Optimal. Leaf size=97 \[ \frac{10 \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{b \sec (c+d x)}}{21 b^3 d}+\frac{10 \sin (c+d x)}{21 b^2 d \sqrt{b \sec (c+d x)}}+\frac{2 \sin (c+d x)}{7 d (b \sec (c+d x))^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0634044, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {16, 3769, 3771, 2641} \[ \frac{10 \sin (c+d x)}{21 b^2 d \sqrt{b \sec (c+d x)}}+\frac{10 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{21 b^3 d}+\frac{2 \sin (c+d x)}{7 d (b \sec (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 16
Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{\cos (c+d x)}{(b \sec (c+d x))^{5/2}} \, dx &=b \int \frac{1}{(b \sec (c+d x))^{7/2}} \, dx\\ &=\frac{2 \sin (c+d x)}{7 d (b \sec (c+d x))^{5/2}}+\frac{5 \int \frac{1}{(b \sec (c+d x))^{3/2}} \, dx}{7 b}\\ &=\frac{2 \sin (c+d x)}{7 d (b \sec (c+d x))^{5/2}}+\frac{10 \sin (c+d x)}{21 b^2 d \sqrt{b \sec (c+d x)}}+\frac{5 \int \sqrt{b \sec (c+d x)} \, dx}{21 b^3}\\ &=\frac{2 \sin (c+d x)}{7 d (b \sec (c+d x))^{5/2}}+\frac{10 \sin (c+d x)}{21 b^2 d \sqrt{b \sec (c+d x)}}+\frac{\left (5 \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{21 b^3}\\ &=\frac{10 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{21 b^3 d}+\frac{2 \sin (c+d x)}{7 d (b \sec (c+d x))^{5/2}}+\frac{10 \sin (c+d x)}{21 b^2 d \sqrt{b \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0689545, size = 66, normalized size = 0.68 \[ \frac{\sqrt{b \sec (c+d x)} \left (40 \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+26 \sin (2 (c+d x))+3 \sin (4 (c+d x))\right )}{84 b^3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.178, size = 153, normalized size = 1.6 \begin{align*} -{\frac{2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) }{21\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}} \left ( 5\,i{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sin \left ( dx+c \right ) -3\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}-5\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+5\,\cos \left ( dx+c \right ) \right ) \left ({\frac{b}{\cos \left ( dx+c \right ) }} \right ) ^{-{\frac{5}{2}}}} \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 \frac{\cos \left (d x + c\right )}{\left (b \sec \left (d x + c\right )\right )^{\frac{5}{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 (\frac{\sqrt{b \sec \left (d x + c\right )} \cos \left (d x + c\right )}{b^{3} \sec \left (d x + c\right )^{3}}, 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 \frac{\cos \left (d x + c\right )}{\left (b \sec \left (d x + c\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]