Optimal. Leaf size=120 \[ \frac{130\ 13^{3/4} \text{EllipticF}\left (\frac{1}{2} \left (c+d x-\tan ^{-1}\left (\frac{3}{2}\right )\right ),2\right )}{21 d}-\frac{2 (3 \cos (c+d x)-2 \sin (c+d x)) (3 \sin (c+d x)+2 \cos (c+d x))^{5/2}}{7 d}-\frac{130 (3 \cos (c+d x)-2 \sin (c+d x)) \sqrt{3 \sin (c+d x)+2 \cos (c+d x)}}{21 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0699545, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3073, 3077, 2641} \[ -\frac{2 (3 \cos (c+d x)-2 \sin (c+d x)) (3 \sin (c+d x)+2 \cos (c+d x))^{5/2}}{7 d}-\frac{130 (3 \cos (c+d x)-2 \sin (c+d x)) \sqrt{3 \sin (c+d x)+2 \cos (c+d x)}}{21 d}+\frac{130\ 13^{3/4} F\left (\left .\frac{1}{2} \left (c+d x-\tan ^{-1}\left (\frac{3}{2}\right )\right )\right |2\right )}{21 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3073
Rule 3077
Rule 2641
Rubi steps
\begin{align*} \int (2 \cos (c+d x)+3 \sin (c+d x))^{7/2} \, dx &=-\frac{2 (3 \cos (c+d x)-2 \sin (c+d x)) (2 \cos (c+d x)+3 \sin (c+d x))^{5/2}}{7 d}+\frac{65}{7} \int (2 \cos (c+d x)+3 \sin (c+d x))^{3/2} \, dx\\ &=-\frac{130 (3 \cos (c+d x)-2 \sin (c+d x)) \sqrt{2 \cos (c+d x)+3 \sin (c+d x)}}{21 d}-\frac{2 (3 \cos (c+d x)-2 \sin (c+d x)) (2 \cos (c+d x)+3 \sin (c+d x))^{5/2}}{7 d}+\frac{845}{21} \int \frac{1}{\sqrt{2 \cos (c+d x)+3 \sin (c+d x)}} \, dx\\ &=-\frac{130 (3 \cos (c+d x)-2 \sin (c+d x)) \sqrt{2 \cos (c+d x)+3 \sin (c+d x)}}{21 d}-\frac{2 (3 \cos (c+d x)-2 \sin (c+d x)) (2 \cos (c+d x)+3 \sin (c+d x))^{5/2}}{7 d}+\frac{1}{21} \left (65\ 13^{3/4}\right ) \int \frac{1}{\sqrt{\cos \left (c+d x-\tan ^{-1}\left (\frac{3}{2}\right )\right )}} \, dx\\ &=\frac{130\ 13^{3/4} F\left (\left .\frac{1}{2} \left (c+d x-\tan ^{-1}\left (\frac{3}{2}\right )\right )\right |2\right )}{21 d}-\frac{130 (3 \cos (c+d x)-2 \sin (c+d x)) \sqrt{2 \cos (c+d x)+3 \sin (c+d x)}}{21 d}-\frac{2 (3 \cos (c+d x)-2 \sin (c+d x)) (2 \cos (c+d x)+3 \sin (c+d x))^{5/2}}{7 d}\\ \end{align*}
Mathematica [C] time = 0.514321, size = 153, normalized size = 1.27 \[ \frac{260\ 13^{3/4} \sqrt{-\left (\sin \left (c+d x+\tan ^{-1}\left (\frac{2}{3}\right )\right )-1\right ) \sin \left (c+d x+\tan ^{-1}\left (\frac{2}{3}\right )\right )} \sqrt{\sin \left (c+d x+\tan ^{-1}\left (\frac{2}{3}\right )\right )+1} \sec \left (c+d x+\tan ^{-1}\left (\frac{2}{3}\right )\right ) \text{HypergeometricPFQ}\left (\left \{\frac{1}{4},\frac{1}{2}\right \},\left \{\frac{5}{4}\right \},\sin ^2\left (c+d x+\tan ^{-1}\left (\frac{2}{3}\right )\right )\right )-\sqrt{3 \sin (c+d x)+2 \cos (c+d x)} (-598 \sin (c+d x)+138 \sin (3 (c+d x))+897 \cos (c+d x)+27 \cos (3 (c+d x)))}{42 d} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 1.334, size = 128, normalized size = 1.1 \begin{align*}{\frac{1}{\cos \left ( dx+c+\arctan \left ({\frac{2}{3}} \right ) \right ) d} \left ({\frac{338\,\sin \left ( dx+c+\arctan \left ( 2/3 \right ) \right ) \left ( \cos \left ( dx+c+\arctan \left ( 2/3 \right ) \right ) \right ) ^{4}}{7}}+{\frac{845}{21}\sqrt{1+\sin \left ( dx+c+\arctan \left ({\frac{2}{3}} \right ) \right ) }\sqrt{-2\,\sin \left ( dx+c+\arctan \left ( 2/3 \right ) \right ) +2}\sqrt{-\sin \left ( dx+c+\arctan \left ({\frac{2}{3}} \right ) \right ) }{\it EllipticF} \left ( \sqrt{1+\sin \left ( dx+c+\arctan \left ({\frac{2}{3}} \right ) \right ) },{\frac{\sqrt{2}}{2}} \right ) }-{\frac{2704\,\sin \left ( dx+c+\arctan \left ( 2/3 \right ) \right ) \left ( \cos \left ( dx+c+\arctan \left ( 2/3 \right ) \right ) \right ) ^{2}}{21}} \right ){\frac{1}{\sqrt{\sqrt{13}\sin \left ( dx+c+\arctan \left ({\frac{2}{3}} \right ) \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )\right )}^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (46 \, \cos \left (d x + c\right )^{3} - 9 \,{\left (\cos \left (d x + c\right )^{2} + 3\right )} \sin \left (d x + c\right ) - 54 \, \cos \left (d x + c\right )\right )} \sqrt{2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )}, x\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )\right )}^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]