Optimal. Leaf size=160 \[ \frac{2 \left (7 a^2+5 b^2\right ) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{21 d}+\frac{2 \left (7 a^2+5 b^2\right ) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{12 a b E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{4 a b \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{12 a b \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^2 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.182658, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {4264, 3788, 3768, 3771, 2639, 4046, 2641} \[ \frac{2 \left (7 a^2+5 b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 \left (7 a^2+5 b^2\right ) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{12 a b E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{4 a b \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{12 a b \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^2 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4264
Rule 3788
Rule 3768
Rule 3771
Rule 2639
Rule 4046
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+b \sec (c+d x))^2}{\cos ^{\frac{5}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2 \, dx\\ &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{5}{2}}(c+d x) \left (a^2+b^2 \sec ^2(c+d x)\right ) \, dx+\left (2 a b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{7}{2}}(c+d x) \, dx\\ &=\frac{2 b^2 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a b \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{1}{5} \left (6 a b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{3}{2}}(c+d x) \, dx+\frac{1}{7} \left (\left (7 a^2+5 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{5}{2}}(c+d x) \, dx\\ &=\frac{2 b^2 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a b \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (7 a^2+5 b^2\right ) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{12 a b \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}-\frac{1}{5} \left (6 a b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{21} \left (\left (7 a^2+5 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{2 b^2 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a b \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (7 a^2+5 b^2\right ) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{12 a b \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}-\frac{1}{5} (6 a b) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{21} \left (7 a^2+5 b^2\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{12 a b E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 \left (7 a^2+5 b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 b^2 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a b \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (7 a^2+5 b^2\right ) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{12 a b \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.535485, size = 142, normalized size = 0.89 \[ \frac{10 \left (7 a^2+5 b^2\right ) \cos ^{\frac{5}{2}}(c+d x) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+35 a^2 \sin (2 (c+d x))+84 a b \sin (c+d x)+252 a b \sin (c+d x) \cos ^2(c+d x)-252 a b \cos ^{\frac{5}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+25 b^2 \sin (2 (c+d x))+30 b^2 \tan (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 5.016, size = 689, normalized size = 4.3 \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 \frac{{\left (b \sec \left (d x + c\right ) + a\right )}^{2}}{\cos \left (d x + c\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{b^{2} \sec \left (d x + c\right )^{2} + 2 \, a b \sec \left (d x + c\right ) + a^{2}}{\cos \left (d x + c\right )^{\frac{5}{2}}}, 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{{\left (b \sec \left (d x + c\right ) + a\right )}^{2}}{\cos \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]