Optimal. Leaf size=16 \[ -A \tanh ^{-1}\left (\frac {A \tan (z)}{B}\right )-B z \]
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Rubi [A] time = 0.09, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {12, 3191, 391, 203, 206} \[ -A \tanh ^{-1}\left (\frac {A \tan (z)}{B}\right )-B z \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 206
Rule 391
Rule 3191
Rubi steps
\begin {align*} \int \frac {\left (-A^2-B^2\right ) \cos ^2(z)}{B \left (1-\frac {\left (A^2+B^2\right ) \sin ^2(z)}{B^2}\right )} \, dz &=-\frac {\left (A^2+B^2\right ) \int \frac {\cos ^2(z)}{1-\frac {\left (A^2+B^2\right ) \sin ^2(z)}{B^2}} \, dz}{B}\\ &=-\frac {\left (A^2+B^2\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+z^2\right ) \left (1+\left (1-\frac {A^2+B^2}{B^2}\right ) z^2\right )} \, dz,z,\tan (z)\right )}{B}\\ &=-\frac {A^2 \operatorname {Subst}\left (\int \frac {1}{1+\left (1-\frac {A^2+B^2}{B^2}\right ) z^2} \, dz,z,\tan (z)\right )}{B}-B \operatorname {Subst}\left (\int \frac {1}{1+z^2} \, dz,z,\tan (z)\right )\\ &=-B z-A \tanh ^{-1}\left (\frac {A \tan (z)}{B}\right )\\ \end {align*}
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Mathematica [B] time = 0.10, size = 35, normalized size = 2.19 \[ -\frac {B \left (A^2+B^2\right ) \left (A \tanh ^{-1}\left (\frac {A \tan (z)}{B}\right )+B z\right )}{A^2 B+B^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 67, normalized size = 4.19 \[ -B z - \frac {1}{4} \, A \log \left (2 \, A B \cos \relax (z) \sin \relax (z) - {\left (A^{2} - B^{2}\right )} \cos \relax (z)^{2} + A^{2}\right ) + \frac {1}{4} \, A \log \left (-2 \, A B \cos \relax (z) \sin \relax (z) - {\left (A^{2} - B^{2}\right )} \cos \relax (z)^{2} + A^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.23, size = 83, normalized size = 5.19 \[ -\frac {{\left (\frac {A^{3} B \log \left ({\left | A \tan \relax (z) + B \right |}\right )}{A^{4} + A^{2} B^{2}} - \frac {A^{3} B \log \left ({\left | A \tan \relax (z) - B \right |}\right )}{A^{4} + A^{2} B^{2}} + \frac {2 \, B^{2} z}{A^{2} + B^{2}}\right )} {\left (A^{2} + B^{2}\right )}}{2 \, B} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 127, normalized size = 7.94 \[ \frac {A^{3} \ln \left (A \tan \relax (z )-B \right )}{2 A^{2}+2 B^{2}}-\frac {A^{3} \ln \left (A \tan \relax (z )+B \right )}{2 \left (A^{2}+B^{2}\right )}-\frac {A^{2} B \arctan \left (\tan \relax (z )\right )}{A^{2}+B^{2}}+\frac {A \,B^{2} \ln \left (A \tan \relax (z )-B \right )}{2 A^{2}+2 B^{2}}-\frac {A \,B^{2} \ln \left (A \tan \relax (z )+B \right )}{2 \left (A^{2}+B^{2}\right )}-\frac {B^{3} \arctan \left (\tan \relax (z )\right )}{A^{2}+B^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.51, size = 69, normalized size = 4.31 \[ -\frac {{\left (A^{2} + B^{2}\right )} {\left (\frac {2 \, B^{2} z}{A^{2} + B^{2}} + \frac {A B \log \left (A \tan \relax (z) + B\right )}{A^{2} + B^{2}} - \frac {A B \log \left (A \tan \relax (z) - B\right )}{A^{2} + B^{2}}\right )}}{2 \, B} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.50, size = 360, normalized size = 22.50 \[ -A\,\mathrm {atanh}\left (\frac {2\,A^{13}\,\mathrm {tan}\relax (z)}{2\,A^{12}\,B+6\,A^{10}\,B^3+6\,A^8\,B^5+2\,A^6\,B^7}+\frac {2\,A^7\,B^6\,\mathrm {tan}\relax (z)}{2\,A^{12}\,B+6\,A^{10}\,B^3+6\,A^8\,B^5+2\,A^6\,B^7}+\frac {6\,A^9\,B^4\,\mathrm {tan}\relax (z)}{2\,A^{12}\,B+6\,A^{10}\,B^3+6\,A^8\,B^5+2\,A^6\,B^7}+\frac {6\,A^{11}\,B^2\,\mathrm {tan}\relax (z)}{2\,A^{12}\,B+6\,A^{10}\,B^3+6\,A^8\,B^5+2\,A^6\,B^7}\right )-B\,\mathrm {atan}\left (\frac {2\,A^4\,B^9\,\mathrm {tan}\relax (z)}{2\,A^{10}\,B^3+6\,A^8\,B^5+6\,A^6\,B^7+2\,A^4\,B^9}+\frac {6\,A^6\,B^7\,\mathrm {tan}\relax (z)}{2\,A^{10}\,B^3+6\,A^8\,B^5+6\,A^6\,B^7+2\,A^4\,B^9}+\frac {6\,A^8\,B^5\,\mathrm {tan}\relax (z)}{2\,A^{10}\,B^3+6\,A^8\,B^5+6\,A^6\,B^7+2\,A^4\,B^9}+\frac {2\,A^{10}\,B^3\,\mathrm {tan}\relax (z)}{2\,A^{10}\,B^3+6\,A^8\,B^5+6\,A^6\,B^7+2\,A^4\,B^9}\right ) \]
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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