5*a*b-126*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*a^2+42*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*b^2-189*(s
in(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^2-42*(sin(
1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*b^2-720*a*b*sin
(1/2*d*x+1/2*c)^3+90*sin(1/2*d*x+1/2*c)*a*b)/d

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.11 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.08 \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \, dx=\frac {21 i \, \sqrt {2} {\left (9 \, a^{2} + 2 \, b^{2}\right )} e^{\frac {5}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} {\left (9 \, a^{2} + 2 \, b^{2}\right )} e^{\frac {5}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (90 \, a b e^{2} \cos \left (d x + c\right )^{3} + 7 \, {\left (5 \, b^{2} e^{2} \cos \left (d x + c\right )^{3} - {\left (9 \, a^{2} + 2 \, b^{2}\right )} e^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{315 \, d} \]

[In]

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

[Out]

1/315*(21*I*sqrt(2)*(9*a^2 + 2*b^2)*e^(5/2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I
*sin(d*x + c))) - 21*I*sqrt(2)*(9*a^2 + 2*b^2)*e^(5/2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d
*x + c) - I*sin(d*x + c))) - 2*(90*a*b*e^2*cos(d*x + c)^3 + 7*(5*b^2*e^2*cos(d*x + c)^3 - (9*a^2 + 2*b^2)*e^2*
cos(d*x + c))*sin(d*x + c))*sqrt(e*cos(d*x + c)))/d

Sympy [F(-1)]

Timed out. \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^2 \,d x \]

[In]

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

[Out]

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