\(\int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}} \, dx\) [1277]

Optimal result

Integrand size = 37, antiderivative size = 159 \[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}} \, dx=\frac {2 \left (a^2+b^2\right ) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}+\frac {\left (2 a^2-b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (4 e-\pi )+f x,2\right ) \sqrt {\sin (2 e+2 f x)}}{3 f g^2 \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}} \]

[Out]

4/3*a*b*(d*sin(f*x+e))^(3/2)/d^2/f/g/(g*cos(f*x+e))^(3/2)+2/3*(a^2+b^2)*(d*sin(f*x+e))^(1/2)/d/f/g/(g*cos(f*x+
e))^(3/2)-1/3*(2*a^2-b^2)*(sin(e+1/4*Pi+f*x)^2)^(1/2)/sin(e+1/4*Pi+f*x)*EllipticF(cos(e+1/4*Pi+f*x),2^(1/2))*s
in(2*f*x+2*e)^(1/2)/f/g^2/(g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.243, Rules used = {2990, 2643, 3281, 468, 335, 243, 342, 281, 238} \[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}} \, dx=-\frac {2 \left (2 a^2-b^2\right ) \left (1-\csc ^2(e+f x)\right )^{3/4} (d \sin (e+f x))^{3/2} \operatorname {EllipticF}\left (\frac {1}{2} \csc ^{-1}(\sin (e+f x)),2\right )}{3 d^2 f g (g \cos (e+f x))^{3/2}}+\frac {2 \left (a^2+b^2\right ) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}} \]

[In]

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

[Out]

(2*(a^2 + b^2)*Sqrt[d*Sin[e + f*x]])/(3*d*f*g*(g*Cos[e + f*x])^(3/2)) + (4*a*b*(d*Sin[e + f*x])^(3/2))/(3*d^2*
f*g*(g*Cos[e + f*x])^(3/2)) - (2*(2*a^2 - b^2)*(1 - Csc[e + f*x]^2)^(3/4)*EllipticF[ArcCsc[Sin[e + f*x]]/2, 2]
*(d*Sin[e + f*x])^(3/2))/(3*d^2*f*g*(g*Cos[e + f*x])^(3/2))

Rule 238

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[-b/a, 2]))*EllipticF[(1/2)*ArcSin[Rt[-b/a,
2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b/a]

Rule 243

Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Dist[x^3*((1 + a/(b*x^4))^(3/4)/(a + b*x^4)^(3/4)), Int[1/(x^3*
(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ[{a, b}, x]

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 342

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rule 468

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d
))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b*e*n*(p + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a
*b*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0]
&& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0]
&& LeQ[-1, m, (-n)*(p + 1)]))

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 2990

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

Rule 3281

Int[(cos[(e_.) + (f_.)*(x_)]*(c_.))^(m_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)
*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff*c^(2*IntPart[(m - 1)/2] + 1)*(
(c*Cos[e + f*x])^(2*FracPart[(m - 1)/2])/(f*(Cos[e + f*x]^2)^FracPart[(m - 1)/2])), Subst[Int[(d*ff*x)^n*(1 -
ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x
] &&  !IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \frac {(2 a b) \int \frac {\sqrt {d \sin (e+f x)}}{(g \cos (e+f x))^{5/2}} \, dx}{d}+\int \frac {a^2+b^2 \sin ^2(e+f x)}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}} \, dx \\ & = \frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}+\frac {\cos ^2(e+f x)^{3/4} \text {Subst}\left (\int \frac {a^2+b^2 x^2}{\sqrt {d x} \left (1-x^2\right )^{7/4}} \, dx,x,\sin (e+f x)\right )}{f g (g \cos (e+f x))^{3/2}} \\ & = \frac {2 \left (a^2+b^2\right ) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}-\frac {\left (\left (-2 a^2+b^2\right ) \cos ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {d x} \left (1-x^2\right )^{3/4}} \, dx,x,\sin (e+f x)\right )}{3 f g (g \cos (e+f x))^{3/2}} \\ & = \frac {2 \left (a^2+b^2\right ) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}-\frac {\left (2 \left (-2 a^2+b^2\right ) \cos ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {x^4}{d^2}\right )^{3/4}} \, dx,x,\sqrt {d \sin (e+f x)}\right )}{3 d f g (g \cos (e+f x))^{3/2}} \\ & = \frac {2 \left (a^2+b^2\right ) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}-\frac {\left (2 \left (-2 a^2+b^2\right ) \left (1-\csc ^2(e+f x)\right )^{3/4} (d \sin (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {d^2}{x^4}\right )^{3/4} x^3} \, dx,x,\sqrt {d \sin (e+f x)}\right )}{3 d f g (g \cos (e+f x))^{3/2}} \\ & = \frac {2 \left (a^2+b^2\right ) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}+\frac {\left (2 \left (-2 a^2+b^2\right ) \left (1-\csc ^2(e+f x)\right )^{3/4} (d \sin (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {x}{\left (1-d^2 x^4\right )^{3/4}} \, dx,x,\frac {1}{\sqrt {d \sin (e+f x)}}\right )}{3 d f g (g \cos (e+f x))^{3/2}} \\ & = \frac {2 \left (a^2+b^2\right ) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}+\frac {\left (\left (-2 a^2+b^2\right ) \left (1-\csc ^2(e+f x)\right )^{3/4} (d \sin (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-d^2 x^2\right )^{3/4}} \, dx,x,\frac {\csc (e+f x)}{d}\right )}{3 d f g (g \cos (e+f x))^{3/2}} \\ & = \frac {2 \left (a^2+b^2\right ) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}-\frac {2 \left (2 a^2-b^2\right ) \left (1-\csc ^2(e+f x)\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin (\csc (e+f x)),2\right ) (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.57 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}} \, dx=\frac {2 \left (15 a^2 \cos ^2(e+f x)^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {7}{4},\frac {5}{4},\sin ^2(e+f x)\right )+b \sin (e+f x) \left (10 a+3 b \cos ^2(e+f x)^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {7}{4},\frac {9}{4},\sin ^2(e+f x)\right ) \sin (e+f x)\right )\right ) \tan (e+f x)}{15 f g^2 \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}} \]

[In]

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

[Out]

(2*(15*a^2*(Cos[e + f*x]^2)^(3/4)*Hypergeometric2F1[1/4, 7/4, 5/4, Sin[e + f*x]^2] + b*Sin[e + f*x]*(10*a + 3*
b*(Cos[e + f*x]^2)^(3/4)*Hypergeometric2F1[5/4, 7/4, 9/4, Sin[e + f*x]^2]*Sin[e + f*x]))*Tan[e + f*x])/(15*f*g
^2*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(424\) vs. \(2(159)=318\).

Time = 2.62 (sec) , antiderivative size = 425, normalized size of antiderivative = 2.67

method	result	size
default	\(\frac {\sqrt {2}\, \left (2 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) a^{2} \cos \left (f x +e \right )-\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) b^{2} \cos \left (f x +e \right )+2 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) a^{2}-\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) b^{2}-2 \sqrt {2}\, a b \cos \left (f x +e \right )+\sqrt {2}\, a^{2} \tan \left (f x +e \right )+\sqrt {2}\, b^{2} \tan \left (f x +e \right )+2 \sqrt {2}\, a b \sec \left (f x +e \right )\right )}{3 g^{2} f \sqrt {g \cos \left (f x +e \right )}\, \sqrt {d \sin \left (f x +e \right )}}\)	\(425\)
parts	\(\frac {a^{2} \sqrt {2}\, \left (2 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )+2 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \tan \left (f x +e \right )\right )}{3 f \sqrt {d \sin \left (f x +e \right )}\, \sqrt {g \cos \left (f x +e \right )}\, g^{2}}+\frac {b^{2} \sqrt {2}\, \left (-\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )-\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \tan \left (f x +e \right )\right )}{3 f \sqrt {d \sin \left (f x +e \right )}\, \sqrt {g \cos \left (f x +e \right )}\, g^{2}}+\frac {4 a b \sin \left (f x +e \right ) \tan \left (f x +e \right )}{3 f \sqrt {d \sin \left (f x +e \right )}\, \sqrt {g \cos \left (f x +e \right )}\, g^{2}}\)	\(462\)

[In]

int((a+b*sin(f*x+e))^2/(g*cos(f*x+e))^(5/2)/(d*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3/g^2/f*2^(1/2)/(g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2)*(2*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(-csc(f*x+e)+c
ot(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^2*
cos(f*x+e)-(-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)*El
lipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*b^2*cos(f*x+e)+2*(-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^2-(-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)*Ellip
ticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*b^2-2*2^(1/2)*a*b*cos(f*x+e)+2^(1/2)*a^2*tan(f*x+e)+2^(1/2)
*b^2*tan(f*x+e)+2*2^(1/2)*a*b*sec(f*x+e))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}} \, dx=-\frac {{\left (2 \, a^{2} - b^{2}\right )} \sqrt {i \, d g} \cos \left (f x + e\right )^{2} F(\arcsin \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\,|\,-1) + {\left (2 \, a^{2} - b^{2}\right )} \sqrt {-i \, d g} \cos \left (f x + e\right )^{2} F(\arcsin \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\,|\,-1) - 2 \, {\left (2 \, a b \sin \left (f x + e\right ) + a^{2} + b^{2}\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {d \sin \left (f x + e\right )}}{3 \, d f g^{3} \cos \left (f x + e\right )^{2}} \]

[In]

integrate((a+b*sin(f*x+e))^2/(g*cos(f*x+e))^(5/2)/(d*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

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

[In]

integrate((a+b*sin(f*x+e))**2/(g*cos(f*x+e))**(5/2)/(d*sin(f*x+e))**(1/2),x)

[Out]

Timed out

Maxima [F]

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

[In]

integrate((a+b*sin(f*x+e))^2/(g*cos(f*x+e))^(5/2)/(d*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F(-1)]

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

[In]

integrate((a+b*sin(f*x+e))^2/(g*cos(f*x+e))^(5/2)/(d*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

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

[In]

int((a + b*sin(e + f*x))^2/((g*cos(e + f*x))^(5/2)*(d*sin(e + f*x))^(1/2)),x)

[Out]

int((a + b*sin(e + f*x))^2/((g*cos(e + f*x))^(5/2)*(d*sin(e + f*x))^(1/2)), x)