Optimal. Leaf size=61 \[ \frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^2}+\frac {(a+3 b) \tanh ^{-1}(\sin (x))}{2 (a+b)^2}+\frac {\sec (x) \tan (x)}{2 (a+b)} \]
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Rubi [A]
time = 0.06, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3269, 425, 536,
212, 211} \begin {gather*} \frac {b^{3/2} \text {ArcTan}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^2}+\frac {(a+3 b) \tanh ^{-1}(\sin (x))}{2 (a+b)^2}+\frac {\tan (x) \sec (x)}{2 (a+b)} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 212
Rule 425
Rule 536
Rule 3269
Rubi steps
\begin {align*} \int \frac {\sec ^3(x)}{a+b \sin ^2(x)} \, dx &=\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2 \left (a+b x^2\right )} \, dx,x,\sin (x)\right )\\ &=\frac {\sec (x) \tan (x)}{2 (a+b)}+\frac {\text {Subst}\left (\int \frac {a+2 b+b x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\sin (x)\right )}{2 (a+b)}\\ &=\frac {\sec (x) \tan (x)}{2 (a+b)}+\frac {b^2 \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sin (x)\right )}{(a+b)^2}+\frac {(a+3 b) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (x)\right )}{2 (a+b)^2}\\ &=\frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^2}+\frac {(a+3 b) \tanh ^{-1}(\sin (x))}{2 (a+b)^2}+\frac {\sec (x) \tan (x)}{2 (a+b)}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(147\) vs. \(2(61)=122\).
time = 0.24, size = 147, normalized size = 2.41 \begin {gather*} \frac {-\frac {2 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \csc (x)}{\sqrt {b}}\right )}{\sqrt {a}}+\frac {2 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a}}-2 (a+3 b) \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+2 (a+3 b) \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )+\frac {a+b}{\left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^2}-\frac {a+b}{\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2}}{4 (a+b)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 96, normalized size = 1.57
method | result | size |
default | \(\frac {b^{2} \arctan \left (\frac {b \sin \left (x \right )}{\sqrt {a b}}\right )}{\left (a +b \right )^{2} \sqrt {a b}}-\frac {1}{\left (4 a +4 b \right ) \left (\sin \left (x \right )-1\right )}+\frac {\left (-a -3 b \right ) \ln \left (\sin \left (x \right )-1\right )}{4 \left (a +b \right )^{2}}-\frac {1}{\left (4 a +4 b \right ) \left (1+\sin \left (x \right )\right )}+\frac {\left (a +3 b \right ) \ln \left (1+\sin \left (x \right )\right )}{4 \left (a +b \right )^{2}}\) | \(96\) |
risch | \(-\frac {i \left ({\mathrm e}^{3 i x}-{\mathrm e}^{i x}\right )}{\left ({\mathrm e}^{2 i x}+1\right )^{2} \left (a +b \right )}-\frac {\ln \left ({\mathrm e}^{i x}-i\right ) a}{2 \left (a^{2}+2 a b +b^{2}\right )}-\frac {3 \ln \left ({\mathrm e}^{i x}-i\right ) b}{2 \left (a^{2}+2 a b +b^{2}\right )}+\frac {\ln \left ({\mathrm e}^{i x}+i\right ) a}{2 a^{2}+4 a b +2 b^{2}}+\frac {3 \ln \left ({\mathrm e}^{i x}+i\right ) b}{2 \left (a^{2}+2 a b +b^{2}\right )}+\frac {\sqrt {-a b}\, b \ln \left ({\mathrm e}^{2 i x}+\frac {2 i \sqrt {-a b}\, {\mathrm e}^{i x}}{b}-1\right )}{2 a \left (a +b \right )^{2}}-\frac {\sqrt {-a b}\, b \ln \left ({\mathrm e}^{2 i x}-\frac {2 i \sqrt {-a b}\, {\mathrm e}^{i x}}{b}-1\right )}{2 a \left (a +b \right )^{2}}\) | \(216\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 104 vs.
\(2 (49) = 98\).
time = 0.48, size = 104, normalized size = 1.70 \begin {gather*} \frac {b^{2} \arctan \left (\frac {b \sin \left (x\right )}{\sqrt {a b}}\right )}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a b}} + \frac {{\left (a + 3 \, b\right )} \log \left (\sin \left (x\right ) + 1\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac {{\left (a + 3 \, b\right )} \log \left (\sin \left (x\right ) - 1\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac {\sin \left (x\right )}{2 \, {\left ({\left (a + b\right )} \sin \left (x\right )^{2} - a - b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.46, size = 203, normalized size = 3.33 \begin {gather*} \left [\frac {2 \, b \sqrt {-\frac {b}{a}} \cos \left (x\right )^{2} \log \left (-\frac {b \cos \left (x\right )^{2} - 2 \, a \sqrt {-\frac {b}{a}} \sin \left (x\right ) + a - b}{b \cos \left (x\right )^{2} - a - b}\right ) + {\left (a + 3 \, b\right )} \cos \left (x\right )^{2} \log \left (\sin \left (x\right ) + 1\right ) - {\left (a + 3 \, b\right )} \cos \left (x\right )^{2} \log \left (-\sin \left (x\right ) + 1\right ) + 2 \, {\left (a + b\right )} \sin \left (x\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (x\right )^{2}}, \frac {4 \, b \sqrt {\frac {b}{a}} \arctan \left (\sqrt {\frac {b}{a}} \sin \left (x\right )\right ) \cos \left (x\right )^{2} + {\left (a + 3 \, b\right )} \cos \left (x\right )^{2} \log \left (\sin \left (x\right ) + 1\right ) - {\left (a + 3 \, b\right )} \cos \left (x\right )^{2} \log \left (-\sin \left (x\right ) + 1\right ) + 2 \, {\left (a + b\right )} \sin \left (x\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (x\right )^{2}}\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 {\sec ^{3}{\left (x \right )}}{a + b \sin ^{2}{\left (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. 102 vs.
\(2 (49) = 98\).
time = 0.42, size = 102, normalized size = 1.67 \begin {gather*} \frac {b^{2} \arctan \left (\frac {b \sin \left (x\right )}{\sqrt {a b}}\right )}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a b}} + \frac {{\left (a + 3 \, b\right )} \log \left (\sin \left (x\right ) + 1\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac {{\left (a + 3 \, b\right )} \log \left (-\sin \left (x\right ) + 1\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac {\sin \left (x\right )}{2 \, {\left (\sin \left (x\right )^{2} - 1\right )} {\left (a + b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 15.36, size = 1139, normalized size = 18.67 \begin {gather*} \frac {\sin \left (x\right )}{2\,{\cos \left (x\right )}^2\,\left (a+b\right )}-\ln \left (\sin \left (x\right )-1\right )\,\left (\frac {b}{2\,{\left (a+b\right )}^2}+\frac {1}{4\,\left (a+b\right )}\right )+\frac {\ln \left (\sin \left (x\right )+1\right )\,\left (a+3\,b\right )}{4\,{\left (a+b\right )}^2}+\frac {\mathrm {atan}\left (\frac {\frac {\sqrt {-a\,b^3}\,\left (\frac {\sin \left (x\right )\,\left (a^2\,b^3+6\,a\,b^4+13\,b^5\right )}{4\,\left (a^2+2\,a\,b+b^2\right )}+\frac {\left (\frac {2\,a^5\,b^2+12\,a^4\,b^3+28\,a^3\,b^4+32\,a^2\,b^5+18\,a\,b^6+4\,b^7}{2\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}-\frac {\sin \left (x\right )\,\sqrt {-a\,b^3}\,\left (-16\,a^5\,b^2-48\,a^4\,b^3-32\,a^3\,b^4+32\,a^2\,b^5+48\,a\,b^6+16\,b^7\right )}{8\,\left (a^2+2\,a\,b+b^2\right )\,\left (a^3+2\,a^2\,b+a\,b^2\right )}\right )\,\sqrt {-a\,b^3}}{2\,\left (a^3+2\,a^2\,b+a\,b^2\right )}\right )\,1{}\mathrm {i}}{a^3+2\,a^2\,b+a\,b^2}+\frac {\sqrt {-a\,b^3}\,\left (\frac {\sin \left (x\right )\,\left (a^2\,b^3+6\,a\,b^4+13\,b^5\right )}{4\,\left (a^2+2\,a\,b+b^2\right )}-\frac {\left (\frac {2\,a^5\,b^2+12\,a^4\,b^3+28\,a^3\,b^4+32\,a^2\,b^5+18\,a\,b^6+4\,b^7}{2\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}+\frac {\sin \left (x\right )\,\sqrt {-a\,b^3}\,\left (-16\,a^5\,b^2-48\,a^4\,b^3-32\,a^3\,b^4+32\,a^2\,b^5+48\,a\,b^6+16\,b^7\right )}{8\,\left (a^2+2\,a\,b+b^2\right )\,\left (a^3+2\,a^2\,b+a\,b^2\right )}\right )\,\sqrt {-a\,b^3}}{2\,\left (a^3+2\,a^2\,b+a\,b^2\right )}\right )\,1{}\mathrm {i}}{a^3+2\,a^2\,b+a\,b^2}}{\frac {\frac {3\,b^5}{2}+\frac {a\,b^4}{2}}{a^3+3\,a^2\,b+3\,a\,b^2+b^3}-\frac {\sqrt {-a\,b^3}\,\left (\frac {\sin \left (x\right )\,\left (a^2\,b^3+6\,a\,b^4+13\,b^5\right )}{4\,\left (a^2+2\,a\,b+b^2\right )}+\frac {\left (\frac {2\,a^5\,b^2+12\,a^4\,b^3+28\,a^3\,b^4+32\,a^2\,b^5+18\,a\,b^6+4\,b^7}{2\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}-\frac {\sin \left (x\right )\,\sqrt {-a\,b^3}\,\left (-16\,a^5\,b^2-48\,a^4\,b^3-32\,a^3\,b^4+32\,a^2\,b^5+48\,a\,b^6+16\,b^7\right )}{8\,\left (a^2+2\,a\,b+b^2\right )\,\left (a^3+2\,a^2\,b+a\,b^2\right )}\right )\,\sqrt {-a\,b^3}}{2\,\left (a^3+2\,a^2\,b+a\,b^2\right )}\right )}{a^3+2\,a^2\,b+a\,b^2}+\frac {\sqrt {-a\,b^3}\,\left (\frac {\sin \left (x\right )\,\left (a^2\,b^3+6\,a\,b^4+13\,b^5\right )}{4\,\left (a^2+2\,a\,b+b^2\right )}-\frac {\left (\frac {2\,a^5\,b^2+12\,a^4\,b^3+28\,a^3\,b^4+32\,a^2\,b^5+18\,a\,b^6+4\,b^7}{2\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}+\frac {\sin \left (x\right )\,\sqrt {-a\,b^3}\,\left (-16\,a^5\,b^2-48\,a^4\,b^3-32\,a^3\,b^4+32\,a^2\,b^5+48\,a\,b^6+16\,b^7\right )}{8\,\left (a^2+2\,a\,b+b^2\right )\,\left (a^3+2\,a^2\,b+a\,b^2\right )}\right )\,\sqrt {-a\,b^3}}{2\,\left (a^3+2\,a^2\,b+a\,b^2\right )}\right )}{a^3+2\,a^2\,b+a\,b^2}}\right )\,\sqrt {-a\,b^3}\,1{}\mathrm {i}}{a^3+2\,a^2\,b+a\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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