Optimal. Leaf size=43 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)}}\right )}{\sqrt {a+b} f} \]
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Rubi [A]
time = 0.05, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4219, 385, 213}
\begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)}}\right )}{f \sqrt {a+b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 385
Rule 4219
Rubi steps
\begin {align*} \int \frac {\csc (e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac {\text {Subst}\left (\int \frac {1}{-1-(-a-b) x^2} \, dx,x,\frac {\sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)}}\right )}{f}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)}}\right )}{\sqrt {a+b} f}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 86, normalized size = 2.00 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b-a \sin ^2(e+f x)}}{\sqrt {a+b}}\right ) \sqrt {a+2 b+a \cos (2 e+2 f x)} \sec (e+f x)}{\sqrt {2} \sqrt {a+b} f \sqrt {a+b \sec ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(279\) vs.
\(2(37)=74\).
time = 0.15, size = 280, normalized size = 6.51
method | result | size |
default | \(\frac {\sqrt {\frac {b +a \left (\cos ^{2}\left (f x +e \right )\right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \left (\ln \left (-\frac {2 \left (\cos \left (f x +e \right )-1\right ) \left (\sqrt {\frac {b +a \left (\cos ^{2}\left (f x +e \right )\right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \cos \left (f x +e \right ) \sqrt {a +b}+\sqrt {\frac {b +a \left (\cos ^{2}\left (f x +e \right )\right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \sqrt {a +b}-\cos \left (f x +e \right ) a +b \right )}{\sin \left (f x +e \right )^{2} \sqrt {a +b}}\right )+\ln \left (-\frac {4 \left (\sqrt {\frac {b +a \left (\cos ^{2}\left (f x +e \right )\right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \cos \left (f x +e \right ) \sqrt {a +b}+\cos \left (f x +e \right ) a +\sqrt {\frac {b +a \left (\cos ^{2}\left (f x +e \right )\right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \sqrt {a +b}+b \right )}{\cos \left (f x +e \right )-1}\right )\right ) \left (\sin ^{2}\left (f x +e \right )\right )}{2 f \sqrt {\frac {b +a \left (\cos ^{2}\left (f x +e \right )\right )}{\cos \left (f x +e \right )^{2}}}\, \cos \left (f x +e \right ) \left (\cos \left (f x +e \right )-1\right ) \sqrt {a +b}}\) | \(280\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.69, size = 148, normalized size = 3.44 \begin {gather*} \left [\frac {\log \left (\frac {2 \, {\left (a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {a + b} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right ) + a + 2 \, b\right )}}{\cos \left (f x + e\right )^{2} - 1}\right )}{2 \, \sqrt {a + b} f}, \frac {\sqrt {-a - b} \arctan \left (\frac {\sqrt {-a - b} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a + b}\right )}{{\left (a + b\right )} f}\right ] \end {gather*}
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 \begin {gather*} \int \frac {\csc {\left (e + f x \right )}}{\sqrt {a + b \sec ^{2}{\left (e + f x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 305 vs.
\(2 (37) = 74\).
time = 0.79, size = 305, normalized size = 7.09 \begin {gather*} -\frac {\frac {\log \left ({\left | -\sqrt {a + b} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + \sqrt {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a + b} + \sqrt {a + b} \right |}\right )}{\sqrt {a + b}} - \frac {\log \left ({\left | -\sqrt {a + b} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + \sqrt {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a + b} - \sqrt {a + b} \right |}\right )}{\sqrt {a + b}} - \frac {\log \left ({\left | -{\left (\sqrt {a + b} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a + b}\right )} {\left (a + b\right )} + \sqrt {a + b} {\left (a - b\right )} \right |}\right )}{\sqrt {a + b}}}{2 \, f \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sin \left (e+f\,x\right )\,\sqrt {a+\frac {b}{{\cos \left (e+f\,x\right )}^2}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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