Optimal. Leaf size=122 \[ -\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a-b \cos ^2(e+f x)+b}}\right )}{f}-\frac {b \cos (e+f x) \sqrt {a-b \cos ^2(e+f x)+b}}{2 f}-\frac {\sqrt {b} (3 a+b) \tan ^{-1}\left (\frac {\sqrt {b} \cos (e+f x)}{\sqrt {a-b \cos ^2(e+f x)+b}}\right )}{2 f} \]
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Rubi [A] time = 0.14, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3186, 416, 523, 217, 203, 377, 206} \[ -\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a-b \cos ^2(e+f x)+b}}\right )}{f}-\frac {b \cos (e+f x) \sqrt {a-b \cos ^2(e+f x)+b}}{2 f}-\frac {\sqrt {b} (3 a+b) \tan ^{-1}\left (\frac {\sqrt {b} \cos (e+f x)}{\sqrt {a-b \cos ^2(e+f x)+b}}\right )}{2 f} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 217
Rule 377
Rule 416
Rule 523
Rule 3186
Rubi steps
\begin {align*} \int \csc (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (a+b-b x^2\right )^{3/2}}{1-x^2} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac {b \cos (e+f x) \sqrt {a+b-b \cos ^2(e+f x)}}{2 f}+\frac {\operatorname {Subst}\left (\int \frac {-(a+b) (2 a+b)+b (3 a+b) x^2}{\left (1-x^2\right ) \sqrt {a+b-b x^2}} \, dx,x,\cos (e+f x)\right )}{2 f}\\ &=-\frac {b \cos (e+f x) \sqrt {a+b-b \cos ^2(e+f x)}}{2 f}-\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b-b x^2}} \, dx,x,\cos (e+f x)\right )}{f}-\frac {(b (3 a+b)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b-b x^2}} \, dx,x,\cos (e+f x)\right )}{2 f}\\ &=-\frac {b \cos (e+f x) \sqrt {a+b-b \cos ^2(e+f x)}}{2 f}-\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\cos (e+f x)}{\sqrt {a+b-b \cos ^2(e+f x)}}\right )}{f}-\frac {(b (3 a+b)) \operatorname {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\cos (e+f x)}{\sqrt {a+b-b \cos ^2(e+f x)}}\right )}{2 f}\\ &=-\frac {\sqrt {b} (3 a+b) \tan ^{-1}\left (\frac {\sqrt {b} \cos (e+f x)}{\sqrt {a+b-b \cos ^2(e+f x)}}\right )}{2 f}-\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+b-b \cos ^2(e+f x)}}\right )}{f}-\frac {b \cos (e+f x) \sqrt {a+b-b \cos ^2(e+f x)}}{2 f}\\ \end {align*}
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Mathematica [A] time = 0.92, size = 141, normalized size = 1.16 \[ -\frac {4 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \cos (e+f x)}{\sqrt {2 a-b \cos (2 (e+f x))+b}}\right )+\sqrt {2} b \cos (e+f x) \sqrt {2 a-b \cos (2 (e+f x))+b}-2 \sqrt {-b} (3 a+b) \log \left (\sqrt {2 a-b \cos (2 (e+f x))+b}+\sqrt {2} \sqrt {-b} \cos (e+f x)\right )}{4 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.88, size = 1282, normalized size = 10.51 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.08, size = 255, normalized size = 2.09 \[ \frac {\sqrt {\left (\cos ^{2}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}\, \left (b^{\frac {3}{2}} \arctan \left (\frac {-2 b \left (\cos ^{2}\left (f x +e \right )\right )+a +b}{2 \sqrt {b}\, \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}\right )-2 a^{\frac {3}{2}} \ln \left (\frac {-\left (a -b \right ) \left (\cos ^{2}\left (f x +e \right )\right )-2 \sqrt {a}\, \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}-a -b}{-1+\cos ^{2}\left (f x +e \right )}\right )+3 \sqrt {b}\, a \arctan \left (\frac {-2 b \left (\cos ^{2}\left (f x +e \right )\right )+a +b}{2 \sqrt {b}\, \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}\right )-2 b \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\right )}{4 \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 179, normalized size = 1.47 \[ -\frac {3 \, a \sqrt {b} \arcsin \left (\frac {b \cos \left (f x + e\right )}{\sqrt {a b + b^{2}}}\right ) + b^{\frac {3}{2}} \arcsin \left (\frac {b \cos \left (f x + e\right )}{\sqrt {a b + b^{2}}}\right ) + \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} b \cos \left (f x + e\right ) + a^{\frac {3}{2}} \log \left (b - \frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a}}{\cos \left (f x + e\right ) - 1} - \frac {a}{\cos \left (f x + e\right ) - 1}\right ) - a^{\frac {3}{2}} \log \left (-b + \frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a}}{\cos \left (f x + e\right ) + 1} + \frac {a}{\cos \left (f x + e\right ) + 1}\right )}{2 \, f} \]
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 \[ \int \frac {{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2}}{\sin \left (e+f\,x\right )} \,d x \]
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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