Optimal. Leaf size=392 \[ \frac {b (A b-a B) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}-\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\sqrt {b} \left (-3 a^3 B+5 a^2 A b+a b^2 B+A b^3\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{3/2} d \left (a^2+b^2\right )^2} \]
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Rubi [A] time = 0.94, antiderivative size = 392, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.394, Rules used = {3581, 3605, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ \frac {b (A b-a B) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}-\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (a^2 (-(A+B))+2 a b (A-B)+b^2 (A+B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\left (a^2 (-(A+B))+2 a b (A-B)+b^2 (A+B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\sqrt {b} \left (5 a^2 A b-3 a^3 B+a b^2 B+A b^3\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{3/2} d \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 63
Rule 204
Rule 205
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rule 3534
Rule 3581
Rule 3605
Rule 3634
Rule 3653
Rubi steps
\begin {align*} \int \frac {\sqrt {\cot (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx &=\int \frac {\cot ^{\frac {3}{2}}(c+d x) (B+A \cot (c+d x))}{(b+a \cot (c+d x))^2} \, dx\\ &=\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac {\int \frac {-\frac {1}{2} b (A b-a B)+a (A b-a B) \cot (c+d x)-\frac {1}{2} \left (2 a^2 A+A b^2+a b B\right ) \cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{a \left (a^2+b^2\right )}\\ &=\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac {\int \frac {a \left (2 a A b-a^2 B+b^2 B\right )-a \left (a^2 A-A b^2+2 a b B\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{a \left (a^2+b^2\right )^2}+\frac {\left (b \left (5 a^2 A b+A b^3-3 a^3 B+a b^2 B\right )\right ) \int \frac {1+\cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{2 a \left (a^2+b^2\right )^2}\\ &=\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac {2 \operatorname {Subst}\left (\int \frac {-a \left (2 a A b-a^2 B+b^2 B\right )+a \left (a^2 A-A b^2+2 a b B\right ) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {\left (b \left (5 a^2 A b+A b^3-3 a^3 B+a b^2 B\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-x} (b-a x)} \, dx,x,-\cot (c+d x)\right )}{2 a \left (a^2+b^2\right )^2 d}\\ &=\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac {\left (b \left (5 a^2 A b+A b^3-3 a^3 B+a b^2 B\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}\\ &=-\frac {\sqrt {b} \left (5 a^2 A b+A b^3-3 a^3 B+a b^2 B\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{3/2} \left (a^2+b^2\right )^2 d}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d}+\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d}\\ &=-\frac {\sqrt {b} \left (5 a^2 A b+A b^3-3 a^3 B+a b^2 B\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{3/2} \left (a^2+b^2\right )^2 d}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}\\ &=-\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\sqrt {b} \left (5 a^2 A b+A b^3-3 a^3 B+a b^2 B\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{3/2} \left (a^2+b^2\right )^2 d}+\frac {b (A b-a B) \sqrt {\cot (c+d x)}}{a \left (a^2+b^2\right ) d (b+a \cot (c+d x))}-\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}\\ \end {align*}
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Mathematica [A] time = 2.70, size = 341, normalized size = 0.87 \[ \frac {\sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (2 \sqrt {2} \left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \left (\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )\right )+\frac {8 \sqrt {b} \left (a^2 (-B)+2 a A b+b^2 B\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {4 b \left (a^2+b^2\right ) (A b-a B) \sqrt {\tan (c+d x)}}{a (a+b \tan (c+d x))}-\sqrt {2} \left (a^2 (A-B)+2 a b (A+B)+b^2 (B-A)\right ) \left (\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )-\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )\right )+\frac {4 \sqrt {b} \left (a^2+b^2\right ) (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2}}\right )}{4 d \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \sqrt {\cot \left (d x + c\right )}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 5.76, size = 36065, normalized size = 92.00 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 362, normalized size = 0.92 \[ \frac {\frac {4 \, {\left (3 \, B a^{3} b - 5 \, A a^{2} b^{2} - B a b^{3} - A b^{4}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left ({\left (A + B\right )} a^{2} - 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left ({\left (A + B\right )} a^{2} - 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} {\left ({\left (A - B\right )} a^{2} + 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} {\left ({\left (A - B\right )} a^{2} + 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {4 \, {\left (B a b - A b^{2}\right )}}{{\left (a^{3} b + a b^{3} + \frac {a^{4} + a^{2} b^{2}}{\tan \left (d x + c\right )}\right )} \sqrt {\tan \left (d x + c\right )}}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {\mathrm {cot}\left (c+d\,x\right )}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )}{{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^2} \,d x \]
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 \[ \int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \sqrt {\cot {\left (c + d x \right )}}}{\left (a + b \tan {\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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