Optimal. Leaf size=76 \[ \frac {2 (a \sin (c+d x)+a)^{5/2}}{3 d e (e \cos (c+d x))^{7/2}}-\frac {4 (a \sin (c+d x)+a)^{7/2}}{21 a d e (e \cos (c+d x))^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2672, 2671} \[ \frac {2 (a \sin (c+d x)+a)^{5/2}}{3 d e (e \cos (c+d x))^{7/2}}-\frac {4 (a \sin (c+d x)+a)^{7/2}}{21 a d e (e \cos (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2671
Rule 2672
Rubi steps
\begin {align*} \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{9/2}} \, dx &=\frac {2 (a+a \sin (c+d x))^{5/2}}{3 d e (e \cos (c+d x))^{7/2}}-\frac {2 \int \frac {(a+a \sin (c+d x))^{7/2}}{(e \cos (c+d x))^{9/2}} \, dx}{3 a}\\ &=\frac {2 (a+a \sin (c+d x))^{5/2}}{3 d e (e \cos (c+d x))^{7/2}}-\frac {4 (a+a \sin (c+d x))^{7/2}}{21 a d e (e \cos (c+d x))^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.17, size = 54, normalized size = 0.71 \[ -\frac {2 (2 \sin (c+d x)-5) \sec ^4(c+d x) (a (\sin (c+d x)+1))^{5/2} \sqrt {e \cos (c+d x)}}{21 d e^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.16, size = 75, normalized size = 0.99 \[ \frac {2 \, {\left (2 \, a^{2} \sin \left (d x + c\right ) - 5 \, a^{2}\right )} \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{21 \, {\left (d e^{5} \cos \left (d x + c\right )^{2} + 2 \, d e^{5} \sin \left (d x + c\right ) - 2 \, d e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [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]
________________________________________________________________________________________
maple [A] time = 0.20, size = 44, normalized size = 0.58 \[ -\frac {2 \left (2 \sin \left (d x +c \right )-5\right ) \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {5}{2}} \cos \left (d x +c \right )}{21 d \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.00, size = 207, normalized size = 2.72 \[ \frac {2 \, {\left (5 \, a^{\frac {5}{2}} \sqrt {e} - \frac {4 \, a^{\frac {5}{2}} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {4 \, a^{\frac {5}{2}} \sqrt {e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {5 \, a^{\frac {5}{2}} \sqrt {e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{2}}{21 \, {\left (e^{5} + \frac {2 \, e^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {e^{5} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} d {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 6.34, size = 96, normalized size = 1.26 \[ \frac {4\,a^2\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (\cos \left (3\,c+3\,d\,x\right )-11\,\cos \left (c+d\,x\right )+7\,\sin \left (2\,c+2\,d\,x\right )\right )}{21\,d\,e^4\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (15\,\sin \left (c+d\,x\right )+6\,\cos \left (2\,c+2\,d\,x\right )-\sin \left (3\,c+3\,d\,x\right )-10\right )} \]
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]
________________________________________________________________________________________