\(\int \frac {\cos ^2(e+f x)}{(a+a \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx\) [937]
Optimal result
Integrand size = 37, antiderivative size = 141 \[
\int \frac {\cos ^2(e+f x)}{(a+a \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{a^{3/2} \sqrt {d} f}-\frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{a^{3/2} \sqrt {c-d} f}
\]
[Out]
-2*arctanh(1/2*cos(f*x+e)*a^(1/2)*(c-d)^(1/2)*2^(1/2)/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2))*2^(1/2)/a
^(3/2)/f/(c-d)^(1/2)+2*arctan(cos(f*x+e)*a^(1/2)*d^(1/2)/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2))/a^(3/2
)/f/d^(1/2)
Rubi [A] (verified)
Time = 0.48 (sec) , antiderivative size = 141, 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 = {2995, 3061, 2861, 214, 2854,
211} \[
\int \frac {\cos ^2(e+f x)}{(a+a \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{a^{3/2} \sqrt {d} f}-\frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{a^{3/2} f \sqrt {c-d}}
\]
[In]
Int[Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]),x]
[Out]
(2*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(a^(3/2)*Sqrt[d
]*f) - (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[
e + f*x]])])/(a^(3/2)*Sqrt[c - d]*f)
Rule 211
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 214
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]
Rule 2854
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[
-2*(b/f), Subst[Int[1/(b + d*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]))
], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Rule 2861
Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> D
ist[-2*(a/f), Subst[Int[1/(2*b^2 - (a*c - b*d)*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c +
d*Sin[e + f*x]]))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 -
d^2, 0]
Rule 2995
Int[cos[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_), x_Symbol] :> Dist[1/b^2, Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*(a - b*Sin[e + f*x])
, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n]
Rule 3061
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin
[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[(A*b - a*B)/b, Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*
x]]), x], x] + Dist[B/b, Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e
, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Rubi steps \begin{align*}
\text {integral}& = \frac {\int \frac {a-a \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{a^2} \\ & = -\frac {\int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{a^2}+\frac {2 \int \frac {1}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{a} \\ & = -\frac {4 \text {Subst}\left (\int \frac {1}{2 a^2-(a c-a d) x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{f}+\frac {2 \text {Subst}\left (\int \frac {1}{a+d x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{a f} \\ & = \frac {2 \arctan \left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{a^{3/2} \sqrt {d} f}-\frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{a^{3/2} \sqrt {c-d} f} \\
\end{align*}
Mathematica [C] (warning: unable to verify)
Result contains complex when optimal does not.
Time = 15.51 (sec) , antiderivative size = 1338, normalized size of antiderivative = 9.49
\[
\int \frac {\cos ^2(e+f x)}{(a+a \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx=\frac {\left (\frac {2 \sqrt {2} \log \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {c-d}}+\frac {i \log \left (\frac {2 i \left (i c+d+(1+i) \sqrt {2} \sqrt {d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(c+i d) \tan \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {d} \left (i+\tan \left (\frac {1}{2} (e+f x)\right )\right )}\right )}{\sqrt {d}}-\frac {2 \sqrt {2} \log \left (c-d+2 \sqrt {c-d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {c-d}}-\frac {i \log \left (-\frac {2 \left (c+i d+(1+i) \sqrt {2} \sqrt {d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(i c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {d} \left (-i+\tan \left (\frac {1}{2} (e+f x)\right )\right )}\right )}{\sqrt {d}}\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (\frac {1}{\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)}}-\frac {\sin (e+f x)}{\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)}}\right )}{f (a (1+\sin (e+f x)))^{3/2} \left (\frac {\sqrt {2} \sec ^2\left (\frac {1}{2} (e+f x)\right )}{\sqrt {c-d} \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )}-\frac {2 \sqrt {2} \left (\frac {1}{2} (-c+d) \sec ^2\left (\frac {1}{2} (e+f x)\right )+\frac {\sqrt {c-d} d \cos (e+f x) \sqrt {\frac {1}{1+\cos (e+f x)}}}{\sqrt {c+d \sin (e+f x)}}+\sqrt {c-d} \left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} \sin (e+f x) \sqrt {c+d \sin (e+f x)}\right )}{\sqrt {c-d} \left (c-d+2 \sqrt {c-d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )}+\frac {\left (i+\tan \left (\frac {1}{2} (e+f x)\right )\right ) \left (\frac {2 i \left (\frac {1}{2} (c+i d) \sec ^2\left (\frac {1}{2} (e+f x)\right )+\frac {(1+i) d^{3/2} \cos (e+f x) \sqrt {\frac {1}{1+\cos (e+f x)}}}{\sqrt {2} \sqrt {c+d \sin (e+f x)}}+\frac {(1+i) \sqrt {d} \left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} \sin (e+f x) \sqrt {c+d \sin (e+f x)}}{\sqrt {2}}\right )}{\sqrt {d} \left (i+\tan \left (\frac {1}{2} (e+f x)\right )\right )}-\frac {i \sec ^2\left (\frac {1}{2} (e+f x)\right ) \left (i c+d+(1+i) \sqrt {2} \sqrt {d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(c+i d) \tan \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {d} \left (i+\tan \left (\frac {1}{2} (e+f x)\right )\right )^2}\right )}{2 \left (i c+d+(1+i) \sqrt {2} \sqrt {d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(c+i d) \tan \left (\frac {1}{2} (e+f x)\right )\right )}+\frac {i \left (-i+\tan \left (\frac {1}{2} (e+f x)\right )\right ) \left (-\frac {2 \left (\frac {1}{2} (i c+d) \sec ^2\left (\frac {1}{2} (e+f x)\right )+\frac {(1+i) d^{3/2} \cos (e+f x) \sqrt {\frac {1}{1+\cos (e+f x)}}}{\sqrt {2} \sqrt {c+d \sin (e+f x)}}+\frac {(1+i) \sqrt {d} \left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} \sin (e+f x) \sqrt {c+d \sin (e+f x)}}{\sqrt {2}}\right )}{\sqrt {d} \left (-i+\tan \left (\frac {1}{2} (e+f x)\right )\right )}+\frac {\sec ^2\left (\frac {1}{2} (e+f x)\right ) \left (c+i d+(1+i) \sqrt {2} \sqrt {d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(i c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {d} \left (-i+\tan \left (\frac {1}{2} (e+f x)\right )\right )^2}\right )}{2 \left (c+i d+(1+i) \sqrt {2} \sqrt {d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(i c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )}\right )}
\]
[In]
Integrate[Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]),x]
[Out]
(((2*Sqrt[2]*Log[1 + Tan[(e + f*x)/2]])/Sqrt[c - d] + (I*Log[((2*I)*(I*c + d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1
+ Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (c + I*d)*Tan[(e + f*x)/2]))/(Sqrt[d]*(I + Tan[(e + f*x)/2])
)])/Sqrt[d] - (2*Sqrt[2]*Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-
c + d)*Tan[(e + f*x)/2]])/Sqrt[c - d] - (I*Log[(-2*(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^
(-1)]*Sqrt[c + d*Sin[e + f*x]] + (I*c + d)*Tan[(e + f*x)/2]))/(Sqrt[d]*(-I + Tan[(e + f*x)/2]))])/Sqrt[d])*(Co
s[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(1/((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) - Sin
[e + f*x]/((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]])))/(f*(a*(1 + Sin[e + f*x]))^(3/2)*(
(Sqrt[2]*Sec[(e + f*x)/2]^2)/(Sqrt[c - d]*(1 + Tan[(e + f*x)/2])) - (2*Sqrt[2]*(((-c + d)*Sec[(e + f*x)/2]^2)/
2 + (Sqrt[c - d]*d*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/Sqrt[c + d*Sin[e + f*x]] + Sqrt[c - d]*((1 + Co
s[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]]))/(Sqrt[c - d]*(c - d + 2*Sqrt[c - d]*Sqrt[(1 +
Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2])) + ((I + Tan[(e + f*x)/2])*(((2*I)*(
((c + I*d)*Sec[(e + f*x)/2]^2)/2 + ((1 + I)*d^(3/2)*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/(Sqrt[2]*Sqrt[
c + d*Sin[e + f*x]]) + ((1 + I)*Sqrt[d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]])
/Sqrt[2]))/(Sqrt[d]*(I + Tan[(e + f*x)/2])) - (I*Sec[(e + f*x)/2]^2*(I*c + d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1
+ Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (c + I*d)*Tan[(e + f*x)/2]))/(Sqrt[d]*(I + Tan[(e + f*x)/2])
^2)))/(2*(I*c + d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (c + I*d)
*Tan[(e + f*x)/2])) + ((I/2)*(-I + Tan[(e + f*x)/2])*((-2*(((I*c + d)*Sec[(e + f*x)/2]^2)/2 + ((1 + I)*d^(3/2)
*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/(Sqrt[2]*Sqrt[c + d*Sin[e + f*x]]) + ((1 + I)*Sqrt[d]*((1 + Cos[e
+ f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[2]))/(Sqrt[d]*(-I + Tan[(e + f*x)/2])) + (Sec
[(e + f*x)/2]^2*(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (I
*c + d)*Tan[(e + f*x)/2]))/(Sqrt[d]*(-I + Tan[(e + f*x)/2])^2)))/(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 +
Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (I*c + d)*Tan[(e + f*x)/2])))
Maple [B] (warning: unable to verify)
Leaf count of result is larger than twice the leaf count of optimal. \(3379\) vs. \(2(114)=228\).
Time = 0.52 (sec) , antiderivative size = 3380, normalized size of antiderivative =
23.97
\[\text {output too large to display}\]
[In]
int(cos(f*x+e)^2/(a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(1/2),x)
[Out]
-1/f/d^2/(2*c-2*d)^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2)/(c^2-2*c*d+d^2)*(2*2^(1/2)*(-(d^2/c^2)^(1/2)*c)^(1/2)*((c+
d*sin(f*x+e))/(1+cos(f*x+e)))^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(1+cos(f*x+e)))^(1/2)*sin(
f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-cos(f*x+e)*d-c+d)/(-cos(f*x+e)+1+sin(f*x+e)))*c^2*cos(f*x+e)*d^2
+2*2^(1/2)*((c+d*sin(f*x+e))/(1+cos(f*x+e)))^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(1+cos(f*x+
e)))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-cos(f*x+e)*d-c+d)/(-cos(f*x+e)+1+sin(f*x+e)))*(-(
d^2/c^2)^(1/2)*c)^(1/2)*c^2*d^2*sin(f*x+e)-4*2^(1/2)*(-(d^2/c^2)^(1/2)*c)^(1/2)*((c+d*sin(f*x+e))/(1+cos(f*x+e
)))^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(
f*x+e)+c*cos(f*x+e)-cos(f*x+e)*d-c+d)/(-cos(f*x+e)+1+sin(f*x+e)))*c*cos(f*x+e)*d^3-4*2^(1/2)*((c+d*sin(f*x+e))
/(1+cos(f*x+e)))^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)+c*sin(
f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-cos(f*x+e)*d-c+d)/(-cos(f*x+e)+1+sin(f*x+e)))*(-(d^2/c^2)^(1/2)*c)^(1/2)*c*d^
3*sin(f*x+e)+2*2^(1/2)*(-(d^2/c^2)^(1/2)*c)^(1/2)*((c+d*sin(f*x+e))/(1+cos(f*x+e)))^(1/2)*ln(2*((2*c-2*d)^(1/2
)*2^(1/2)*((c+d*sin(f*x+e))/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-cos(f*x+e)
*d-c+d)/(-cos(f*x+e)+1+sin(f*x+e)))*cos(f*x+e)*d^4+2*2^(1/2)*((c+d*sin(f*x+e))/(1+cos(f*x+e)))^(1/2)*ln(2*((2*
c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)
-cos(f*x+e)*d-c+d)/(-cos(f*x+e)+1+sin(f*x+e)))*(-(d^2/c^2)^(1/2)*c)^(1/2)*d^4*sin(f*x+e)-(d^2/c^2)^(1/2)*arcta
n(1/(-(d^2/c^2)^(1/2)*c)^(1/2)*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2))*(d*(c+d*sin(f*x+e)
)/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*(2*c-2*d)^(1/2)*c^3*d*cos(f*x+e)+2*(d^2/c^2)^(1/2)*arctan(1/(-(d^2/c
^2)^(1/2)*c)^(1/2)*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2))*(d*(c+d*sin(f*x+e))/((d^2/c^2)
^(1/2)*c*sin(f*x+e)+d))^(1/2)*(2*c-2*d)^(1/2)*c^2*d^2*cos(f*x+e)-(d^2/c^2)^(1/2)*arctan(1/(-(d^2/c^2)^(1/2)*c)
^(1/2)*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2))*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin
(f*x+e)+d))^(1/2)*(2*c-2*d)^(1/2)*c*d^3*cos(f*x+e)+2*2^(1/2)*(-(d^2/c^2)^(1/2)*c)^(1/2)*((c+d*sin(f*x+e))/(1+c
os(f*x+e)))^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)+c*sin(f*x+e
)-d*sin(f*x+e)+c*cos(f*x+e)-cos(f*x+e)*d-c+d)/(-cos(f*x+e)+1+sin(f*x+e)))*c^2*d^2-4*2^(1/2)*(-(d^2/c^2)^(1/2)*
c)^(1/2)*((c+d*sin(f*x+e))/(1+cos(f*x+e)))^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(1+cos(f*x+e)
))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-cos(f*x+e)*d-c+d)/(-cos(f*x+e)+1+sin(f*x+e)))*c*d^3
+2*2^(1/2)*(-(d^2/c^2)^(1/2)*c)^(1/2)*((c+d*sin(f*x+e))/(1+cos(f*x+e)))^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((
c+d*sin(f*x+e))/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-cos(f*x+e)*d-c+d)/(-co
s(f*x+e)+1+sin(f*x+e)))*d^4+(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*d^2*c^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^
4)*c)^(1/2)*(d^2/c^2)^(1/2)*arctan(((d^2/c^2)^(1/2)*c^2-d^2)*c*((d^2/c^2)^(1/2)-1)/(((d^2/c^2)^(1/2)*c^4+6*(d^
2/c^2)^(1/2)*d^2*c^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2)*((d^2/c^2)^(1/2)*c*sin(f*x+e)+cos(f*x+e)*d-
d)/(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/((d^2/c^2)^(1/2)*c*sin(f*x+e)-cos(f*x+e)*d+d))*
(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*(-(d^2/c^2)^(1/2)*c)^(1/2)*(2*c-2*d)^(1/2)*c*sin(f
*x+e)+arctan(1/(-(d^2/c^2)^(1/2)*c)^(1/2)*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2))*(d*(c+d
*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*(2*c-2*d)^(1/2)*c^2*d^2*cos(f*x+e)-2*arctan(1/(-(d^2/c^2)
^(1/2)*c)^(1/2)*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2))*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1
/2)*c*sin(f*x+e)+d))^(1/2)*(2*c-2*d)^(1/2)*c*d^3*cos(f*x+e)+arctan(1/(-(d^2/c^2)^(1/2)*c)^(1/2)*(d*(c+d*sin(f*
x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2))*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*(2*
c-2*d)^(1/2)*d^4*cos(f*x+e)+(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*d^2*c^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^
4)*c)^(1/2)*(d^2/c^2)^(1/2)*arctan(((d^2/c^2)^(1/2)*c^2-d^2)*c*((d^2/c^2)^(1/2)-1)/(((d^2/c^2)^(1/2)*c^4+6*(d^
2/c^2)^(1/2)*d^2*c^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2)*((d^2/c^2)^(1/2)*c*sin(f*x+e)+cos(f*x+e)*d-
d)/(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/((d^2/c^2)^(1/2)*c*sin(f*x+e)-cos(f*x+e)*d+d))*
(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*(-(d^2/c^2)^(1/2)*c)^(1/2)*(2*c-2*d)^(1/2)*c+(((d^
2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*d^2*c^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2)*arctan(((d^2/c^2)^(1/
2)*c^2-d^2)*c*((d^2/c^2)^(1/2)-1)/(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*d^2*c^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^
2-4*d^4)*c)^(1/2)*((d^2/c^2)^(1/2)*c*sin(f*x+e)+cos(f*x+e)*d-d)/(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x
+e)+d))^(1/2)/((d^2/c^2)^(1/2)*c*sin(f*x+e)-cos(f*x+e)*d+d))*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)
+d))^(1/2)*(-(d^2/c^2)^(1/2)*c)^(1/2)*(2*c-2*d)^(1/2)*d*sin(f*x+e)+(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*d^2
*c^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2)*arctan(((d^2/c^2)^(1/2)*c^2-d^2)*c*((d^2/c^2)^(1/2)-1)/(((d
^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*d^2*c^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2)*((d^2/c^2)^(1/2)*c*s
in(f*x+e)+cos(f*x+e)*d-d)/(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/((d^2/c^2)^(1/2)*c*sin(f
*x+e)-cos(f*x+e)*d+d))*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*(-(d^2/c^2)^(1/2)*c)^(1/2)*
(2*c-2*d)^(1/2)*d)/(c+d*sin(f*x+e))^(1/2)/(a*(1+sin(f*x+e)))^(1/2)/a
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (114) = 228\).
Time = 0.76 (sec) , antiderivative size = 2071, normalized size of antiderivative = 14.69
\[
\int \frac {\cos ^2(e+f x)}{(a+a \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx=\text {Too large to display}
\]
[In]
integrate(cos(f*x+e)^2/(a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(1/2),x, algorithm="fricas")
[Out]
[1/4*(4*sqrt(2)*a*d*log(-((c - 3*d)*cos(f*x + e)^2 - 2*sqrt(2)*((c - d)*cos(f*x + e) - (c - d)*sin(f*x + e) +
c - d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/sqrt(a*c - a*d) + (3*c - d)*cos(f*x + e) - ((c - 3*d)
*cos(f*x + e) - 2*c - 2*d)*sin(f*x + e) + 2*c + 2*d)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f
*x + e) - 2))/sqrt(a*c - a*d) - sqrt(-a*d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4
*a*c*d^3 + a*d^4 + 128*(2*a*c*d^3 - a*d^4)*cos(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)*cos(f*x +
e)^3 - 32*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16*d^3*cos(f*x + e)^4 + 24*(c*d^2
- d^3)*cos(f*x + e)^3 - c^3 + 17*c^2*d - 59*c*d^2 + 51*d^3 - 2*(5*c^2*d - 26*c*d^2 + 33*d^3)*cos(f*x + e)^2 -
(c^3 - 7*c^2*d + 31*c*d^2 - 25*d^3)*cos(f*x + e) + (16*d^3*cos(f*x + e)^3 + c^3 - 17*c^2*d + 59*c*d^2 - 51*d^
3 - 8*(3*c*d^2 - 5*d^3)*cos(f*x + e)^2 - 2*(5*c^2*d - 14*c*d^2 + 13*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(-a*d
)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c) + (a*c^4 - 28*a*c^3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*
a*d^4)*cos(f*x + e) + (128*a*d^4*cos(f*x + e)^4 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*(a
*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2
*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)))/(a^2*d*f), 1/2*(2
*sqrt(2)*a*d*log(-((c - 3*d)*cos(f*x + e)^2 - 2*sqrt(2)*((c - d)*cos(f*x + e) - (c - d)*sin(f*x + e) + c - d)*
sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/sqrt(a*c - a*d) + (3*c - d)*cos(f*x + e) - ((c - 3*d)*cos(f*
x + e) - 2*c - 2*d)*sin(f*x + e) + 2*c + 2*d)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e)
- 2))/sqrt(a*c - a*d) - sqrt(a*d)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(
f*x + e))*sqrt(a*d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(2*a*d^3*cos(f*x + e)^3 - (3*a*c*d^2 - a
*d^3)*cos(f*x + e)*sin(f*x + e) - (a*c^2*d - a*c*d^2 + 2*a*d^3)*cos(f*x + e))))/(a^2*d*f), 1/4*(8*sqrt(2)*a*d*
sqrt(-1/(a*c - a*d))*arctan(sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-1/(a*c - a*d))/cos
(f*x + e)) - sqrt(-a*d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 +
128*(2*a*c*d^3 - a*d^4)*cos(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 32*(a*c^3*d
- 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16*d^3*cos(f*x + e)^4 + 24*(c*d^2 - d^3)*cos(f*x + e
)^3 - c^3 + 17*c^2*d - 59*c*d^2 + 51*d^3 - 2*(5*c^2*d - 26*c*d^2 + 33*d^3)*cos(f*x + e)^2 - (c^3 - 7*c^2*d + 3
1*c*d^2 - 25*d^3)*cos(f*x + e) + (16*d^3*cos(f*x + e)^3 + c^3 - 17*c^2*d + 59*c*d^2 - 51*d^3 - 8*(3*c*d^2 - 5*
d^3)*cos(f*x + e)^2 - 2*(5*c^2*d - 14*c*d^2 + 13*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(-a*d)*sqrt(a*sin(f*x +
e) + a)*sqrt(d*sin(f*x + e) + c) + (a*c^4 - 28*a*c^3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^4)*cos(f*x + e)
+ (128*a*d^4*cos(f*x + e)^4 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*(a*c*d^3 - a*d^4)*cos
(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2 + 15*a*c*d^3 -
9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)))/(a^2*d*f), 1/2*(4*sqrt(2)*a*d*sqrt(-
1/(a*c - a*d))*arctan(sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-1/(a*c - a*d))/cos(f*x +
e)) - sqrt(a*d)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*d
)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(2*a*d^3*cos(f*x + e)^3 - (3*a*c*d^2 - a*d^3)*cos(f*x + e)
*sin(f*x + e) - (a*c^2*d - a*c*d^2 + 2*a*d^3)*cos(f*x + e))))/(a^2*d*f)]
Sympy [F]
\[
\int \frac {\cos ^2(e+f x)}{(a+a \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {\cos ^{2}{\left (e + f x \right )}}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \sqrt {c + d \sin {\left (e + f x \right )}}}\, dx
\]
[In]
integrate(cos(f*x+e)**2/(a+a*sin(f*x+e))**(3/2)/(c+d*sin(f*x+e))**(1/2),x)
[Out]
Integral(cos(e + f*x)**2/((a*(sin(e + f*x) + 1))**(3/2)*sqrt(c + d*sin(e + f*x))), x)
Maxima [F]
\[
\int \frac {\cos ^2(e+f x)}{(a+a \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {\cos \left (f x + e\right )^{2}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sqrt {d \sin \left (f x + e\right ) + c}} \,d x }
\]
[In]
integrate(cos(f*x+e)^2/(a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(1/2),x, algorithm="maxima")
[Out]
integrate(cos(f*x + e)^2/((a*sin(f*x + e) + a)^(3/2)*sqrt(d*sin(f*x + e) + c)), x)
Giac [F(-1)]
Timed out. \[
\int \frac {\cos ^2(e+f x)}{(a+a \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx=\text {Timed out}
\]
[In]
integrate(cos(f*x+e)^2/(a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(1/2),x, algorithm="giac")
[Out]
Timed out
Mupad [F(-1)]
Timed out. \[
\int \frac {\cos ^2(e+f x)}{(a+a \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^2}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x
\]
[In]
int(cos(e + f*x)^2/((a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^(1/2)),x)
[Out]
int(cos(e + f*x)^2/((a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^(1/2)), x)