Optimal. Leaf size=221 \[ -\frac {(a-b) \sin (e+f x) \cos (e+f x)}{3 a f (a+b)^2 \sqrt {a+b \sin ^2(e+f x)}}-\frac {\sin (e+f x) \cos (e+f x)}{3 f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1} F\left (e+f x\left |-\frac {b}{a}\right .\right )}{3 b f (a+b) \sqrt {a+b \sin ^2(e+f x)}}-\frac {(a-b) \sqrt {a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{3 a b f (a+b)^2 \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.29, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3173, 3172, 3178, 3177, 3183, 3182} \[ -\frac {(a-b) \sin (e+f x) \cos (e+f x)}{3 a f (a+b)^2 \sqrt {a+b \sin ^2(e+f x)}}-\frac {\sin (e+f x) \cos (e+f x)}{3 f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1} F\left (e+f x\left |-\frac {b}{a}\right .\right )}{3 b f (a+b) \sqrt {a+b \sin ^2(e+f x)}}-\frac {(a-b) \sqrt {a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{3 a b f (a+b)^2 \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3172
Rule 3173
Rule 3177
Rule 3178
Rule 3182
Rule 3183
Rubi steps
\begin {align*} \int \frac {\sin ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx &=-\frac {\cos (e+f x) \sin (e+f x)}{3 (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {\int \frac {a+a \sin ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx}{3 a (a+b)}\\ &=-\frac {\cos (e+f x) \sin (e+f x)}{3 (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {(a-b) \cos (e+f x) \sin (e+f x)}{3 a (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\int \frac {2 a^2-a (a-b) \sin ^2(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx}{3 a^2 (a+b)^2}\\ &=-\frac {\cos (e+f x) \sin (e+f x)}{3 (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {(a-b) \cos (e+f x) \sin (e+f x)}{3 a (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(a-b) \int \sqrt {a+b \sin ^2(e+f x)} \, dx}{3 a b (a+b)^2}+\frac {\int \frac {1}{\sqrt {a+b \sin ^2(e+f x)}} \, dx}{3 b (a+b)}\\ &=-\frac {\cos (e+f x) \sin (e+f x)}{3 (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {(a-b) \cos (e+f x) \sin (e+f x)}{3 a (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left ((a-b) \sqrt {a+b \sin ^2(e+f x)}\right ) \int \sqrt {1+\frac {b \sin ^2(e+f x)}{a}} \, dx}{3 a b (a+b)^2 \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {\sqrt {1+\frac {b \sin ^2(e+f x)}{a}} \int \frac {1}{\sqrt {1+\frac {b \sin ^2(e+f x)}{a}}} \, dx}{3 b (a+b) \sqrt {a+b \sin ^2(e+f x)}}\\ &=-\frac {\cos (e+f x) \sin (e+f x)}{3 (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {(a-b) \cos (e+f x) \sin (e+f x)}{3 a (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(a-b) E\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {a+b \sin ^2(e+f x)}}{3 a b (a+b)^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {F\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 b (a+b) f \sqrt {a+b \sin ^2(e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.53, size = 174, normalized size = 0.79 \[ \frac {-\sqrt {2} b \sin (2 (e+f x)) \left (4 a^2+b (b-a) \cos (2 (e+f x))+a b-b^2\right )+2 a^2 (a+b) \left (\frac {2 a-b \cos (2 (e+f x))+b}{a}\right )^{3/2} F\left (e+f x\left |-\frac {b}{a}\right .\right )-2 a^2 (a-b) \left (\frac {2 a-b \cos (2 (e+f x))+b}{a}\right )^{3/2} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{6 a b f (a+b)^2 (2 a-b \cos (2 (e+f x))+b)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} {\left (\cos \left (f x + e\right )^{2} - 1\right )}}{b^{3} \cos \left (f x + e\right )^{6} - 3 \, {\left (a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left (a^{2} b + 2 \, a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.75, size = 483, normalized size = 2.19 \[ \frac {\left (a \,b^{2}-b^{3}\right ) \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )+\left (-2 a^{2} b -a \,b^{2}+b^{3}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-\sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, a b \left (\EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a +\EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b -\EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a +\EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b \right ) \left (\cos ^{2}\left (f x +e \right )\right )+\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}+2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b +\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}-\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}+\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}}{3 \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}} \left (a +b \right )^{2} a b \cos \left (f x +e \right ) f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\sin \left (e+f\,x\right )}^2}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________