Integrand size = 21, antiderivative size = 142 \[ \int \frac {\cos ^5(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=-\frac {2 \cos ^5(a+b x)}{b d \sqrt {d \tan (a+b x)}}-\frac {77 \cos (a+b x) E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {d \tan (a+b x)}}{20 b d^2 \sqrt {\sin (2 a+2 b x)}}-\frac {77 \cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{30 b d^3}-\frac {11 \cos ^5(a+b x) (d \tan (a+b x))^{3/2}}{5 b d^3} \]
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Time = 0.21 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2689, 2692, 2695, 2652, 2719} \[ \int \frac {\cos ^5(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=-\frac {11 \cos ^5(a+b x) (d \tan (a+b x))^{3/2}}{5 b d^3}-\frac {77 \cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{30 b d^3}-\frac {77 \cos (a+b x) E\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {d \tan (a+b x)}}{20 b d^2 \sqrt {\sin (2 a+2 b x)}}-\frac {2 \cos ^5(a+b x)}{b d \sqrt {d \tan (a+b x)}} \]
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Rule 2652
Rule 2689
Rule 2692
Rule 2695
Rule 2719
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos ^5(a+b x)}{b d \sqrt {d \tan (a+b x)}}-\frac {11 \int \cos ^5(a+b x) \sqrt {d \tan (a+b x)} \, dx}{d^2} \\ & = -\frac {2 \cos ^5(a+b x)}{b d \sqrt {d \tan (a+b x)}}-\frac {11 \cos ^5(a+b x) (d \tan (a+b x))^{3/2}}{5 b d^3}-\frac {77 \int \cos ^3(a+b x) \sqrt {d \tan (a+b x)} \, dx}{10 d^2} \\ & = -\frac {2 \cos ^5(a+b x)}{b d \sqrt {d \tan (a+b x)}}-\frac {77 \cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{30 b d^3}-\frac {11 \cos ^5(a+b x) (d \tan (a+b x))^{3/2}}{5 b d^3}-\frac {77 \int \cos (a+b x) \sqrt {d \tan (a+b x)} \, dx}{20 d^2} \\ & = -\frac {2 \cos ^5(a+b x)}{b d \sqrt {d \tan (a+b x)}}-\frac {77 \cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{30 b d^3}-\frac {11 \cos ^5(a+b x) (d \tan (a+b x))^{3/2}}{5 b d^3}-\frac {\left (77 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)} \, dx}{20 d^2 \sqrt {\sin (a+b x)}} \\ & = -\frac {2 \cos ^5(a+b x)}{b d \sqrt {d \tan (a+b x)}}-\frac {77 \cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{30 b d^3}-\frac {11 \cos ^5(a+b x) (d \tan (a+b x))^{3/2}}{5 b d^3}-\frac {\left (77 \cos (a+b x) \sqrt {d \tan (a+b x)}\right ) \int \sqrt {\sin (2 a+2 b x)} \, dx}{20 d^2 \sqrt {\sin (2 a+2 b x)}} \\ & = -\frac {2 \cos ^5(a+b x)}{b d \sqrt {d \tan (a+b x)}}-\frac {77 \cos (a+b x) E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {d \tan (a+b x)}}{20 b d^2 \sqrt {\sin (2 a+2 b x)}}-\frac {77 \cos ^3(a+b x) (d \tan (a+b x))^{3/2}}{30 b d^3}-\frac {11 \cos ^5(a+b x) (d \tan (a+b x))^{3/2}}{5 b d^3} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.91 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.63 \[ \int \frac {\cos ^5(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\frac {\sin (a+b x) \left (-277+34 \cos (2 (a+b x))+3 \cos (4 (a+b x))-308 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\tan ^2(a+b x)\right ) \sqrt {\sec ^2(a+b x)} \tan ^2(a+b x)\right )}{120 b (d \tan (a+b x))^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(392\) vs. \(2(151)=302\).
Time = 1.39 (sec) , antiderivative size = 393, normalized size of antiderivative = 2.77
method | result | size |
default | \(\frac {\left (12 \left (\cos ^{5}\left (b x +a \right )\right ) \sqrt {2}+22 \left (\cos ^{3}\left (b x +a \right )\right ) \sqrt {2}+462 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-231 \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+462 \sec \left (b x +a \right ) \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-231 \sec \left (b x +a \right ) \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+77 \sqrt {2}\, \cos \left (b x +a \right )-231 \sqrt {2}\right ) \sqrt {2}}{120 b \sqrt {d \tan \left (b x +a \right )}\, d}\) | \(393\) |
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\[ \int \frac {\cos ^5(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int { \frac {\cos \left (b x + a\right )^{5}}{\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^5(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos ^5(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int { \frac {\cos \left (b x + a\right )^{5}}{\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\cos ^5(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int { \frac {\cos \left (b x + a\right )^{5}}{\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^5(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^5}{{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2}} \,d x \]
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