Optimal. Leaf size=106 \[ -\frac {14 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{15 b}-\frac {14 \cos (2 a+2 b x)}{45 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {\csc ^2(a+b x)}{9 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {14 \cos (2 a+2 b x)}{15 b \sqrt {\sin (2 a+2 b x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {4385, 2716,
2719} \begin {gather*} -\frac {14 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{15 b}-\frac {14 \cos (2 a+2 b x)}{45 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {14 \cos (2 a+2 b x)}{15 b \sqrt {\sin (2 a+2 b x)}}-\frac {\csc ^2(a+b x)}{9 b \sin ^{\frac {5}{2}}(2 a+2 b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2716
Rule 2719
Rule 4385
Rubi steps
\begin {align*} \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {7}{2}}(2 a+2 b x)} \, dx &=-\frac {\csc ^2(a+b x)}{9 b \sin ^{\frac {5}{2}}(2 a+2 b x)}+\frac {14}{9} \int \frac {1}{\sin ^{\frac {7}{2}}(2 a+2 b x)} \, dx\\ &=-\frac {14 \cos (2 a+2 b x)}{45 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {\csc ^2(a+b x)}{9 b \sin ^{\frac {5}{2}}(2 a+2 b x)}+\frac {14}{15} \int \frac {1}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx\\ &=-\frac {14 \cos (2 a+2 b x)}{45 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {\csc ^2(a+b x)}{9 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {14 \cos (2 a+2 b x)}{15 b \sqrt {\sin (2 a+2 b x)}}-\frac {14}{15} \int \sqrt {\sin (2 a+2 b x)} \, dx\\ &=-\frac {14 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{15 b}-\frac {14 \cos (2 a+2 b x)}{45 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {\csc ^2(a+b x)}{9 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {14 \cos (2 a+2 b x)}{15 b \sqrt {\sin (2 a+2 b x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.85, size = 85, normalized size = 0.80 \begin {gather*} -\frac {336 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )+\frac {(-9+98 \cos (2 (a+b x))-28 \cos (4 (a+b x))-42 \cos (6 (a+b x))+21 \cos (8 (a+b x))) \csc ^2(a+b x)}{\sin ^{\frac {5}{2}}(2 (a+b x))}}{360 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\csc ^{2}\left (x b +a \right )}{\sin \left (2 x b +2 a \right )^{\frac {7}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\sin \left (a+b\,x\right )}^2\,{\sin \left (2\,a+2\,b\,x\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________