\(\int \frac {(a+a \sin (c+d x))^4}{\sqrt {e \cos (c+d x)}} \, dx\) [226]

Optimal result

Integrand size = 25, antiderivative size = 178 \[ \int \frac {(a+a \sin (c+d x))^4}{\sqrt {e \cos (c+d x)}} \, dx=-\frac {78 a^4 \sqrt {e \cos (c+d x)}}{7 d e}+\frac {78 a^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{7 d \sqrt {e \cos (c+d x)}}-\frac {2 a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}{7 d e}-\frac {26 \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}{35 d e}-\frac {78 \sqrt {e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}{35 d e} \]

[Out]

78/7*a^4*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2
)/d/(e*cos(d*x+c))^(1/2)-78/7*a^4*(e*cos(d*x+c))^(1/2)/d/e-2/7*a*(a+a*sin(d*x+c))^3*(e*cos(d*x+c))^(1/2)/d/e-2
6/35*(a^2+a^2*sin(d*x+c))^2*(e*cos(d*x+c))^(1/2)/d/e-78/35*(a^4+a^4*sin(d*x+c))*(e*cos(d*x+c))^(1/2)/d/e

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2757, 2748, 2721, 2720} \[ \int \frac {(a+a \sin (c+d x))^4}{\sqrt {e \cos (c+d x)}} \, dx=-\frac {78 a^4 \sqrt {e \cos (c+d x)}}{7 d e}-\frac {78 \left (a^4 \sin (c+d x)+a^4\right ) \sqrt {e \cos (c+d x)}}{35 d e}+\frac {78 a^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{7 d \sqrt {e \cos (c+d x)}}-\frac {26 \left (a^2 \sin (c+d x)+a^2\right )^2 \sqrt {e \cos (c+d x)}}{35 d e}-\frac {2 a (a \sin (c+d x)+a)^3 \sqrt {e \cos (c+d x)}}{7 d e} \]

[In]

Int[(a + a*Sin[c + d*x])^4/Sqrt[e*Cos[c + d*x]],x]

[Out]

(-78*a^4*Sqrt[e*Cos[c + d*x]])/(7*d*e) + (78*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(7*d*Sqrt[e*Cos
[c + d*x]]) - (2*a*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3)/(7*d*e) - (26*Sqrt[e*Cos[c + d*x]]*(a^2 + a^2*
Sin[c + d*x])^2)/(35*d*e) - (78*Sqrt[e*Cos[c + d*x]]*(a^4 + a^4*Sin[c + d*x]))/(35*d*e)

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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 2757

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[a*((2*m + p - 1)/(m + p)), Int
[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2,
0] && GtQ[m, 0] && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}{7 d e}+\frac {1}{7} (13 a) \int \frac {(a+a \sin (c+d x))^3}{\sqrt {e \cos (c+d x)}} \, dx \\ & = -\frac {2 a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}{7 d e}-\frac {26 \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}{35 d e}+\frac {1}{35} \left (117 a^2\right ) \int \frac {(a+a \sin (c+d x))^2}{\sqrt {e \cos (c+d x)}} \, dx \\ & = -\frac {2 a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}{7 d e}-\frac {26 \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}{35 d e}-\frac {78 \sqrt {e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}{35 d e}+\frac {1}{7} \left (39 a^3\right ) \int \frac {a+a \sin (c+d x)}{\sqrt {e \cos (c+d x)}} \, dx \\ & = -\frac {78 a^4 \sqrt {e \cos (c+d x)}}{7 d e}-\frac {2 a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}{7 d e}-\frac {26 \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}{35 d e}-\frac {78 \sqrt {e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}{35 d e}+\frac {1}{7} \left (39 a^4\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx \\ & = -\frac {78 a^4 \sqrt {e \cos (c+d x)}}{7 d e}-\frac {2 a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}{7 d e}-\frac {26 \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}{35 d e}-\frac {78 \sqrt {e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}{35 d e}+\frac {\left (39 a^4 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{7 \sqrt {e \cos (c+d x)}} \\ & = -\frac {78 a^4 \sqrt {e \cos (c+d x)}}{7 d e}+\frac {78 a^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{7 d \sqrt {e \cos (c+d x)}}-\frac {2 a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}{7 d e}-\frac {26 \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}{35 d e}-\frac {78 \sqrt {e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}{35 d e} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.36 \[ \int \frac {(a+a \sin (c+d x))^4}{\sqrt {e \cos (c+d x)}} \, dx=-\frac {32 \sqrt [4]{2} a^4 \sqrt {e \cos (c+d x)} \operatorname {Hypergeometric2F1}\left (-\frac {13}{4},\frac {1}{4},\frac {5}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{d e \sqrt [4]{1+\sin (c+d x)}} \]

[In]

Integrate[(a + a*Sin[c + d*x])^4/Sqrt[e*Cos[c + d*x]],x]

[Out]

(-32*2^(1/4)*a^4*Sqrt[e*Cos[c + d*x]]*Hypergeometric2F1[-13/4, 1/4, 5/4, (1 - Sin[c + d*x])/2])/(d*e*(1 + Sin[
c + d*x])^(1/4))

Maple [A] (verified)

Time = 5.70 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.25

method	result	size
default	\(-\frac {2 a^{4} \left (80 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+224 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-280 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-336 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+160 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+195 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-392 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+252 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\)	\(222\)
parts	\(\frac {2 a^{4} \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \operatorname {am}^{-1}\left (\frac {d x}{2}+\frac {c}{2}| \sqrt {2}\right )}{d \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}}-\frac {8 a^{4} \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 )}\, \left (4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\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}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{7 \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 {8 a^{4} \sqrt {e \cos \left (d x +c \right )}}{d e}+\frac {8 a^{4} \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 )}\, \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\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}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{\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 {8 a^{4} \left (\frac {\left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-e^{2} \sqrt {e \cos \left (d x +c \right )}\right )}{d \,e^{3}}\)	\(518\)

[In]

int((a+a*sin(d*x+c))^4/(e*cos(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/35/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*a^4*(80*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8-1
20*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+224*sin(1/2*d*x+1/2*c)^7-280*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c
)-336*sin(1/2*d*x+1/2*c)^5+160*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+195*(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))-392*sin(1/2*d*x+1/2*c)^3+252*sin(1/2*d*x+1/2*
c))/d

Fricas [C] (verification not implemented)

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

Time = 0.12 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.70 \[ \int \frac {(a+a \sin (c+d x))^4}{\sqrt {e \cos (c+d x)}} \, dx=\frac {-195 i \, \sqrt {2} a^{4} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 195 i \, \sqrt {2} a^{4} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (28 \, a^{4} \cos \left (d x + c\right )^{2} - 280 \, a^{4} + 5 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - 17 \, a^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{35 \, d e} \]

[In]

integrate((a+a*sin(d*x+c))^4/(e*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

1/35*(-195*I*sqrt(2)*a^4*sqrt(e)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 195*I*sqrt(2)*a^4
*sqrt(e)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 2*(28*a^4*cos(d*x + c)^2 - 280*a^4 + 5*(a
^4*cos(d*x + c)^2 - 17*a^4)*sin(d*x + c))*sqrt(e*cos(d*x + c)))/(d*e)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+a \sin (c+d x))^4}{\sqrt {e \cos (c+d x)}} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sin(d*x+c))**4/(e*cos(d*x+c))**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(a+a \sin (c+d x))^4}{\sqrt {e \cos (c+d x)}} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}{\sqrt {e \cos \left (d x + c\right )}} \,d x } \]

[In]

integrate((a+a*sin(d*x+c))^4/(e*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate((a*sin(d*x + c) + a)^4/sqrt(e*cos(d*x + c)), x)

Giac [F]

\[ \int \frac {(a+a \sin (c+d x))^4}{\sqrt {e \cos (c+d x)}} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}{\sqrt {e \cos \left (d x + c\right )}} \,d x } \]

[In]

integrate((a+a*sin(d*x+c))^4/(e*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate((a*sin(d*x + c) + a)^4/sqrt(e*cos(d*x + c)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+a \sin (c+d x))^4}{\sqrt {e \cos (c+d x)}} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4}{\sqrt {e\,\cos \left (c+d\,x\right )}} \,d x \]

[In]

int((a + a*sin(c + d*x))^4/(e*cos(c + d*x))^(1/2),x)

[Out]

int((a + a*sin(c + d*x))^4/(e*cos(c + d*x))^(1/2), x)