Optimal. Leaf size=112 \[ -\frac {2 \sqrt {\sin (2 a+2 b x)} \csc (a+b x) F\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {d \tan (a+b x)}}{21 b d^2}-\frac {2 \csc ^3(a+b x)}{7 b d \sqrt {d \tan (a+b x)}}+\frac {2 \csc (a+b x)}{21 b d \sqrt {d \tan (a+b x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2597, 2599, 2601, 2573, 2641} \[ -\frac {2 \sqrt {\sin (2 a+2 b x)} \csc (a+b x) F\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {d \tan (a+b x)}}{21 b d^2}-\frac {2 \csc ^3(a+b x)}{7 b d \sqrt {d \tan (a+b x)}}+\frac {2 \csc (a+b x)}{21 b d \sqrt {d \tan (a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2573
Rule 2597
Rule 2599
Rule 2601
Rule 2641
Rubi steps
\begin {align*} \int \frac {\csc ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx &=-\frac {2 \csc ^3(a+b x)}{7 b d \sqrt {d \tan (a+b x)}}-\frac {\int \csc ^3(a+b x) \sqrt {d \tan (a+b x)} \, dx}{7 d^2}\\ &=\frac {2 \csc (a+b x)}{21 b d \sqrt {d \tan (a+b x)}}-\frac {2 \csc ^3(a+b x)}{7 b d \sqrt {d \tan (a+b x)}}-\frac {2 \int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx}{21 d^2}\\ &=\frac {2 \csc (a+b x)}{21 b d \sqrt {d \tan (a+b x)}}-\frac {2 \csc ^3(a+b x)}{7 b d \sqrt {d \tan (a+b x)}}-\frac {\left (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 d^2 \sqrt {\sin (a+b x)}}\\ &=\frac {2 \csc (a+b x)}{21 b d \sqrt {d \tan (a+b x)}}-\frac {2 \csc ^3(a+b x)}{7 b d \sqrt {d \tan (a+b x)}}-\frac {\left (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}{21 d^2}\\ &=\frac {2 \csc (a+b x)}{21 b d \sqrt {d \tan (a+b x)}}-\frac {2 \csc ^3(a+b x)}{7 b d \sqrt {d \tan (a+b x)}}-\frac {2 \csc (a+b x) F\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}{21 b d^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 1.71, size = 136, normalized size = 1.21 \[ \frac {\csc ^3(a+b x) \left ((10 \cos (2 (a+b x))+\cos (4 (a+b x))+1) \sec ^2(a+b x)^{3/2}-8 \sqrt [4]{-1} \cos (2 (a+b x)) \tan ^{\frac {7}{2}}(a+b x) F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt {\tan (a+b x)}\right )\right |-1\right )\right )}{42 b d \left (\tan ^2(a+b x)-1\right ) \sqrt {\sec ^2(a+b x)} \sqrt {d \tan (a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \tan \left (b x + a\right )} \csc \left (b x + a\right )^{3}}{d^{2} \tan \left (b x + a\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (b x + a\right )^{3}}{\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.59, size = 558, normalized size = 4.98 \[ \frac {\left (-1+\cos \left (b x +a \right )\right )^{2} \left (2 \left (\cos ^{3}\left (b x +a \right )\right ) \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sin \left (b x +a \right )+2 \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \left (\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )-2 \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right ) \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sin \left (b x +a \right )-2 \sin \left (b x +a \right ) \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}-\left (\cos ^{3}\left (b x +a \right )\right ) \sqrt {2}-2 \cos \left (b x +a \right ) \sqrt {2}\right ) \left (\cos \left (b x +a \right )+1\right )^{2} \sqrt {2}}{21 b \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{6} \left (\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (b x + a\right )^{3}}{\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\sin \left (a+b\,x\right )}^3\,{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{3}{\left (a + b x \right )}}{\left (d \tan {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________