Integrand size = 25, antiderivative size = 317 \[ \int \frac {\tan ^{\frac {9}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=-\frac {i \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(i a-b)^{5/2} d}-\frac {5 a \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{b^{7/2} d}+\frac {i \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(i a+b)^{5/2} d}-\frac {2 a^2 \tan ^{\frac {5}{2}}(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 a^2 \left (5 a^2+11 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {\left (5 a^4+10 a^2 b^2+b^4\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{b^3 \left (a^2+b^2\right )^2 d} \]
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Time = 2.70 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3646, 3726, 3728, 3736, 6857, 65, 223, 212, 95, 211, 214} \[ \int \frac {\tan ^{\frac {9}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=-\frac {2 a^2 \tan ^{\frac {5}{2}}(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {2 a^2 \left (5 a^2+11 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b^2 d \left (a^2+b^2\right )^2 \sqrt {a+b \tan (c+d x)}}+\frac {\left (5 a^4+10 a^2 b^2+b^4\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{b^3 d \left (a^2+b^2\right )^2}-\frac {i \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d (-b+i a)^{5/2}}-\frac {5 a \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{b^{7/2} d}+\frac {i \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d (b+i a)^{5/2}} \]
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Rule 65
Rule 95
Rule 211
Rule 212
Rule 214
Rule 223
Rule 3646
Rule 3726
Rule 3728
Rule 3736
Rule 6857
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2 \tan ^{\frac {5}{2}}(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac {2 \int \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {5 a^2}{2}-\frac {3}{2} a b \tan (c+d x)+\frac {1}{2} \left (5 a^2+3 b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^{3/2}} \, dx}{3 b \left (a^2+b^2\right )} \\ & = -\frac {2 a^2 \tan ^{\frac {5}{2}}(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 a^2 \left (5 a^2+11 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {4 \int \frac {\sqrt {\tan (c+d x)} \left (\frac {3}{4} a^2 \left (5 a^2+11 b^2\right )-\frac {3}{2} a b^3 \tan (c+d x)+\frac {3}{4} \left (5 a^4+10 a^2 b^2+b^4\right ) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{3 b^2 \left (a^2+b^2\right )^2} \\ & = -\frac {2 a^2 \tan ^{\frac {5}{2}}(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 a^2 \left (5 a^2+11 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {\left (5 a^4+10 a^2 b^2+b^4\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{b^3 \left (a^2+b^2\right )^2 d}+\frac {4 \int \frac {-\frac {3}{8} a \left (5 a^4+10 a^2 b^2+b^4\right )+\frac {3}{4} b^3 \left (a^2-b^2\right ) \tan (c+d x)-\frac {15}{8} a \left (a^2+b^2\right )^2 \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{3 b^3 \left (a^2+b^2\right )^2} \\ & = -\frac {2 a^2 \tan ^{\frac {5}{2}}(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 a^2 \left (5 a^2+11 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {\left (5 a^4+10 a^2 b^2+b^4\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{b^3 \left (a^2+b^2\right )^2 d}+\frac {4 \text {Subst}\left (\int \frac {-\frac {3}{8} a \left (5 a^4+10 a^2 b^2+b^4\right )+\frac {3}{4} b^3 \left (a^2-b^2\right ) x-\frac {15}{8} a \left (a^2+b^2\right )^2 x^2}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{3 b^3 \left (a^2+b^2\right )^2 d} \\ & = -\frac {2 a^2 \tan ^{\frac {5}{2}}(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 a^2 \left (5 a^2+11 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {\left (5 a^4+10 a^2 b^2+b^4\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{b^3 \left (a^2+b^2\right )^2 d}+\frac {4 \text {Subst}\left (\int \left (-\frac {15 a \left (a^2+b^2\right )^2}{8 \sqrt {x} \sqrt {a+b x}}+\frac {3 \left (2 a b^4+b^3 \left (a^2-b^2\right ) x\right )}{4 \sqrt {x} \sqrt {a+b x} \left (1+x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{3 b^3 \left (a^2+b^2\right )^2 d} \\ & = -\frac {2 a^2 \tan ^{\frac {5}{2}}(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 a^2 \left (5 a^2+11 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {\left (5 a^4+10 a^2 b^2+b^4\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{b^3 \left (a^2+b^2\right )^2 d}-\frac {(5 a) \text {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 b^3 d}+\frac {\text {Subst}\left (\int \frac {2 a b^4+b^3 \left (a^2-b^2\right ) x}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{b^3 \left (a^2+b^2\right )^2 d} \\ & = -\frac {2 a^2 \tan ^{\frac {5}{2}}(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 a^2 \left (5 a^2+11 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {\left (5 a^4+10 a^2 b^2+b^4\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{b^3 \left (a^2+b^2\right )^2 d}-\frac {(5 a) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {\tan (c+d x)}\right )}{b^3 d}+\frac {\text {Subst}\left (\int \left (\frac {2 i a b^4-b^3 \left (a^2-b^2\right )}{2 (i-x) \sqrt {x} \sqrt {a+b x}}+\frac {2 i a b^4+b^3 \left (a^2-b^2\right )}{2 \sqrt {x} (i+x) \sqrt {a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{b^3 \left (a^2+b^2\right )^2 d} \\ & = -\frac {2 a^2 \tan ^{\frac {5}{2}}(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 a^2 \left (5 a^2+11 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {\left (5 a^4+10 a^2 b^2+b^4\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{b^3 \left (a^2+b^2\right )^2 d}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {x} (i+x) \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 (a-i b)^2 d}-\frac {\text {Subst}\left (\int \frac {1}{(i-x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 (a+i b)^2 d}-\frac {(5 a) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{b^3 d} \\ & = -\frac {5 a \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{b^{7/2} d}-\frac {2 a^2 \tan ^{\frac {5}{2}}(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 a^2 \left (5 a^2+11 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {\left (5 a^4+10 a^2 b^2+b^4\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{b^3 \left (a^2+b^2\right )^2 d}+\frac {\text {Subst}\left (\int \frac {1}{i-(-a+i b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(a-i b)^2 d}-\frac {\text {Subst}\left (\int \frac {1}{i-(a+i b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(a+i b)^2 d} \\ & = -\frac {i \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(i a-b)^{5/2} d}-\frac {5 a \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{b^{7/2} d}+\frac {i \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(i a+b)^{5/2} d}-\frac {2 a^2 \tan ^{\frac {5}{2}}(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 a^2 \left (5 a^2+11 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {\left (5 a^4+10 a^2 b^2+b^4\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{b^3 \left (a^2+b^2\right )^2 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 6.30 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.15 \[ \int \frac {\tan ^{\frac {9}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=-\frac {\sqrt [4]{-1} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(-a+i b)^{5/2} d}+\frac {\sqrt [4]{-1} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(a+i b)^{5/2} d}-\frac {\tan ^{\frac {3}{2}}(c+d x)}{3 (a-i b) d (a+b \tan (c+d x))^{3/2}}-\frac {\tan ^{\frac {3}{2}}(c+d x)}{3 (a+i b) d (a+b \tan (c+d x))^{3/2}}+\frac {i \sqrt {\tan (c+d x)}}{(a-i b)^2 d \sqrt {a+b \tan (c+d x)}}-\frac {i \sqrt {\tan (c+d x)}}{(a+i b)^2 d \sqrt {a+b \tan (c+d x)}}+\frac {2 \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {7}{2},\frac {9}{2},-\frac {b \tan (c+d x)}{a}\right ) \tan ^{\frac {7}{2}}(c+d x) \sqrt {1+\frac {b \tan (c+d x)}{a}}}{7 a^2 d \sqrt {a+b \tan (c+d x)}} \]
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result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 1.32 (sec) , antiderivative size = 1492946, normalized size of antiderivative = 4709.61
\[\text {output too large to display}\]
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Leaf count of result is larger than twice the leaf count of optimal. 11619 vs. \(2 (265) = 530\).
Time = 4.37 (sec) , antiderivative size = 23240, normalized size of antiderivative = 73.31 \[ \int \frac {\tan ^{\frac {9}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\tan ^{\frac {9}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\tan ^{\frac {9}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\int { \frac {\tan \left (d x + c\right )^{\frac {9}{2}}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\tan ^{\frac {9}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\tan ^{\frac {9}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\int \frac {{\mathrm {tan}\left (c+d\,x\right )}^{9/2}}{{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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