Optimal. Leaf size=65 \[ -\frac {2 d^5}{11 b (d \tan (a+b x))^{11/2}}-\frac {4 d^3}{7 b (d \tan (a+b x))^{7/2}}-\frac {2 d}{3 b (d \tan (a+b x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2591, 270} \[ -\frac {2 d^5}{11 b (d \tan (a+b x))^{11/2}}-\frac {4 d^3}{7 b (d \tan (a+b x))^{7/2}}-\frac {2 d}{3 b (d \tan (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 270
Rule 2591
Rubi steps
\begin {align*} \int \frac {\csc ^6(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx &=\frac {d \operatorname {Subst}\left (\int \frac {\left (d^2+x^2\right )^2}{x^{13/2}} \, dx,x,d \tan (a+b x)\right )}{b}\\ &=\frac {d \operatorname {Subst}\left (\int \left (\frac {d^4}{x^{13/2}}+\frac {2 d^2}{x^{9/2}}+\frac {1}{x^{5/2}}\right ) \, dx,x,d \tan (a+b x)\right )}{b}\\ &=-\frac {2 d^5}{11 b (d \tan (a+b x))^{11/2}}-\frac {4 d^3}{7 b (d \tan (a+b x))^{7/2}}-\frac {2 d}{3 b (d \tan (a+b x))^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.16, size = 50, normalized size = 0.77 \[ \frac {2 d (28 \cos (2 (a+b x))-4 \cos (4 (a+b x))-45) \csc ^4(a+b x)}{231 b (d \tan (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.75, size = 93, normalized size = 1.43 \[ \frac {2 \, {\left (32 \, \cos \left (b x + a\right )^{6} - 88 \, \cos \left (b x + a\right )^{4} + 77 \, \cos \left (b x + a\right )^{2}\right )} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{231 \, {\left (b d \cos \left (b x + a\right )^{6} - 3 \, b d \cos \left (b x + a\right )^{4} + 3 \, b d \cos \left (b x + a\right )^{2} - b d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.44, size = 58, normalized size = 0.89 \[ -\frac {2 \, {\left (77 \, d^{5} \tan \left (b x + a\right )^{4} + 66 \, d^{5} \tan \left (b x + a\right )^{2} + 21 \, d^{5}\right )}}{231 \, \sqrt {d \tan \left (b x + a\right )} b d^{5} \tan \left (b x + a\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.68, size = 60, normalized size = 0.92 \[ -\frac {2 \left (32 \left (\cos ^{4}\left (b x +a \right )\right )-88 \left (\cos ^{2}\left (b x +a \right )\right )+77\right ) \cos \left (b x +a \right )}{231 b \sin \left (b x +a \right )^{5} \sqrt {\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 48, normalized size = 0.74 \[ -\frac {2 \, {\left (77 \, d^{4} \tan \left (b x + a\right )^{4} + 66 \, d^{4} \tan \left (b x + a\right )^{2} + 21 \, d^{4}\right )} d}{231 \, \left (d \tan \left (b x + a\right )\right )^{\frac {11}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 12.40, size = 831, normalized size = 12.78 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{6}{\left (a + b x \right )}}{\sqrt {d \tan {\left (a + b x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________