Optimal. Leaf size=127 \[ \frac {b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{f}+\frac {(a-2 b) \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a+b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{2 f}+\frac {(a+b) \tan (e+f x) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{2 f} \]
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Rubi [A] time = 0.15, antiderivative size = 127, 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 = {3190, 413, 523, 217, 206, 377} \[ \frac {b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{f}+\frac {(a-2 b) \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a+b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{2 f}+\frac {(a+b) \tan (e+f x) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{2 f} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 377
Rule 413
Rule 523
Rule 3190
Rubi steps
\begin {align*} \int \sec ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^{3/2}}{\left (1-x^2\right )^2} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {(a+b) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{2 f}-\frac {\operatorname {Subst}\left (\int \frac {-a (a-b)+2 b^2 x^2}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{2 f}\\ &=\frac {(a+b) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{2 f}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{f}+\frac {((a-2 b) (a+b)) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{2 f}\\ &=\frac {(a+b) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{2 f}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{f}+\frac {((a-2 b) (a+b)) \operatorname {Subst}\left (\int \frac {1}{1-(a+b) x^2} \, dx,x,\frac {\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{2 f}\\ &=\frac {b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{f}+\frac {(a-2 b) \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a+b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{2 f}+\frac {(a+b) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{2 f}\\ \end {align*}
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Mathematica [A] time = 0.95, size = 210, normalized size = 1.65 \[ \frac {2 \left (a^2-a b-b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2 a+2 b} \sin (e+f x)}{\sqrt {2 a-b \cos (2 (e+f x))+b}}\right )+\sqrt {a+b} \left (4 b^{3/2} \log \left (\sqrt {2 a-b \cos (2 (e+f x))+b}+\sqrt {2} \sqrt {b} \sin (e+f x)\right )+\sqrt {2} (a+b) \tan (e+f x) \sec (e+f x) \sqrt {2 a-b \cos (2 (e+f x))+b}\right )-2 b^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a+b} \sin (e+f x)}{\sqrt {2 a-b \cos (2 (e+f x))+b}}\right )}{4 f \sqrt {a+b}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.04, size = 1471, normalized size = 11.58 \[ \text {result too large to display} \]
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 \[ \int {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \sec \left (f x + e\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.87, size = 402, normalized size = 3.17 \[ \frac {2 \sin \left (f x +e \right ) \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (a +b \right )^{\frac {5}{2}}-\left (-4 b^{\frac {3}{2}} \ln \left (\sin \left (f x +e \right ) \sqrt {b}+\sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}\right ) \left (a +b \right )^{\frac {3}{2}}-\ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}+2 b \sin \left (f x +e \right )+2 a}{\sin \left (f x +e \right )-1}\right ) a^{3}+3 \ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}+2 b \sin \left (f x +e \right )+2 a}{\sin \left (f x +e \right )-1}\right ) a \,b^{2}+2 \ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}+2 b \sin \left (f x +e \right )+2 a}{\sin \left (f x +e \right )-1}\right ) b^{3}+\ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}-2 b \sin \left (f x +e \right )+2 a}{1+\sin \left (f x +e \right )}\right ) a^{3}-3 \ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}-2 b \sin \left (f x +e \right )+2 a}{1+\sin \left (f x +e \right )}\right ) a \,b^{2}-2 \ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}-2 b \sin \left (f x +e \right )+2 a}{1+\sin \left (f x +e \right )}\right ) b^{3}\right ) \left (\cos ^{2}\left (f x +e \right )\right )}{4 \left (a +b \right )^{\frac {3}{2}} \cos \left (f x +e \right )^{2} f} \]
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 \[ \int {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \sec \left (f x + e\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2}}{{\cos \left (e+f\,x\right )}^3} \,d x \]
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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