Integrand size = 21, antiderivative size = 140 \[ \int \csc ^7(a+b x) (d \tan (a+b x))^{5/2} \, dx=-\frac {40 d^3 \csc (a+b x)}{21 b \sqrt {d \tan (a+b x)}}-\frac {20 d^3 \csc ^3(a+b x)}{21 b \sqrt {d \tan (a+b x)}}+\frac {40 d^2 \csc (a+b x) \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}{21 b}+\frac {2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b} \]
[Out]
Time = 0.21 (sec) , antiderivative size = 140, 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 = {2673, 2679, 2681, 2653, 2720} \[ \int \csc ^7(a+b x) (d \tan (a+b x))^{5/2} \, dx=-\frac {20 d^3 \csc ^3(a+b x)}{21 b \sqrt {d \tan (a+b x)}}-\frac {40 d^3 \csc (a+b x)}{21 b \sqrt {d \tan (a+b x)}}+\frac {40 d^2 \sqrt {\sin (2 a+2 b x)} \csc (a+b x) \operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right ) \sqrt {d \tan (a+b x)}}{21 b}+\frac {2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b} \]
[In]
[Out]
Rule 2653
Rule 2673
Rule 2679
Rule 2681
Rule 2720
Rubi steps \begin{align*} \text {integral}& = \frac {2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b}+\frac {1}{3} \left (10 d^2\right ) \int \csc ^5(a+b x) \sqrt {d \tan (a+b x)} \, dx \\ & = -\frac {20 d^3 \csc ^3(a+b x)}{21 b \sqrt {d \tan (a+b x)}}+\frac {2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b}+\frac {1}{7} \left (20 d^2\right ) \int \csc ^3(a+b x) \sqrt {d \tan (a+b x)} \, dx \\ & = -\frac {40 d^3 \csc (a+b x)}{21 b \sqrt {d \tan (a+b x)}}-\frac {20 d^3 \csc ^3(a+b x)}{21 b \sqrt {d \tan (a+b x)}}+\frac {2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b}+\frac {1}{21} \left (40 d^2\right ) \int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx \\ & = -\frac {40 d^3 \csc (a+b x)}{21 b \sqrt {d \tan (a+b x)}}-\frac {20 d^3 \csc ^3(a+b x)}{21 b \sqrt {d \tan (a+b x)}}+\frac {2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b}+\frac {\left (40 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}{21 \sqrt {\sin (a+b x)}} \\ & = -\frac {40 d^3 \csc (a+b x)}{21 b \sqrt {d \tan (a+b x)}}-\frac {20 d^3 \csc ^3(a+b x)}{21 b \sqrt {d \tan (a+b x)}}+\frac {2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b}+\frac {1}{21} \left (40 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 {40 d^3 \csc (a+b x)}{21 b \sqrt {d \tan (a+b x)}}-\frac {20 d^3 \csc ^3(a+b x)}{21 b \sqrt {d \tan (a+b x)}}+\frac {40 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)}}{21 b}+\frac {2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.84 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.93 \[ \int \csc ^7(a+b x) (d \tan (a+b x))^{5/2} \, dx=-\frac {d^2 \csc (a+b x) \left ((1+10 \cos (2 (a+b x))-5 \cos (4 (a+b x))) \csc ^3(a+b x) \sec (a+b x) \sqrt {\sec ^2(a+b x)}+80 \sqrt [4]{-1} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (a+b x)}\right ),-1\right ) \sqrt {\tan (a+b x)}\right ) \sqrt {d \tan (a+b x)}}{21 b \sqrt {\sec ^2(a+b x)}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(455\) vs. \(2(147)=294\).
Time = 151.87 (sec) , antiderivative size = 456, normalized size of antiderivative = 3.26
method | result | size |
default | \(\frac {\sin \left (b x +a \right ) \tan \left (b x +a \right ) \left (40 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \left (\cos ^{4}\left (b x +a \right )\right ) \sin \left (b x +a \right )+40 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \left (\cos ^{3}\left (b x +a \right )\right ) \sin \left (b x +a \right )-40 \sin \left (b x +a \right ) \left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-40 \sin \left (b x +a \right ) \cos \left (b x +a \right ) \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-20 \sqrt {2}\, \left (\cos ^{4}\left (b x +a \right )\right )+30 \left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {2}-7 \sqrt {2}\right ) \sqrt {d \tan \left (b x +a \right )}\, d^{2} \sqrt {2}}{21 b \left (-1+\cos \left (b x +a \right )\right )^{3} \left (\cos \left (b x +a \right )+1\right )^{3}}\) | \(456\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.49 \[ \int \csc ^7(a+b x) (d \tan (a+b x))^{5/2} \, dx=-\frac {2 \, {\left (20 \, {\left (d^{2} \cos \left (b x + a\right )^{5} - 2 \, d^{2} \cos \left (b x + a\right )^{3} + d^{2} \cos \left (b x + a\right )\right )} \sqrt {i \, d} F(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) + 20 \, {\left (d^{2} \cos \left (b x + a\right )^{5} - 2 \, d^{2} \cos \left (b x + a\right )^{3} + d^{2} \cos \left (b x + a\right )\right )} \sqrt {-i \, d} F(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) - {\left (20 \, d^{2} \cos \left (b x + a\right )^{4} - 30 \, d^{2} \cos \left (b x + a\right )^{2} + 7 \, d^{2}\right )} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}\right )}}{21 \, {\left (b \cos \left (b x + a\right )^{5} - 2 \, b \cos \left (b x + a\right )^{3} + b \cos \left (b x + a\right )\right )}} \]
[In]
[Out]
Timed out. \[ \int \csc ^7(a+b x) (d \tan (a+b x))^{5/2} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \csc ^7(a+b x) (d \tan (a+b x))^{5/2} \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{\frac {5}{2}} \csc \left (b x + a\right )^{7} \,d x } \]
[In]
[Out]
\[ \int \csc ^7(a+b x) (d \tan (a+b x))^{5/2} \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{\frac {5}{2}} \csc \left (b x + a\right )^{7} \,d x } \]
[In]
[Out]
Timed out. \[ \int \csc ^7(a+b x) (d \tan (a+b x))^{5/2} \, dx=\int \frac {{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{5/2}}{{\sin \left (a+b\,x\right )}^7} \,d x \]
[In]
[Out]