Optimal. Leaf size=195 \[ \frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{2 d \left (a^3 \sec (c+d x)+a^3\right )}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{2 a^3 d}-\frac{9 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 a d (a \sec (c+d x)+a)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.356466, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {3238, 3817, 4019, 4020, 3787, 3771, 2639, 2641} \[ \frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{2 d \left (a^3 \sec (c+d x)+a^3\right )}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{2 a^3 d}-\frac{9 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 a d (a \sec (c+d x)+a)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3238
Rule 3817
Rule 4019
Rule 4020
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{(a+a \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x)} \, dx &=\int \frac{\sqrt{\sec (c+d x)}}{(a+a \sec (c+d x))^3} \, dx\\ &=-\frac{\sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{\int \frac{\sqrt{\sec (c+d x)} \left (-\frac{9 a}{2}+\frac{3}{2} a \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{\sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{2 \sqrt{\sec (c+d x)} \sin (c+d x)}{5 a d (a+a \sec (c+d x))^2}-\frac{\int \frac{3 a^2-\frac{9}{2} a^2 \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))} \, dx}{15 a^4}\\ &=-\frac{\sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{2 \sqrt{\sec (c+d x)} \sin (c+d x)}{5 a d (a+a \sec (c+d x))^2}+\frac{\sqrt{\sec (c+d x)} \sin (c+d x)}{2 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{\int \frac{\frac{27 a^3}{4}-\frac{15}{4} a^3 \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{15 a^6}\\ &=-\frac{\sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{2 \sqrt{\sec (c+d x)} \sin (c+d x)}{5 a d (a+a \sec (c+d x))^2}+\frac{\sqrt{\sec (c+d x)} \sin (c+d x)}{2 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac{\int \sqrt{\sec (c+d x)} \, dx}{4 a^3}-\frac{9 \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{20 a^3}\\ &=-\frac{\sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{2 \sqrt{\sec (c+d x)} \sin (c+d x)}{5 a d (a+a \sec (c+d x))^2}+\frac{\sqrt{\sec (c+d x)} \sin (c+d x)}{2 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{4 a^3}-\frac{\left (9 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{20 a^3}\\ &=-\frac{9 \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{10 a^3 d}+\frac{\sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{2 a^3 d}-\frac{\sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{2 \sqrt{\sec (c+d x)} \sin (c+d x)}{5 a d (a+a \sec (c+d x))^2}+\frac{\sqrt{\sec (c+d x)} \sin (c+d x)}{2 d \left (a^3+a^3 \sec (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 2.82389, size = 272, normalized size = 1.39 \[ \frac{e^{-i d x} \cos \left (\frac{1}{2} (c+d x)\right ) \sqrt{\sec (c+d x)} \left (\cos \left (\frac{1}{2} (c+3 d x)\right )+i \sin \left (\frac{1}{2} (c+3 d x)\right )\right ) \left (i \left (3 e^{-2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left (1+e^{i (c+d x)}\right )^5 \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right )+6 i \sin (c+d x)+8 i \sin (2 (c+d x))+6 i \sin (3 (c+d x))-128 \cos (c+d x)-68 \cos (2 (c+d x))-24 \cos (3 (c+d x))-68\right )+160 \sqrt{\cos (c+d x)} \cos ^5\left (\frac{1}{2} (c+d x)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (\cos \left (\frac{1}{2} (c+d x)\right )-i \sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{40 a^3 d (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 2.234, size = 270, normalized size = 1.4 \begin{align*} -{\frac{1}{20\,{a}^{3}d}\sqrt{ \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 36\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}+10\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}+18\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1} \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -66\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}+38\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-9\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ){\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}}}} \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{1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \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{1}{{\left (a^{3} \cos \left (d x + c\right )^{3} + 3 \, a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )} \sec \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{1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]