Optimal. Leaf size=121 \[ -\frac {b \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{2 f}+\frac {(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{f}-\frac {\sqrt {b} (3 a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{2 f} \]
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Rubi [A] time = 0.14, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3190, 416, 523, 217, 206, 377} \[ -\frac {b \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{2 f}+\frac {(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{f}-\frac {\sqrt {b} (3 a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{2 f} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 377
Rule 416
Rule 523
Rule 3190
Rubi steps
\begin {align*} \int \sec (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}}{1-x^2} \, dx,x,\sin (e+f x)\right )}{f}\\ &=-\frac {b \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{2 f}-\frac {\operatorname {Subst}\left (\int \frac {-a (2 a+b)-b (3 a+2 b) x^2}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{2 f}\\ &=-\frac {b \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{2 f}+\frac {(a+b)^2 \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{f}-\frac {(b (3 a+2 b)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{2 f}\\ &=-\frac {b \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{2 f}+\frac {(a+b)^2 \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 )}{f}-\frac {(b (3 a+2 b)) \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 )}{2 f}\\ &=-\frac {\sqrt {b} (3 a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{2 f}+\frac {(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{f}-\frac {b \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{2 f}\\ \end {align*}
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Mathematica [A] time = 0.67, size = 233, normalized size = 1.93 \[ \frac {\sqrt {2} \left (4 a^2+5 a b+2 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2 a+2 b} \sin (e+f x)}{\sqrt {2 a-b \cos (2 (e+f x))+b}}\right )-2 \sqrt {b} \sqrt {a+b} \left (\sqrt {b} \sin (e+f x) \sqrt {2 a-b \cos (2 (e+f x))+b}+\sqrt {2} (3 a+2 b) \log \left (\sqrt {2 a-b \cos (2 (e+f x))+b}+\sqrt {2} \sqrt {b} \sin (e+f x)\right )\right )+\sqrt {2} b (3 a+2 b) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a+b} \sin (e+f x)}{\sqrt {2 a-b \cos (2 (e+f x))+b}}\right )}{4 \sqrt {2} f \sqrt {a+b}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.95, size = 1381, normalized size = 11.41 \[ \text {result too large to display} \]
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 \[ \int {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \sec \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.71, size = 451, normalized size = 3.73 \[ -\frac {b^{\frac {3}{2}} \ln \left (\sin \left (f x +e \right ) \sqrt {b}+\sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}\right )}{f}-\frac {b \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}\, \sin \left (f x +e \right )}{2 f}-\frac {3 a \sqrt {b}\, \ln \left (\sin \left (f x +e \right ) \sqrt {b}+\sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}\right )}{2 f}+\frac {\ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}+2 b \sin \left (f x +e \right )+2 a}{\sin \left (f x +e \right )-1}\right ) a^{2}}{2 \sqrt {a +b}\, f}+\frac {\ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}+2 b \sin \left (f x +e \right )+2 a}{\sin \left (f x +e \right )-1}\right ) a b}{\sqrt {a +b}\, f}+\frac {\ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}+2 b \sin \left (f x +e \right )+2 a}{\sin \left (f x +e \right )-1}\right ) b^{2}}{2 \sqrt {a +b}\, f}-\frac {\ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}-2 b \sin \left (f x +e \right )+2 a}{1+\sin \left (f x +e \right )}\right ) a^{2}}{2 \sqrt {a +b}\, f}-\frac {\ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}-2 b \sin \left (f x +e \right )+2 a}{1+\sin \left (f x +e \right )}\right ) a b}{\sqrt {a +b}\, f}-\frac {\ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}-2 b \sin \left (f x +e \right )+2 a}{1+\sin \left (f x +e \right )}\right ) b^{2}}{2 \sqrt {a +b}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 168, normalized size = 1.39 \[ -\frac {3 \, a \sqrt {b} \operatorname {arsinh}\left (\frac {b \sin \left (f x + e\right )}{\sqrt {a b}}\right ) + 2 \, b^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {b \sin \left (f x + e\right )}{\sqrt {a b}}\right ) - {\left (a + b\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {b \sin \left (f x + e\right )}{\sqrt {a b} {\left (\sin \left (f x + e\right ) + 1\right )}} - \frac {a}{\sqrt {a b} {\left (\sin \left (f x + e\right ) + 1\right )}}\right ) - {\left (a + b\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (-\frac {b \sin \left (f x + e\right )}{\sqrt {a b} {\left (\sin \left (f x + e\right ) - 1\right )}} - \frac {a}{\sqrt {a b} {\left (\sin \left (f x + e\right ) - 1\right )}}\right ) + \sqrt {b \sin \left (f x + e\right )^{2} + a} b \sin \left (f x + e\right )}{2 \, f} \]
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 \[ \int \frac {{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2}}{\cos \left (e+f\,x\right )} \,d x \]
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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