Optimal. Leaf size=31 \[ \frac {16 \cos ^7(a+b x)}{7 b}-\frac {16 \cos ^5(a+b x)}{5 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4288, 2565, 14} \[ \frac {16 \cos ^7(a+b x)}{7 b}-\frac {16 \cos ^5(a+b x)}{5 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 2565
Rule 4288
Rubi steps
\begin {align*} \int \csc (a+b x) \sin ^4(2 a+2 b x) \, dx &=16 \int \cos ^4(a+b x) \sin ^3(a+b x) \, dx\\ &=-\frac {16 \operatorname {Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {16 \operatorname {Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {16 \cos ^5(a+b x)}{5 b}+\frac {16 \cos ^7(a+b x)}{7 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 59, normalized size = 1.90 \[ -\frac {3 \cos (a+b x)}{4 b}-\frac {\cos (3 (a+b x))}{4 b}+\frac {\cos (5 (a+b x))}{20 b}+\frac {\cos (7 (a+b x))}{28 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 26, normalized size = 0.84 \[ \frac {16 \, {\left (5 \, \cos \left (b x + a\right )^{7} - 7 \, \cos \left (b x + a\right )^{5}\right )}}{35 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.56, size = 138, normalized size = 4.45 \[ -\frac {64 \, {\left (\frac {7 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {14 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {70 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac {35 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + \frac {35 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} - 1\right )}}{35 \, b {\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.66, size = 35, normalized size = 1.13 \[ \frac {-\frac {16 \left (\sin ^{2}\left (b x +a \right )\right ) \left (\cos ^{5}\left (b x +a \right )\right )}{7}-\frac {32 \left (\cos ^{5}\left (b x +a \right )\right )}{35}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 47, normalized size = 1.52 \[ \frac {5 \, \cos \left (7 \, b x + 7 \, a\right ) + 7 \, \cos \left (5 \, b x + 5 \, a\right ) - 35 \, \cos \left (3 \, b x + 3 \, a\right ) - 105 \, \cos \left (b x + a\right )}{140 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 26, normalized size = 0.84 \[ -\frac {16\,\left (7\,{\cos \left (a+b\,x\right )}^5-5\,{\cos \left (a+b\,x\right )}^7\right )}{35\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________