Integrand size = 25, antiderivative size = 149 \[ \int \frac {\sqrt {\tan (c+d x)}}{(a+b \tan (c+d x))^{3/2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(i a-b)^{3/2} d}+\frac {\text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(i a+b)^{3/2} d}-\frac {2 b \sqrt {\tan (c+d x)}}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}} \]
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Time = 0.54 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3649, 3697, 3696, 95, 209, 212} \[ \int \frac {\sqrt {\tan (c+d x)}}{(a+b \tan (c+d x))^{3/2}} \, dx=-\frac {2 b \sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}-\frac {\arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d (-b+i a)^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d (b+i a)^{3/2}} \]
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Rule 95
Rule 209
Rule 212
Rule 3649
Rule 3696
Rule 3697
Rubi steps \begin{align*} \text {integral}& = -\frac {2 b \sqrt {\tan (c+d x)}}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}-\frac {2 \int \frac {-\frac {b}{2}-\frac {1}{2} a \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{a^2+b^2} \\ & = -\frac {2 b \sqrt {\tan (c+d x)}}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}-\frac {\int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{2 (i a-b)}+\frac {\int \frac {1+i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{2 (i a+b)} \\ & = -\frac {2 b \sqrt {\tan (c+d x)}}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}-\frac {\text {Subst}\left (\int \frac {1}{(1+i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 (i a-b) d}+\frac {\text {Subst}\left (\int \frac {1}{(1-i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 (i a+b) d} \\ & = -\frac {2 b \sqrt {\tan (c+d x)}}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}-\frac {\text {Subst}\left (\int \frac {1}{1-(-i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(i a-b) d}+\frac {\text {Subst}\left (\int \frac {1}{1-(i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(i a+b) d} \\ & = -\frac {\arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(i a-b)^{3/2} d}+\frac {\text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(i a+b)^{3/2} d}-\frac {2 b \sqrt {\tan (c+d x)}}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}} \\ \end{align*}
Time = 1.30 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {\tan (c+d x)}}{(a+b \tan (c+d x))^{3/2}} \, dx=\frac {\frac {\sqrt [4]{-1} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(-a+i b)^{3/2}}+\frac {\sqrt [4]{-1} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(a+i b)^{3/2}}-\frac {2 b \sqrt {\tan (c+d x)}}{\left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{d} \]
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result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 0.93 (sec) , antiderivative size = 798213, normalized size of antiderivative = 5357.13
\[\text {output too large to display}\]
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Leaf count of result is larger than twice the leaf count of optimal. 7445 vs. \(2 (121) = 242\).
Time = 1.39 (sec) , antiderivative size = 7445, normalized size of antiderivative = 49.97 \[ \int \frac {\sqrt {\tan (c+d x)}}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {\sqrt {\tan (c+d x)}}{(a+b \tan (c+d x))^{3/2}} \, dx=\int \frac {\sqrt {\tan {\left (c + d x \right )}}}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\sqrt {\tan (c+d x)}}{(a+b \tan (c+d x))^{3/2}} \, dx=\int { \frac {\sqrt {\tan \left (d x + c\right )}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {\tan (c+d x)}}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\sqrt {\tan (c+d x)}}{(a+b \tan (c+d x))^{3/2}} \, dx=\int \frac {\sqrt {\mathrm {tan}\left (c+d\,x\right )}}{{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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