Optimal. Leaf size=61 \[ \frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^2}+\frac {(a+3 b) \tanh ^{-1}(\sin (x))}{2 (a+b)^2}+\frac {\tan (x) \sec (x)}{2 (a+b)} \]
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Rubi [A] time = 0.09, 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 = {3190, 414, 522, 206, 205} \[ \frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^2}+\frac {(a+3 b) \tanh ^{-1}(\sin (x))}{2 (a+b)^2}+\frac {\tan (x) \sec (x)}{2 (a+b)} \]
Antiderivative was successfully verified.
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Rule 205
Rule 206
Rule 414
Rule 522
Rule 3190
Rubi steps
\begin {align*} \int \frac {\sec ^3(x)}{a+b \sin ^2(x)} \, dx &=\operatorname {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 {\operatorname {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 \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sin (x)\right )}{(a+b)^2}+\frac {(a+3 b) \operatorname {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] time = 0.32, size = 147, normalized size = 2.41 \[ \frac {\frac {2 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {2 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \csc (x)}{\sqrt {b}}\right )}{\sqrt {a}}+\frac {a+b}{\left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^2}-\frac {a+b}{\left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^2}-2 (a+3 b) \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+2 (a+3 b) \log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )}{4 (a+b)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 203, normalized size = 3.33 \[ \left [\frac {2 \, b \sqrt {-\frac {b}{a}} \cos \relax (x)^{2} \log \left (-\frac {b \cos \relax (x)^{2} - 2 \, a \sqrt {-\frac {b}{a}} \sin \relax (x) + a - b}{b \cos \relax (x)^{2} - a - b}\right ) + {\left (a + 3 \, b\right )} \cos \relax (x)^{2} \log \left (\sin \relax (x) + 1\right ) - {\left (a + 3 \, b\right )} \cos \relax (x)^{2} \log \left (-\sin \relax (x) + 1\right ) + 2 \, {\left (a + b\right )} \sin \relax (x)}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \relax (x)^{2}}, \frac {4 \, b \sqrt {\frac {b}{a}} \arctan \left (\sqrt {\frac {b}{a}} \sin \relax (x)\right ) \cos \relax (x)^{2} + {\left (a + 3 \, b\right )} \cos \relax (x)^{2} \log \left (\sin \relax (x) + 1\right ) - {\left (a + 3 \, b\right )} \cos \relax (x)^{2} \log \left (-\sin \relax (x) + 1\right ) + 2 \, {\left (a + b\right )} \sin \relax (x)}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \relax (x)^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 102, normalized size = 1.67 \[ \frac {b^{2} \arctan \left (\frac {b \sin \relax (x)}{\sqrt {a b}}\right )}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a b}} + \frac {{\left (a + 3 \, b\right )} \log \left (\sin \relax (x) + 1\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac {{\left (a + 3 \, b\right )} \log \left (-\sin \relax (x) + 1\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac {\sin \relax (x)}{2 \, {\left (\sin \relax (x)^{2} - 1\right )} {\left (a + b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.24, size = 112, normalized size = 1.84 \[ -\frac {1}{\left (4 a +4 b \right ) \left (-1+\sin \relax (x )\right )}-\frac {\ln \left (-1+\sin \relax (x )\right ) a}{4 \left (a +b \right )^{2}}-\frac {3 \ln \left (-1+\sin \relax (x )\right ) b}{4 \left (a +b \right )^{2}}+\frac {b^{2} \arctan \left (\frac {\sin \relax (x ) b}{\sqrt {a b}}\right )}{\left (a +b \right )^{2} \sqrt {a b}}-\frac {1}{\left (4 a +4 b \right ) \left (1+\sin \relax (x )\right )}+\frac {\ln \left (1+\sin \relax (x )\right ) a}{4 \left (a +b \right )^{2}}+\frac {3 \ln \left (1+\sin \relax (x )\right ) b}{4 \left (a +b \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.59, size = 104, normalized size = 1.70 \[ \frac {b^{2} \arctan \left (\frac {b \sin \relax (x)}{\sqrt {a b}}\right )}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a b}} + \frac {{\left (a + 3 \, b\right )} \log \left (\sin \relax (x) + 1\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac {{\left (a + 3 \, b\right )} \log \left (\sin \relax (x) - 1\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac {\sin \relax (x)}{2 \, {\left ({\left (a + b\right )} \sin \relax (x)^{2} - a - b\right )}} \]
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 \[ \frac {\sin \relax (x)}{2\,{\cos \relax (x)}^2\,\left (a+b\right )}-\ln \left (\sin \relax (x)-1\right )\,\left (\frac {b}{2\,{\left (a+b\right )}^2}+\frac {1}{4\,\left (a+b\right )}\right )+\frac {\ln \left (\sin \relax (x)+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 \relax (x)\,\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 \relax (x)\,\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 \relax (x)\,\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 \relax (x)\,\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 \relax (x)\,\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 \relax (x)\,\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 \relax (x)\,\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 \relax (x)\,\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} \]
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 \[ \int \frac {\sec ^{3}{\relax (x )}}{a + b \sin ^{2}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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