Integrand size = 21, antiderivative size = 110 \[ \int \csc ^5(a+b x) (d \tan (a+b x))^{5/2} \, dx=-\frac {4 d^3 \csc (a+b x)}{3 b \sqrt {d \tan (a+b x)}}+\frac {4 d^2 \csc (a+b x) \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}{3 b}+\frac {2 d \csc ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b} \]
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Time = 0.16 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2673, 2679, 2681, 2653, 2720} \[ \int \csc ^5(a+b x) (d \tan (a+b x))^{5/2} \, dx=-\frac {4 d^3 \csc (a+b x)}{3 b \sqrt {d \tan (a+b x)}}+\frac {4 d^2 \sqrt {\sin (2 a+2 b x)} \csc (a+b x) \operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right ) \sqrt {d \tan (a+b x)}}{3 b}+\frac {2 d \csc ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b} \]
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Rule 2653
Rule 2673
Rule 2679
Rule 2681
Rule 2720
Rubi steps \begin{align*} \text {integral}& = \frac {2 d \csc ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b}+\left (2 d^2\right ) \int \csc ^3(a+b x) \sqrt {d \tan (a+b x)} \, dx \\ & = -\frac {4 d^3 \csc (a+b x)}{3 b \sqrt {d \tan (a+b x)}}+\frac {2 d \csc ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b}+\frac {1}{3} \left (4 d^2\right ) \int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx \\ & = -\frac {4 d^3 \csc (a+b x)}{3 b \sqrt {d \tan (a+b x)}}+\frac {2 d \csc ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b}+\frac {\left (4 d^2 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}\right ) \int \frac {1}{\sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}} \, dx}{3 \sqrt {\sin (a+b x)}} \\ & = -\frac {4 d^3 \csc (a+b x)}{3 b \sqrt {d \tan (a+b x)}}+\frac {2 d \csc ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b}+\frac {1}{3} \left (4 d^2 \csc (a+b x) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx \\ & = -\frac {4 d^3 \csc (a+b x)}{3 b \sqrt {d \tan (a+b x)}}+\frac {4 d^2 \csc (a+b x) \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}{3 b}+\frac {2 d \csc ^3(a+b x) (d \tan (a+b x))^{3/2}}{3 b} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.70 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00 \[ \int \csc ^5(a+b x) (d \tan (a+b x))^{5/2} \, dx=-\frac {2 d \csc ^3(a+b x) \left (\cos (2 (a+b x)) \sqrt {\sec ^2(a+b x)}+2 \sqrt [4]{-1} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (a+b x)}\right ),-1\right ) \sin (2 (a+b x)) \sqrt {\tan (a+b x)}\right ) (d \tan (a+b x))^{3/2}}{3 b \sqrt {\sec ^2(a+b x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(258\) vs. \(2(121)=242\).
Time = 20.39 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.35
method | result | size |
default | \(-\frac {\sqrt {d \tan \left (b x +a \right )}\, d^{2} \left (-4 \cos \left (b x +a \right ) \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \left (\sin ^{3}\left (b x +a \right )\right ) F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-4 \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \left (\sin ^{3}\left (b x +a \right )\right ) F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+2 \cos \left (b x +a \right ) \sqrt {2}\, \left (\sin ^{2}\left (b x +a \right )\right )-\sin \left (b x +a \right ) \tan \left (b x +a \right ) \sqrt {2}\right ) \sqrt {2}}{3 b \left (-1+\cos \left (b x +a \right )\right )^{2} \left (\cos \left (b x +a \right )+1\right )^{2}}\) | \(259\) |
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Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.47 \[ \int \csc ^5(a+b x) (d \tan (a+b x))^{5/2} \, dx=-\frac {2 \, {\left (2 \, {\left (d^{2} \cos \left (b x + a\right )^{3} - d^{2} \cos \left (b x + a\right )\right )} \sqrt {i \, d} F(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) + 2 \, {\left (d^{2} \cos \left (b x + a\right )^{3} - d^{2} \cos \left (b x + a\right )\right )} \sqrt {-i \, d} F(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) - {\left (2 \, d^{2} \cos \left (b x + a\right )^{2} - d^{2}\right )} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}\right )}}{3 \, {\left (b \cos \left (b x + a\right )^{3} - b \cos \left (b x + a\right )\right )}} \]
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Timed out. \[ \int \csc ^5(a+b x) (d \tan (a+b x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int \csc ^5(a+b x) (d \tan (a+b x))^{5/2} \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{\frac {5}{2}} \csc \left (b x + a\right )^{5} \,d x } \]
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\[ \int \csc ^5(a+b x) (d \tan (a+b x))^{5/2} \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{\frac {5}{2}} \csc \left (b x + a\right )^{5} \,d x } \]
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Timed out. \[ \int \csc ^5(a+b x) (d \tan (a+b x))^{5/2} \, dx=\int \frac {{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{5/2}}{{\sin \left (a+b\,x\right )}^5} \,d x \]
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