Optimal. Leaf size=73 \[ \frac {2 \sin (e+f x)}{3 a^2 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\sin (e+f x) \cos ^2(e+f x)}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3190, 378, 191} \[ \frac {2 \sin (e+f x)}{3 a^2 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\sin (e+f x) \cos ^2(e+f x)}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 378
Rule 3190
Rubi steps
\begin {align*} \int \frac {\cos ^3(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1-x^2}{\left (a+b x^2\right )^{5/2}} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {\cos ^2(e+f x) \sin (e+f x)}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 a f}\\ &=\frac {\cos ^2(e+f x) \sin (e+f x)}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 \sin (e+f x)}{3 a^2 f \sqrt {a+b \sin ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 51, normalized size = 0.70 \[ \frac {3 a \sin (e+f x)-(a-2 b) \sin ^3(e+f x)}{3 a^2 f \left (a+b \sin ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 107, normalized size = 1.47 \[ \frac {{\left ({\left (a - 2 \, b\right )} \cos \left (f x + e\right )^{2} + 2 \, a + 2 \, b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sin \left (f x + e\right )}{3 \, {\left (a^{2} b^{2} f \cos \left (f x + e\right )^{4} - 2 \, {\left (a^{3} b + a^{2} b^{2}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.78, size = 58, normalized size = 0.79 \[ -\frac {{\left (\frac {{\left (a b - 2 \, b^{2}\right )} \sin \left (f x + e\right )^{2}}{a^{2} b} - \frac {3}{a}\right )} \sin \left (f x + e\right )}{3 \, {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 3.89, size = 120, normalized size = 1.64 \[ \frac {\sin \left (f x +e \right ) \sqrt {-b \left (\cos ^{2}\left (f x +e \right )\right )+\frac {a \,b^{2}+b^{3}}{b^{2}}}\, \left (a \left (\cos ^{2}\left (f x +e \right )\right )-2 b \left (\cos ^{2}\left (f x +e \right )\right )+2 a +2 b \right )}{3 a^{2} \left (b^{2} \left (\cos ^{4}\left (f x +e \right )\right )-2 a b \left (\cos ^{2}\left (f x +e \right )\right )-2 b^{2} \left (\cos ^{2}\left (f x +e \right )\right )+a^{2}+2 a b +b^{2}\right ) f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 107, normalized size = 1.47 \[ \frac {\frac {2 \, \sin \left (f x + e\right )}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a^{2}} + \frac {\sin \left (f x + e\right )}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} a} + \frac {\sin \left (f x + e\right )}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} b} - \frac {\sin \left (f x + e\right )}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a b}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 23.81, size = 183, normalized size = 2.51 \[ -\frac {2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )\,\sqrt {a+b\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,\left (a\,1{}\mathrm {i}-b\,2{}\mathrm {i}+a\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,10{}\mathrm {i}+a\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,1{}\mathrm {i}+b\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,4{}\mathrm {i}-b\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,2{}\mathrm {i}\right )}{3\,a^2\,f\,{\left (b-4\,a\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-2\,b\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+b\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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