Optimal. Leaf size=100 \[ \frac{10 \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{b \sec (c+d x)}}{21 b^3 d}+\frac{2 \sin (c+d x) (b \sec (c+d x))^{7/2}}{7 b^6 d}+\frac{10 \sin (c+d x) (b \sec (c+d x))^{3/2}}{21 b^4 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0571708, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {16, 3768, 3771, 2641} \[ \frac{2 \sin (c+d x) (b \sec (c+d x))^{7/2}}{7 b^6 d}+\frac{10 \sin (c+d x) (b \sec (c+d x))^{3/2}}{21 b^4 d}+\frac{10 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{21 b^3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 16
Rule 3768
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{\sec ^7(c+d x)}{(b \sec (c+d x))^{5/2}} \, dx &=\frac{\int (b \sec (c+d x))^{9/2} \, dx}{b^7}\\ &=\frac{2 (b \sec (c+d x))^{7/2} \sin (c+d x)}{7 b^6 d}+\frac{5 \int (b \sec (c+d x))^{5/2} \, dx}{7 b^5}\\ &=\frac{10 (b \sec (c+d x))^{3/2} \sin (c+d x)}{21 b^4 d}+\frac{2 (b \sec (c+d x))^{7/2} \sin (c+d x)}{7 b^6 d}+\frac{5 \int \sqrt{b \sec (c+d x)} \, dx}{21 b^3}\\ &=\frac{10 (b \sec (c+d x))^{3/2} \sin (c+d x)}{21 b^4 d}+\frac{2 (b \sec (c+d x))^{7/2} \sin (c+d x)}{7 b^6 d}+\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{10 (b \sec (c+d x))^{3/2} \sin (c+d x)}{21 b^4 d}+\frac{2 (b \sec (c+d x))^{7/2} \sin (c+d x)}{7 b^6 d}\\ \end{align*}
Mathematica [A] time = 0.132118, size = 64, normalized size = 0.64 \[ \frac{(b \sec (c+d x))^{5/2} \left (10 \cos ^{\frac{5}{2}}(c+d x) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+5 \sin (2 (c+d x))+6 \tan (c+d x)\right )}{21 b^5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.217, size = 152, normalized size = 1.5 \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 ) ^{6}} \left ( 5\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) -5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+5\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-3\,\cos \left ( dx+c \right ) +3 \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{\sec \left (d x + c\right )^{7}}{\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 )} \sec \left (d x + c\right )^{4}}{b^{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{\sec \left (d x + c\right )^{7}}{\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]