Optimal. Leaf size=89 \[ \frac{64 a^3 \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{16 a^2 \sec (c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}-\frac{2 a \sec (c+d x) (a \sin (c+d x)+a)^{5/2}}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.173659, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2674, 2673} \[ \frac{64 a^3 \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{16 a^2 \sec (c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}-\frac{2 a \sec (c+d x) (a \sin (c+d x)+a)^{5/2}}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2674
Rule 2673
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+a \sin (c+d x))^{7/2} \, dx &=-\frac{2 a \sec (c+d x) (a+a \sin (c+d x))^{5/2}}{3 d}+\frac{1}{3} (8 a) \int \sec ^2(c+d x) (a+a \sin (c+d x))^{5/2} \, dx\\ &=-\frac{16 a^2 \sec (c+d x) (a+a \sin (c+d x))^{3/2}}{3 d}-\frac{2 a \sec (c+d x) (a+a \sin (c+d x))^{5/2}}{3 d}+\frac{1}{3} \left (32 a^2\right ) \int \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\\ &=\frac{64 a^3 \sec (c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}-\frac{16 a^2 \sec (c+d x) (a+a \sin (c+d x))^{3/2}}{3 d}-\frac{2 a \sec (c+d x) (a+a \sin (c+d x))^{5/2}}{3 d}\\ \end{align*}
Mathematica [A] time = 5.47737, size = 48, normalized size = 0.54 \[ \frac{a^3 \sec (c+d x) \sqrt{a (\sin (c+d x)+1)} (-20 \sin (c+d x)+\cos (2 (c+d x))+45)}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.101, size = 55, normalized size = 0.6 \begin{align*} -{\frac{2\,{a}^{4} \left ( 1+\sin \left ( dx+c \right ) \right ) \left ( \left ( \sin \left ( dx+c \right ) \right ) ^{2}+10\,\sin \left ( dx+c \right ) -23 \right ) }{3\,d\cos \left ( dx+c \right ) }{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.67178, size = 320, normalized size = 3.6 \begin{align*} -\frac{2 \,{\left (23 \, a^{\frac{7}{2}} - \frac{20 \, a^{\frac{7}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{88 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{60 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{130 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{60 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{88 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{20 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{23 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )}}{3 \, d{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.66787, size = 134, normalized size = 1.51 \begin{align*} \frac{2 \,{\left (a^{3} \cos \left (d x + c\right )^{2} - 10 \, a^{3} \sin \left (d x + c\right ) + 22 \, a^{3}\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{3 \, d \cos \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]