\(\int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{13/2}} \, dx\) [232]

Optimal result

Integrand size = 25, antiderivative size = 169 \[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{13/2}} \, dx=-\frac {2 a^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{77 d e^6 \sqrt {e \cos (c+d x)}}+\frac {4 a^7 \sqrt {e \cos (c+d x)}}{11 d e^7 (a-a \sin (c+d x))^3}-\frac {2 a^8 \sqrt {e \cos (c+d x)}}{77 d e^7 \left (a^2-a^2 \sin (c+d x)\right )^2}-\frac {2 a^8 \sqrt {e \cos (c+d x)}}{77 d e^7 \left (a^4-a^4 \sin (c+d x)\right )} \]

[Out]

-2/77*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^6/(e*cos(d*x+c))^(1/2)+4/11*a^7*(e*cos(d*x+c))^(1/2)/d/e^7/(a-a*sin(d*x+c))^3-2/77*a^8*(e*cos(d*x+c))^(
1/2)/d/e^7/(a^2-a^2*sin(d*x+c))^2-2/77*a^8*(e*cos(d*x+c))^(1/2)/d/e^7/(a^4-a^4*sin(d*x+c))

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2749, 2759, 2760, 2762, 2721, 2720} \[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{13/2}} \, dx=\frac {4 a^7 \sqrt {e \cos (c+d x)}}{11 d e^7 (a-a \sin (c+d x))^3}-\frac {2 a^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{77 d e^6 \sqrt {e \cos (c+d x)}}-\frac {2 a^8 \sqrt {e \cos (c+d x)}}{77 d e^7 \left (a^4-a^4 \sin (c+d x)\right )}-\frac {2 a^8 \sqrt {e \cos (c+d x)}}{77 d e^7 \left (a^2-a^2 \sin (c+d x)\right )^2} \]

[In]

Int[(a + a*Sin[c + d*x])^4/(e*Cos[c + d*x])^(13/2),x]

[Out]

(-2*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(77*d*e^6*Sqrt[e*Cos[c + d*x]]) + (4*a^7*Sqrt[e*Cos[c +
d*x]])/(11*d*e^7*(a - a*Sin[c + d*x])^3) - (2*a^8*Sqrt[e*Cos[c + d*x]])/(77*d*e^7*(a^2 - a^2*Sin[c + d*x])^2)
- (2*a^8*Sqrt[e*Cos[c + d*x]])/(77*d*e^7*(a^4 - a^4*Sin[c + d*x]))

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 2749

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

Rule 2759

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

Rule 2760

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(2*m + p + 1))), x] + Dist[(m + p + 1)/(a*(2*m + p + 1)),
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] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] && IntegersQ[2*m, 2*p]

Rule 2762

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

Rubi steps \begin{align*} \text {integral}& = \frac {a^8 \int \frac {(e \cos (c+d x))^{3/2}}{(a-a \sin (c+d x))^4} \, dx}{e^8} \\ & = \frac {4 a^7 \sqrt {e \cos (c+d x)}}{11 d e^7 (a-a \sin (c+d x))^3}-\frac {a^6 \int \frac {1}{\sqrt {e \cos (c+d x)} (a-a \sin (c+d x))^2} \, dx}{11 e^6} \\ & = \frac {4 a^7 \sqrt {e \cos (c+d x)}}{11 d e^7 (a-a \sin (c+d x))^3}-\frac {2 a^6 \sqrt {e \cos (c+d x)}}{77 d e^7 (a-a \sin (c+d x))^2}-\frac {\left (3 a^5\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} (a-a \sin (c+d x))} \, dx}{77 e^6} \\ & = \frac {4 a^7 \sqrt {e \cos (c+d x)}}{11 d e^7 (a-a \sin (c+d x))^3}-\frac {2 a^6 \sqrt {e \cos (c+d x)}}{77 d e^7 (a-a \sin (c+d x))^2}-\frac {2 a^5 \sqrt {e \cos (c+d x)}}{77 d e^7 (a-a \sin (c+d x))}-\frac {a^4 \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{77 e^6} \\ & = \frac {4 a^7 \sqrt {e \cos (c+d x)}}{11 d e^7 (a-a \sin (c+d x))^3}-\frac {2 a^6 \sqrt {e \cos (c+d x)}}{77 d e^7 (a-a \sin (c+d x))^2}-\frac {2 a^5 \sqrt {e \cos (c+d x)}}{77 d e^7 (a-a \sin (c+d x))}-\frac {\left (a^4 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{77 e^6 \sqrt {e \cos (c+d x)}} \\ & = -\frac {2 a^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{77 d e^6 \sqrt {e \cos (c+d x)}}+\frac {4 a^7 \sqrt {e \cos (c+d x)}}{11 d e^7 (a-a \sin (c+d x))^3}-\frac {2 a^6 \sqrt {e \cos (c+d x)}}{77 d e^7 (a-a \sin (c+d x))^2}-\frac {2 a^5 \sqrt {e \cos (c+d x)}}{77 d e^7 (a-a \sin (c+d x))} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.14 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.39 \[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{13/2}} \, dx=\frac {4 \sqrt [4]{2} a^4 \operatorname {Hypergeometric2F1}\left (-\frac {11}{4},-\frac {1}{4},-\frac {7}{4},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{11/4}}{11 d e (e \cos (c+d x))^{11/2}} \]

[In]

Integrate[(a + a*Sin[c + d*x])^4/(e*Cos[c + d*x])^(13/2),x]

[Out]

(4*2^(1/4)*a^4*Hypergeometric2F1[-11/4, -1/4, -7/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(11/4))/(11*d*e*(
e*Cos[c + d*x])^(11/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(582\) vs. \(2(177)=354\).

Time = 15.88 (sec) , antiderivative size = 583, normalized size of antiderivative = 3.45

method	result	size
default	\(\frac {2 \left (32 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+32 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-80 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-64 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+80 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+176 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-40 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-144 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+10 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+176 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-78 \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 )-176 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{77 \left (32 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-80 \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+80 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-40 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \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}\, e^{6} d}\)	\(583\)
parts	\(\text {Expression too large to display}\)	\(1058\)

[In]

int((a+a*sin(d*x+c))^4/(e*cos(d*x+c))^(13/2),x,method=_RETURNVERBOSE)

[Out]

2/77/(32*sin(1/2*d*x+1/2*c)^10-80*sin(1/2*d*x+1/2*c)^8+80*sin(1/2*d*x+1/2*c)^6-40*sin(1/2*d*x+1/2*c)^4+10*sin(
1/2*d*x+1/2*c)^2-1)/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)/e^6*(32*(sin(1/2*d*x+1/2*c)^2)^(1/2
)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^10+32*sin(1/2*d*x+
1/2*c)^10*cos(1/2*d*x+1/2*c)-80*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*
d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^8-64*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+80*(sin(1/2*d*x+1/2*c)^2
)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^6+176*sin(1/
2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-40*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin
(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^4-144*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+10*(sin(1/2*d*x+1/
2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^2+176*
sin(1/2*d*x+1/2*c)^5-78*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-(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))-176*sin(1/2*d*x+1/2*c)^3-12*sin(1/2*d*x+1/2*c))*a^4/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 = 253, normalized size of antiderivative = 1.50 \[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{13/2}} \, dx=-\frac {{\left (-3 i \, \sqrt {2} a^{4} \cos \left (d x + c\right )^{2} + 4 i \, \sqrt {2} a^{4} + {\left (i \, \sqrt {2} a^{4} \cos \left (d x + c\right )^{2} - 4 i \, \sqrt {2} a^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + {\left (3 i \, \sqrt {2} a^{4} \cos \left (d x + c\right )^{2} - 4 i \, \sqrt {2} a^{4} + {\left (-i \, \sqrt {2} a^{4} \cos \left (d x + c\right )^{2} + 4 i \, \sqrt {2} a^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (a^{4} \cos \left (d x + c\right )^{2} + 3 \, a^{4} \sin \left (d x + c\right ) + 11 \, a^{4}\right )} \sqrt {e \cos \left (d x + c\right )}}{77 \, {\left (3 \, d e^{7} \cos \left (d x + c\right )^{2} - 4 \, d e^{7} - {\left (d e^{7} \cos \left (d x + c\right )^{2} - 4 \, d e^{7}\right )} \sin \left (d x + c\right )\right )}} \]

[In]

integrate((a+a*sin(d*x+c))^4/(e*cos(d*x+c))^(13/2),x, algorithm="fricas")

[Out]

-1/77*((-3*I*sqrt(2)*a^4*cos(d*x + c)^2 + 4*I*sqrt(2)*a^4 + (I*sqrt(2)*a^4*cos(d*x + c)^2 - 4*I*sqrt(2)*a^4)*s
in(d*x + c))*sqrt(e)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + (3*I*sqrt(2)*a^4*cos(d*x + c)
^2 - 4*I*sqrt(2)*a^4 + (-I*sqrt(2)*a^4*cos(d*x + c)^2 + 4*I*sqrt(2)*a^4)*sin(d*x + c))*sqrt(e)*weierstrassPInv
erse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 2*(a^4*cos(d*x + c)^2 + 3*a^4*sin(d*x + c) + 11*a^4)*sqrt(e*cos(d
*x + c)))/(3*d*e^7*cos(d*x + c)^2 - 4*d*e^7 - (d*e^7*cos(d*x + c)^2 - 4*d*e^7)*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{13/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sin(d*x+c))**4/(e*cos(d*x+c))**(13/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{13/2}} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}{\left (e \cos \left (d x + c\right )\right )^{\frac {13}{2}}} \,d x } \]

[In]

integrate((a+a*sin(d*x+c))^4/(e*cos(d*x+c))^(13/2),x, algorithm="maxima")

[Out]

integrate((a*sin(d*x + c) + a)^4/(e*cos(d*x + c))^(13/2), x)

Giac [F(-1)]

Timed out. \[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{13/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sin(d*x+c))^4/(e*cos(d*x+c))^(13/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{13/2}} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{13/2}} \,d x \]

[In]

int((a + a*sin(c + d*x))^4/(e*cos(c + d*x))^(13/2),x)

[Out]

int((a + a*sin(c + d*x))^4/(e*cos(c + d*x))^(13/2), x)