Optimal. Leaf size=322 \[ \frac {2 b (3 a+2 b) \cot (e+f x)}{3 a^2 f (a+b)^2 \sqrt {a+b \sin ^2(e+f x)}}+\frac {(3 a+4 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right )}{3 a^2 f (a+b) \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 f (a+b)^2}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right )}{3 a^3 f (a+b)^2 \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}+\frac {b \cot (e+f x)}{3 a f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}} \]
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Rubi [A] time = 0.41, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3188, 472, 579, 583, 524, 426, 424, 421, 419} \[ -\frac {\left (3 a^2+13 a b+8 b^2\right ) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 f (a+b)^2}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right )}{3 a^3 f (a+b)^2 \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}+\frac {2 b (3 a+2 b) \cot (e+f x)}{3 a^2 f (a+b)^2 \sqrt {a+b \sin ^2(e+f x)}}+\frac {(3 a+4 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right )}{3 a^2 f (a+b) \sqrt {a+b \sin ^2(e+f x)}}+\frac {b \cot (e+f x)}{3 a f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 421
Rule 424
Rule 426
Rule 472
Rule 524
Rule 579
Rule 583
Rule 3188
Rubi steps
\begin {align*} \int \frac {\csc ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-x^2} \left (a+b x^2\right )^{5/2}} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {b \cot (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname {Subst}\left (\int \frac {-3 a-4 b+3 b x^2}{x^2 \sqrt {1-x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 a (a+b) f}\\ &=\frac {b \cot (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 b (3 a+2 b) \cot (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname {Subst}\left (\int \frac {3 a^2+13 a b+8 b^2-2 b (3 a+2 b) x^2}{x^2 \sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^2 (a+b)^2 f}\\ &=\frac {b \cot (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 b (3 a+2 b) \cot (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 (a+b)^2 f}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname {Subst}\left (\int \frac {2 a b (3 a+2 b)+b \left (3 a^2+13 a b+8 b^2\right ) x^2}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^3 (a+b)^2 f}\\ &=\frac {b \cot (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 b (3 a+2 b) \cot (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 (a+b)^2 f}+\frac {\left ((3 a+4 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^2 (a+b) f}-\frac {\left (\left (3 a^2+13 a b+8 b^2\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^3 (a+b)^2 f}\\ &=\frac {b \cot (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 b (3 a+2 b) \cot (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 (a+b)^2 f}-\frac {\left (\left (3 a^2+13 a b+8 b^2\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {b x^2}{a}}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^3 (a+b)^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {\left ((3 a+4 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {b x^2}{a}}} \, dx,x,\sin (e+f x)\right )}{3 a^2 (a+b) f \sqrt {a+b \sin ^2(e+f x)}}\\ &=\frac {b \cot (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 b (3 a+2 b) \cot (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 (a+b)^2 f}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {\cos ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 (a+b)^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {(3 a+4 b) \sqrt {\cos ^2(e+f x)} F\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a^2 (a+b) f \sqrt {a+b \sin ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 2.36, size = 214, normalized size = 0.66 \[ \frac {4 a^2 \left (\frac {2 a-b \cos (2 (e+f x))+b}{a}\right )^{3/2} \left (\left (3 a^2+7 a b+4 b^2\right ) F\left (e+f x\left |-\frac {b}{a}\right .\right )-\left (3 a^2+13 a b+8 b^2\right ) E\left (e+f x\left |-\frac {b}{a}\right .\right )\right )-2 \sqrt {2} \left (2 a b^2 (a+b) \sin (2 (e+f x))+b^2 (7 a+5 b) \sin (2 (e+f x)) (2 a-b \cos (2 (e+f x))+b)+3 (a+b)^2 \cot (e+f x) (2 a-b \cos (2 (e+f x))+b)^2\right )}{12 a^3 f (a+b)^2 (2 a-b \cos (2 (e+f x))+b)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \csc \left (f x + e\right )^{2}}{b^{3} \cos \left (f x + e\right )^{6} - 3 \, {\left (a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left (a^{2} b + 2 \, a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.04, size = 527, normalized size = 1.64 \[ \frac {-\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 (3 \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}+7 \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b +4 \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{2}-3 \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}-13 \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b -8 \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{2}\right ) \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\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 \left (3 \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}+10 \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b +11 \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}+4 \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{3}-3 \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}-16 \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b -21 \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}-8 \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{3}\right ) \sin \left (f x +e \right )+\left (-3 a^{2} b^{2}-13 a \,b^{3}-8 b^{4}\right ) \left (\cos ^{6}\left (f x +e \right )\right )+\left (6 a^{3} b +26 a^{2} b^{2}+38 a \,b^{3}+16 b^{4}\right ) \left (\cos ^{4}\left (f x +e \right )\right )+\left (-3 a^{4}-12 a^{3} b -26 a^{2} b^{2}-25 a \,b^{3}-8 b^{4}\right ) \left (\cos ^{2}\left (f x +e \right )\right )}{3 \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}} \left (a +b \right )^{2} \sin \left (f x +e \right ) a^{3} \cos \left (f x +e \right ) f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\sin \left (e+f\,x\right )}^2\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{2}{\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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