Optimal. Leaf size=59 \[ \frac {b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a+b}}\right )}{a^2 \sqrt {a+b}}+\frac {(a-2 b) \tanh ^{-1}(\sin (x))}{2 a^2}+\frac {\tan (x) \sec (x)}{2 a} \]
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Rubi [A] time = 0.10, antiderivative size = 59, 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 = {3186, 414, 522, 206, 208} \[ \frac {b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a+b}}\right )}{a^2 \sqrt {a+b}}+\frac {(a-2 b) \tanh ^{-1}(\sin (x))}{2 a^2}+\frac {\tan (x) \sec (x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 206
Rule 208
Rule 414
Rule 522
Rule 3186
Rubi steps
\begin {align*} \int \frac {\sec ^3(x)}{a+b \cos ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2 \left (a+b-b x^2\right )} \, dx,x,\sin (x)\right )\\ &=\frac {\sec (x) \tan (x)}{2 a}+\frac {\operatorname {Subst}\left (\int \frac {a-b-b x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\sin (x)\right )}{2 a}\\ &=\frac {\sec (x) \tan (x)}{2 a}+\frac {(a-2 b) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (x)\right )}{2 a^2}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\sin (x)\right )}{a^2}\\ &=\frac {(a-2 b) \tanh ^{-1}(\sin (x))}{2 a^2}+\frac {b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a+b}}\right )}{a^2 \sqrt {a+b}}+\frac {\sec (x) \tan (x)}{2 a}\\ \end {align*}
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Mathematica [B] time = 0.39, size = 152, normalized size = 2.58 \[ \frac {-\frac {2 b^{3/2} \log \left (\sqrt {a+b}-\sqrt {b} \sin (x)\right )}{\sqrt {a+b}}+\frac {2 b^{3/2} \log \left (\sqrt {a+b}+\sqrt {b} \sin (x)\right )}{\sqrt {a+b}}-2 (a-2 b) \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+2 (a-2 b) \log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )+\frac {a}{\left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^2}-\frac {a}{\left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^2}}{4 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 186, normalized size = 3.15 \[ \left [\frac {2 \, b \sqrt {\frac {b}{a + b}} \cos \relax (x)^{2} \log \left (-\frac {b \cos \relax (x)^{2} - 2 \, {\left (a + b\right )} \sqrt {\frac {b}{a + b}} \sin \relax (x) - a - 2 \, b}{b \cos \relax (x)^{2} + a}\right ) + {\left (a - 2 \, b\right )} \cos \relax (x)^{2} \log \left (\sin \relax (x) + 1\right ) - {\left (a - 2 \, b\right )} \cos \relax (x)^{2} \log \left (-\sin \relax (x) + 1\right ) + 2 \, a \sin \relax (x)}{4 \, a^{2} \cos \relax (x)^{2}}, -\frac {4 \, b \sqrt {-\frac {b}{a + b}} \arctan \left (\sqrt {-\frac {b}{a + b}} \sin \relax (x)\right ) \cos \relax (x)^{2} - {\left (a - 2 \, b\right )} \cos \relax (x)^{2} \log \left (\sin \relax (x) + 1\right ) + {\left (a - 2 \, b\right )} \cos \relax (x)^{2} \log \left (-\sin \relax (x) + 1\right ) - 2 \, a \sin \relax (x)}{4 \, a^{2} \cos \relax (x)^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 85, normalized size = 1.44 \[ -\frac {b^{2} \arctan \left (\frac {b \sin \relax (x)}{\sqrt {-a b - b^{2}}}\right )}{\sqrt {-a b - b^{2}} a^{2}} + \frac {{\left (a - 2 \, b\right )} \log \left (\sin \relax (x) + 1\right )}{4 \, a^{2}} - \frac {{\left (a - 2 \, b\right )} \log \left (-\sin \relax (x) + 1\right )}{4 \, a^{2}} - \frac {\sin \relax (x)}{2 \, {\left (\sin \relax (x)^{2} - 1\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 92, normalized size = 1.56 \[ \frac {b^{2} \arctanh \left (\frac {\sin \relax (x ) b}{\sqrt {\left (a +b \right ) b}}\right )}{a^{2} \sqrt {\left (a +b \right ) b}}-\frac {1}{4 a \left (\sin \relax (x )-1\right )}-\frac {\ln \left (\sin \relax (x )-1\right )}{4 a}+\frac {\ln \left (\sin \relax (x )-1\right ) b}{2 a^{2}}-\frac {1}{4 a \left (\sin \relax (x )+1\right )}+\frac {\ln \left (\sin \relax (x )+1\right )}{4 a}-\frac {\ln \left (\sin \relax (x )+1\right ) b}{2 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 92, normalized size = 1.56 \[ -\frac {b^{2} \log \left (\frac {b \sin \relax (x) - \sqrt {{\left (a + b\right )} b}}{b \sin \relax (x) + \sqrt {{\left (a + b\right )} b}}\right )}{2 \, \sqrt {{\left (a + b\right )} b} a^{2}} + \frac {{\left (a - 2 \, b\right )} \log \left (\sin \relax (x) + 1\right )}{4 \, a^{2}} - \frac {{\left (a - 2 \, b\right )} \log \left (\sin \relax (x) - 1\right )}{4 \, a^{2}} - \frac {\sin \relax (x)}{2 \, {\left (a \sin \relax (x)^{2} - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.53, size = 483, normalized size = 8.19 \[ -\frac {a^2\,\sin \relax (x)+a^2\,\mathrm {atanh}\left (\sin \relax (x)\right )-2\,b^2\,\mathrm {atanh}\left (\sin \relax (x)\right )+a\,b\,\sin \relax (x)-a\,b\,\mathrm {atanh}\left (\sin \relax (x)\right )-a^2\,\mathrm {atanh}\left (\sin \relax (x)\right )\,{\sin \relax (x)}^2+2\,b^2\,\mathrm {atanh}\left (\sin \relax (x)\right )\,{\sin \relax (x)}^2+a\,b\,\mathrm {atanh}\left (\sin \relax (x)\right )\,{\sin \relax (x)}^2+\mathrm {atan}\left (\frac {b^5\,\sin \relax (x)\,\sqrt {b^4+a\,b^3}\,8{}\mathrm {i}-a\,\sin \relax (x)\,{\left (b^4+a\,b^3\right )}^{3/2}\,4{}\mathrm {i}-b\,\sin \relax (x)\,{\left (b^4+a\,b^3\right )}^{3/2}\,8{}\mathrm {i}+a\,b^4\,\sin \relax (x)\,\sqrt {b^4+a\,b^3}\,12{}\mathrm {i}+a^4\,b\,\sin \relax (x)\,\sqrt {b^4+a\,b^3}\,1{}\mathrm {i}+a^2\,b^3\,\sin \relax (x)\,\sqrt {b^4+a\,b^3}\,1{}\mathrm {i}-a^3\,b^2\,\sin \relax (x)\,\sqrt {b^4+a\,b^3}\,2{}\mathrm {i}}{-a^5\,b^2+a^4\,b^3+5\,a^3\,b^4+3\,a^2\,b^5}\right )\,\sqrt {b^4+a\,b^3}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {b^5\,\sin \relax (x)\,\sqrt {b^4+a\,b^3}\,8{}\mathrm {i}-a\,\sin \relax (x)\,{\left (b^4+a\,b^3\right )}^{3/2}\,4{}\mathrm {i}-b\,\sin \relax (x)\,{\left (b^4+a\,b^3\right )}^{3/2}\,8{}\mathrm {i}+a\,b^4\,\sin \relax (x)\,\sqrt {b^4+a\,b^3}\,12{}\mathrm {i}+a^4\,b\,\sin \relax (x)\,\sqrt {b^4+a\,b^3}\,1{}\mathrm {i}+a^2\,b^3\,\sin \relax (x)\,\sqrt {b^4+a\,b^3}\,1{}\mathrm {i}-a^3\,b^2\,\sin \relax (x)\,\sqrt {b^4+a\,b^3}\,2{}\mathrm {i}}{-a^5\,b^2+a^4\,b^3+5\,a^3\,b^4+3\,a^2\,b^5}\right )\,{\sin \relax (x)}^2\,\sqrt {b^4+a\,b^3}\,2{}\mathrm {i}}{2\,a^3\,{\sin \relax (x)}^2-2\,a^3+2\,b\,a^2\,{\sin \relax (x)}^2-2\,b\,a^2} \]
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 \[ \int \frac {\sec ^{3}{\relax (x )}}{a + b \cos ^{2}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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