\(\int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx\) [272]

Optimal result

Integrand size = 25, antiderivative size = 225 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx=-\frac {42 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{221 a^4 d e^2 \sqrt {\cos (c+d x)}}+\frac {42 \sin (c+d x)}{221 a^4 d e \sqrt {e \cos (c+d x)}}-\frac {2}{17 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac {18}{221 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )} \]

[Out]

42/221*sin(d*x+c)/a^4/d/e/(e*cos(d*x+c))^(1/2)-2/17/d/e/(a+a*sin(d*x+c))^4/(e*cos(d*x+c))^(1/2)-18/221/a/d/e/(
a+a*sin(d*x+c))^3/(e*cos(d*x+c))^(1/2)-14/221/d/e/(a^2+a^2*sin(d*x+c))^2/(e*cos(d*x+c))^(1/2)-14/221/d/e/(a^4+
a^4*sin(d*x+c))/(e*cos(d*x+c))^(1/2)-42/221*(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)/a^4/d/e^2/cos(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2760, 2762, 2716, 2721, 2719} \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx=-\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{221 a^4 d e^2 \sqrt {\cos (c+d x)}}+\frac {42 \sin (c+d x)}{221 a^4 d e \sqrt {e \cos (c+d x)}}-\frac {14}{221 d e \left (a^4 \sin (c+d x)+a^4\right ) \sqrt {e \cos (c+d x)}}-\frac {14}{221 d e \left (a^2 \sin (c+d x)+a^2\right )^2 \sqrt {e \cos (c+d x)}}-\frac {18}{221 a d e (a \sin (c+d x)+a)^3 \sqrt {e \cos (c+d x)}}-\frac {2}{17 d e (a \sin (c+d x)+a)^4 \sqrt {e \cos (c+d x)}} \]

[In]

Int[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^4),x]

[Out]

(-42*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(221*a^4*d*e^2*Sqrt[Cos[c + d*x]]) + (42*Sin[c + d*x])/(2
21*a^4*d*e*Sqrt[e*Cos[c + d*x]]) - 2/(17*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^4) - 18/(221*a*d*e*Sqrt
[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3) - 14/(221*d*e*Sqrt[e*Cos[c + d*x]]*(a^2 + a^2*Sin[c + d*x])^2) - 14/(
221*d*e*Sqrt[e*Cos[c + d*x]]*(a^4 + a^4*Sin[c + d*x]))

Rule 2716

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1
))), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[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 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 {2}{17 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4}+\frac {9 \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3} \, dx}{17 a} \\ & = -\frac {2}{17 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac {18}{221 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}+\frac {63 \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2} \, dx}{221 a^2} \\ & = -\frac {2}{17 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac {18}{221 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac {35 \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))} \, dx}{221 a^3} \\ & = -\frac {2}{17 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac {18}{221 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}+\frac {21 \int \frac {1}{(e \cos (c+d x))^{3/2}} \, dx}{221 a^4} \\ & = \frac {42 \sin (c+d x)}{221 a^4 d e \sqrt {e \cos (c+d x)}}-\frac {2}{17 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac {18}{221 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}-\frac {21 \int \sqrt {e \cos (c+d x)} \, dx}{221 a^4 e^2} \\ & = \frac {42 \sin (c+d x)}{221 a^4 d e \sqrt {e \cos (c+d x)}}-\frac {2}{17 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac {18}{221 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}-\frac {\left (21 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{221 a^4 e^2 \sqrt {\cos (c+d x)}} \\ & = -\frac {42 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{221 a^4 d e^2 \sqrt {\cos (c+d x)}}+\frac {42 \sin (c+d x)}{221 a^4 d e \sqrt {e \cos (c+d x)}}-\frac {2}{17 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac {18}{221 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.29 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {21}{4},\frac {3}{4},\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt [4]{1+\sin (c+d x)}}{8 \sqrt [4]{2} a^4 d e \sqrt {e \cos (c+d x)}} \]

[In]

Integrate[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^4),x]

[Out]

(Hypergeometric2F1[-1/4, 21/4, 3/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(1/4))/(8*2^(1/4)*a^4*d*e*Sqrt[e*
Cos[c + d*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(877\) vs. \(2(225)=450\).

Time = 13.88 (sec) , antiderivative size = 878, normalized size of antiderivative = 3.90

method	result	size
default	\(\text {Expression too large to display}\)	\(878\)

[In]

int(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

2/221/(256*sin(1/2*d*x+1/2*c)^16-1024*sin(1/2*d*x+1/2*c)^14+1792*sin(1/2*d*x+1/2*c)^12-1792*sin(1/2*d*x+1/2*c)
^10+1120*sin(1/2*d*x+1/2*c)^8-448*sin(1/2*d*x+1/2*c)^6+112*sin(1/2*d*x+1/2*c)^4-16*sin(1/2*d*x+1/2*c)^2+1)/a^4
/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)/e*(10752*sin(1/2*d*x+1/2*c)^18*cos(1/2*d*x+1/2*c)-5376
*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(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)^16-43008*sin(1/2*d*x+1/2*c)^16*cos(1/2*d*x+1/2*c)+21504*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(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)^14+76160*sin(1/2*d*x+1/2*c)^14*cos(
1/2*d*x+1/2*c)-37632*(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))*(sin(1/2*d*x+1/2*c
)^2)^(1/2)*sin(1/2*d*x+1/2*c)^12-77952*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12+37632*(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))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^10+50560*sin(1/
2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)-23520*EllipticE(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)^(1/2)*sin(1/2*d*x+1/2*c)^8-21376*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+9408*Ellipti
cE(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)^(1/2)*sin(1/2*d*x+1/2*c
)^6+5656*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-2352*(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))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^4-792*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+3
36*(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))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2
*d*x+1/2*c)^2+272*sin(1/2*d*x+1/2*c)^5+242*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-21*(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))-272*sin(1/2*d*x+1/2*c)^3-36*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 = 351, normalized size of antiderivative = 1.56 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx=\frac {21 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{5} + 8 i \, \sqrt {2} \cos \left (d x + c\right )^{3} + 4 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{3} - 2 i \, \sqrt {2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 8 i \, \sqrt {2} \cos \left (d x + c\right )\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{5} - 8 i \, \sqrt {2} \cos \left (d x + c\right )^{3} + 4 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{3} + 2 i \, \sqrt {2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 8 i \, \sqrt {2} \cos \left (d x + c\right )\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (84 \, \cos \left (d x + c\right )^{4} - 224 \, \cos \left (d x + c\right )^{2} + {\left (21 \, \cos \left (d x + c\right )^{4} - 161 \, \cos \left (d x + c\right )^{2} + 117\right )} \sin \left (d x + c\right ) + 104\right )} \sqrt {e \cos \left (d x + c\right )}}{221 \, {\left (a^{4} d e^{2} \cos \left (d x + c\right )^{5} - 8 \, a^{4} d e^{2} \cos \left (d x + c\right )^{3} + 8 \, a^{4} d e^{2} \cos \left (d x + c\right ) - 4 \, {\left (a^{4} d e^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{4} d e^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \]

[In]

integrate(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

1/221*(21*(-I*sqrt(2)*cos(d*x + c)^5 + 8*I*sqrt(2)*cos(d*x + c)^3 + 4*(I*sqrt(2)*cos(d*x + c)^3 - 2*I*sqrt(2)*
cos(d*x + c))*sin(d*x + c) - 8*I*sqrt(2)*cos(d*x + c))*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4,
0, cos(d*x + c) + I*sin(d*x + c))) + 21*(I*sqrt(2)*cos(d*x + c)^5 - 8*I*sqrt(2)*cos(d*x + c)^3 + 4*(-I*sqrt(2)
*cos(d*x + c)^3 + 2*I*sqrt(2)*cos(d*x + c))*sin(d*x + c) + 8*I*sqrt(2)*cos(d*x + c))*sqrt(e)*weierstrassZeta(-
4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) + 2*(84*cos(d*x + c)^4 - 224*cos(d*x + c)^2 +
(21*cos(d*x + c)^4 - 161*cos(d*x + c)^2 + 117)*sin(d*x + c) + 104)*sqrt(e*cos(d*x + c)))/(a^4*d*e^2*cos(d*x +
c)^5 - 8*a^4*d*e^2*cos(d*x + c)^3 + 8*a^4*d*e^2*cos(d*x + c) - 4*(a^4*d*e^2*cos(d*x + c)^3 - 2*a^4*d*e^2*cos(
d*x + c))*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*cos(d*x+c))**(3/2)/(a+a*sin(d*x+c))**4,x)

[Out]

Timed out

Maxima [F(-1)]

Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

Timed out

Giac [F]

\[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{4}} \,d x } \]

[In]

integrate(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

integrate(1/((e*cos(d*x + c))^(3/2)*(a*sin(d*x + c) + a)^4), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx=\int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4} \,d x \]

[In]

int(1/((e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^4),x)

[Out]

int(1/((e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^4), x)