\(\int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{7/2} \sqrt {d \sin (e+f x)}} \, dx\) [1278]

Optimal result

Integrand size = 37, antiderivative size = 193 \[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{7/2} \sqrt {d \sin (e+f x)}} \, dx=\frac {8 a^2 \sqrt {d \sin (e+f x)}}{5 d f g^3 \sqrt {g \cos (e+f x)}}+\frac {8 a b (d \sin (e+f x))^{3/2}}{5 d^2 f g^3 \sqrt {g \cos (e+f x)}}+\frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}-\frac {8 a b \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{5 d f g^4 \sqrt {\sin (2 e+2 f x)}} \]

[Out]

8/5*a*b*(d*sin(f*x+e))^(3/2)/d^2/f/g^3/(g*cos(f*x+e))^(1/2)+2/5*(a+b*sin(f*x+e))^2*(d*sin(f*x+e))^(1/2)/d/f/g/
(g*cos(f*x+e))^(5/2)+8/5*a^2*(d*sin(f*x+e))^(1/2)/d/f/g^3/(g*cos(f*x+e))^(1/2)+8/5*a*b*(sin(e+1/4*Pi+f*x)^2)^(
1/2)/sin(e+1/4*Pi+f*x)*EllipticE(cos(e+1/4*Pi+f*x),2^(1/2))*(g*cos(f*x+e))^(1/2)*(d*sin(f*x+e))^(1/2)/d/f/g^4/
sin(2*f*x+2*e)^(1/2)

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {2967, 2917, 2643, 2651, 2652, 2719} \[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{7/2} \sqrt {d \sin (e+f x)}} \, dx=\frac {8 a^2 \sqrt {d \sin (e+f x)}}{5 d f g^3 \sqrt {g \cos (e+f x)}}+\frac {8 a b (d \sin (e+f x))^{3/2}}{5 d^2 f g^3 \sqrt {g \cos (e+f x)}}-\frac {8 a b E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{5 d f g^4 \sqrt {\sin (2 e+2 f x)}}+\frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}} \]

[In]

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

[Out]

(8*a^2*Sqrt[d*Sin[e + f*x]])/(5*d*f*g^3*Sqrt[g*Cos[e + f*x]]) + (8*a*b*(d*Sin[e + f*x])^(3/2))/(5*d^2*f*g^3*Sq
rt[g*Cos[e + f*x]]) + (2*Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^2)/(5*d*f*g*(g*Cos[e + f*x])^(5/2)) - (8*a*
b*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(5*d*f*g^4*Sqrt[Sin[2*e + 2*f*x]])

Rule 2643

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Simp[(a*Sin[e +
f*x])^(m + 1)*((b*Cos[e + f*x])^(n + 1)/(a*b*f*(m + 1))), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n + 2,
0] && NeQ[m, -1]

Rule 2651

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b*Sin[e +
f*x])^(n + 1))*((a*Cos[e + f*x])^(m + 1)/(a*b*f*(m + 1))), x] + Dist[(m + n + 2)/(a^2*(m + 1)), Int[(b*Sin[e +
f*x])^n*(a*Cos[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n]

Rule 2652

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a*Sin[e +
f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]), Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f},
x]

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 2917

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)
*(x_)]), x_Symbol] :> Dist[a, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[(g*Cos[e + f*x
])^p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]

Rule 2967

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_))/Sqrt[(d_.)*sin[(e_.) +
(f_.)*(x_)]], x_Symbol] :> Simp[2*(g*Cos[e + f*x])^(p + 1)*Sqrt[d*Sin[e + f*x]]*((a + b*Sin[e + f*x])^m/(d*f*g
*(2*m + 1))), x] + Dist[2*a*(m/(g^2*(2*m + 1))), Int[(g*Cos[e + f*x])^(p + 2)*((a + b*Sin[e + f*x])^(m - 1)/Sq
rt[d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && EqQ[m + p + 3/2,
0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}+\frac {(4 a) \int \frac {a+b \sin (e+f x)}{(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}} \, dx}{5 g^2} \\ & = \frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}+\frac {\left (4 a^2\right ) \int \frac {1}{(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}} \, dx}{5 g^2}+\frac {(4 a b) \int \frac {\sqrt {d \sin (e+f x)}}{(g \cos (e+f x))^{3/2}} \, dx}{5 d g^2} \\ & = \frac {8 a^2 \sqrt {d \sin (e+f x)}}{5 d f g^3 \sqrt {g \cos (e+f x)}}+\frac {8 a b (d \sin (e+f x))^{3/2}}{5 d^2 f g^3 \sqrt {g \cos (e+f x)}}+\frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}-\frac {(8 a b) \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)} \, dx}{5 d g^4} \\ & = \frac {8 a^2 \sqrt {d \sin (e+f x)}}{5 d f g^3 \sqrt {g \cos (e+f x)}}+\frac {8 a b (d \sin (e+f x))^{3/2}}{5 d^2 f g^3 \sqrt {g \cos (e+f x)}}+\frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}-\frac {\left (8 a b \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}\right ) \int \sqrt {\sin (2 e+2 f x)} \, dx}{5 d g^4 \sqrt {\sin (2 e+2 f x)}} \\ & = \frac {8 a^2 \sqrt {d \sin (e+f x)}}{5 d f g^3 \sqrt {g \cos (e+f x)}}+\frac {8 a b (d \sin (e+f x))^{3/2}}{5 d^2 f g^3 \sqrt {g \cos (e+f x)}}+\frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}-\frac {8 a b \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{5 d f g^4 \sqrt {\sin (2 e+2 f x)}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.70 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.54 \[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{7/2} \sqrt {d \sin (e+f x)}} \, dx=\frac {2 \left (15 a^2+10 a b \cos ^2(e+f x)^{5/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {9}{4},\frac {7}{4},\sin ^2(e+f x)\right ) \sin (e+f x)+3 \left (-4 a^2+b^2\right ) \sin ^2(e+f x)\right ) \tan (e+f x)}{15 f g^2 (g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}} \]

[In]

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

[Out]

(2*(15*a^2 + 10*a*b*(Cos[e + f*x]^2)^(5/4)*Hypergeometric2F1[3/4, 9/4, 7/4, Sin[e + f*x]^2]*Sin[e + f*x] + 3*(
-4*a^2 + b^2)*Sin[e + f*x]^2)*Tan[e + f*x])/(15*f*g^2*(g*Cos[e + f*x])^(3/2)*Sqrt[d*Sin[e + f*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(457\) vs. \(2(192)=384\).

Time = 2.87 (sec) , antiderivative size = 458, normalized size of antiderivative = 2.37

method	result	size
default	\(\frac {\sqrt {2}\, \left (-4 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) a b \cos \left (f x +e \right )+8 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, E\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) a b \cos \left (f x +e \right )-4 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) a b +8 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, E\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) a b +\sqrt {2}\, b^{2} \sin \left (f x +e \right ) \left (\tan ^{2}\left (f x +e \right )\right )+4 \sin \left (f x +e \right ) \sqrt {2}\, a^{2}-4 \sqrt {2}\, a b \cos \left (f x +e \right )+2 a b \sqrt {2}+\sqrt {2}\, a^{2} \tan \left (f x +e \right ) \sec \left (f x +e \right )+2 a b \sqrt {2}\, \left (\sec ^{2}\left (f x +e \right )\right )\right )}{5 g^{3} f \sqrt {g \cos \left (f x +e \right )}\, \sqrt {d \sin \left (f x +e \right )}}\)	\(458\)
parts	\(\frac {2 a^{2} \left (4 \sin \left (f x +e \right )+\sec \left (f x +e \right ) \tan \left (f x +e \right )\right )}{5 f \,g^{3} \sqrt {d \sin \left (f x +e \right )}\, \sqrt {g \cos \left (f x +e \right )}}+\frac {2 b^{2} \sin \left (f x +e \right ) \left (\tan ^{2}\left (f x +e \right )\right )}{5 f \,g^{3} \sqrt {d \sin \left (f x +e \right )}\, \sqrt {g \cos \left (f x +e \right )}}-\frac {2 a b \sqrt {2}\, \left (-4 E\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, \cos \left (f x +e \right )+2 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )-4 E\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}+2 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right )+2 \cos \left (f x +e \right ) \sqrt {2}-\sqrt {2}-\sqrt {2}\, \left (\sec ^{2}\left (f x +e \right )\right )\right )}{5 f \,g^{3} \sqrt {d \sin \left (f x +e \right )}\, \sqrt {g \cos \left (f x +e \right )}}\)	\(491\)

[In]

int((a+b*sin(f*x+e))^2/(g*cos(f*x+e))^(7/2)/(d*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/5/g^3/f*2^(1/2)/(g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2)*(-4*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(-csc(f*x+e)+
cot(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*a*b
*cos(f*x+e)+8*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)
*EllipticE((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*a*b*cos(f*x+e)-4*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(-c
sc(f*x+e)+cot(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^
(1/2))*a*b+8*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*
EllipticE((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*a*b+2^(1/2)*b^2*sin(f*x+e)*tan(f*x+e)^2+4*sin(f*x+e)*2
^(1/2)*a^2-4*2^(1/2)*a*b*cos(f*x+e)+2*a*b*2^(1/2)+2^(1/2)*a^2*tan(f*x+e)*sec(f*x+e)+2*a*b*2^(1/2)*sec(f*x+e)^2
)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.16 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.23 \[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{7/2} \sqrt {d \sin (e+f x)}} \, dx=-\frac {2 \, {\left (2 i \, \sqrt {i \, d g} a b \cos \left (f x + e\right )^{3} E(\arcsin \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\,|\,-1) - 2 i \, \sqrt {-i \, d g} a b \cos \left (f x + e\right )^{3} E(\arcsin \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\,|\,-1) - 2 i \, \sqrt {i \, d g} a b \cos \left (f x + e\right )^{3} F(\arcsin \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\,|\,-1) + 2 i \, \sqrt {-i \, d g} a b \cos \left (f x + e\right )^{3} F(\arcsin \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\,|\,-1) - {\left ({\left (4 \, a^{2} - b^{2}\right )} \cos \left (f x + e\right )^{2} + a^{2} + b^{2} + 2 \, {\left (2 \, a b \cos \left (f x + e\right )^{2} + a b\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {d \sin \left (f x + e\right )}\right )}}{5 \, d f g^{4} \cos \left (f x + e\right )^{3}} \]

[In]

integrate((a+b*sin(f*x+e))^2/(g*cos(f*x+e))^(7/2)/(d*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

-2/5*(2*I*sqrt(I*d*g)*a*b*cos(f*x + e)^3*elliptic_e(arcsin(cos(f*x + e) + I*sin(f*x + e)), -1) - 2*I*sqrt(-I*d
*g)*a*b*cos(f*x + e)^3*elliptic_e(arcsin(cos(f*x + e) - I*sin(f*x + e)), -1) - 2*I*sqrt(I*d*g)*a*b*cos(f*x + e
)^3*elliptic_f(arcsin(cos(f*x + e) + I*sin(f*x + e)), -1) + 2*I*sqrt(-I*d*g)*a*b*cos(f*x + e)^3*elliptic_f(arc
sin(cos(f*x + e) - I*sin(f*x + e)), -1) - ((4*a^2 - b^2)*cos(f*x + e)^2 + a^2 + b^2 + 2*(2*a*b*cos(f*x + e)^2
+ a*b)*sin(f*x + e))*sqrt(g*cos(f*x + e))*sqrt(d*sin(f*x + e)))/(d*f*g^4*cos(f*x + e)^3)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{7/2} \sqrt {d \sin (e+f x)}} \, dx=\text {Timed out} \]

[In]

integrate((a+b*sin(f*x+e))**2/(g*cos(f*x+e))**(7/2)/(d*sin(f*x+e))**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{7/2} \sqrt {d \sin (e+f x)}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{2}}{\left (g \cos \left (f x + e\right )\right )^{\frac {7}{2}} \sqrt {d \sin \left (f x + e\right )}} \,d x } \]

[In]

integrate((a+b*sin(f*x+e))^2/(g*cos(f*x+e))^(7/2)/(d*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{7/2} \sqrt {d \sin (e+f x)}} \, dx=\text {Timed out} \]

[In]

integrate((a+b*sin(f*x+e))^2/(g*cos(f*x+e))^(7/2)/(d*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{7/2} \sqrt {d \sin (e+f x)}} \, dx=\int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2}{{\left (g\,\cos \left (e+f\,x\right )\right )}^{7/2}\,\sqrt {d\,\sin \left (e+f\,x\right )}} \,d x \]

[In]

int((a + b*sin(e + f*x))^2/((g*cos(e + f*x))^(7/2)*(d*sin(e + f*x))^(1/2)),x)

[Out]

int((a + b*sin(e + f*x))^2/((g*cos(e + f*x))^(7/2)*(d*sin(e + f*x))^(1/2)), x)