Optimal. Leaf size=238 \[ \frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left (\left (4 a^2-3 b^2\right ) \sin (c+d x)+a b\right )}{6 d}+\frac{2 a \left (a^2-b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{3 d \sqrt{a+b \sin (c+d x)}}-\frac{\left (4 a^2-3 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.393696, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {2691, 2861, 2752, 2663, 2661, 2655, 2653} \[ \frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left (\left (4 a^2-3 b^2\right ) \sin (c+d x)+a b\right )}{6 d}+\frac{2 a \left (a^2-b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{3 d \sqrt{a+b \sin (c+d x)}}-\frac{\left (4 a^2-3 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2691
Rule 2861
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \sec ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx &=\frac{\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{3 d}-\frac{1}{3} \int \sec ^2(c+d x) \sqrt{a+b \sin (c+d x)} \left (-2 a^2+\frac{3 b^2}{2}-\frac{1}{2} a b \sin (c+d x)\right ) \, dx\\ &=\frac{\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{3 d}+\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left (a b+\left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{6 d}+\frac{1}{3} \int \frac{-\frac{a b^2}{4}-\frac{1}{4} b \left (4 a^2-3 b^2\right ) \sin (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx\\ &=\frac{\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{3 d}+\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left (a b+\left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{6 d}+\frac{1}{3} \left (a \left (a^2-b^2\right )\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx+\frac{1}{12} \left (-4 a^2+3 b^2\right ) \int \sqrt{a+b \sin (c+d x)} \, dx\\ &=\frac{\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{3 d}+\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left (a b+\left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{6 d}+\frac{\left (\left (-4 a^2+3 b^2\right ) \sqrt{a+b \sin (c+d x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \sin (c+d x)}{a+b}} \, dx}{12 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (a \left (a^2-b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}\right ) \int \frac{1}{\sqrt{\frac{a}{a+b}+\frac{b \sin (c+d x)}{a+b}}} \, dx}{3 \sqrt{a+b \sin (c+d x)}}\\ &=\frac{\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{3 d}-\frac{\left (4 a^2-3 b^2\right ) E\left (\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )|\frac{2 b}{a+b}\right ) \sqrt{a+b \sin (c+d x)}}{6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 a \left (a^2-b^2\right ) F\left (\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )|\frac{2 b}{a+b}\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}{3 d \sqrt{a+b \sin (c+d x)}}+\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left (a b+\left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{6 d}\\ \end{align*}
Mathematica [A] time = 3.42777, size = 259, normalized size = 1.09 \[ \frac{-4 a \left (a^2-b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )+\left (4 a^2 b+4 a^3-3 a b^2-3 b^3\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )+\frac{1}{8} \sec ^3(c+d x) \left (-4 \left (3 a^2 b+2 b^3\right ) \cos (2 (c+d x))+\left (3 b^3-4 a^2 b\right ) \cos (4 (c+d x))+40 a^2 b+24 a^3 \sin (c+d x)+8 a^3 \sin (3 (c+d x))+40 a b^2 \sin (c+d x)-8 a b^2 \sin (3 (c+d x))+5 b^3\right )}{6 d \sqrt{a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.605, size = 1249, normalized size = 5.3 \begin{align*} \text{result too large to display} \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 (b \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \sec \left (d x + c\right )^{4}\,{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 (2 \, a b \sec \left (d x + c\right )^{4} \sin \left (d x + c\right ) -{\left (b^{2} \cos \left (d x + c\right )^{2} - a^{2} - b^{2}\right )} \sec \left (d x + c\right )^{4}\right )} \sqrt{b \sin \left (d x + c\right ) + a}, 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(-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]