Optimal. Leaf size=120 \[ \frac{2 a \left (a^2+9 b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 b \left (3 a^2-b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a^2 \sin (c+d x) (a+b \cos (c+d x))}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{16 a^2 b \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.189672, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {2792, 3021, 2748, 2641, 2639} \[ \frac{2 a \left (a^2+9 b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 b \left (3 a^2-b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a^2 \sin (c+d x) (a+b \cos (c+d x))}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{16 a^2 b \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2792
Rule 3021
Rule 2748
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+b \cos (c+d x))^3}{\cos ^{\frac{5}{2}}(c+d x)} \, dx &=\frac{2 a^2 (a+b \cos (c+d x)) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2}{3} \int \frac{4 a^2 b+\frac{1}{2} a \left (a^2+9 b^2\right ) \cos (c+d x)-\frac{1}{2} b \left (a^2-3 b^2\right ) \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{16 a^2 b \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 (a+b \cos (c+d x)) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4}{3} \int \frac{\frac{1}{4} a \left (a^2+9 b^2\right )-\frac{3}{4} b \left (3 a^2-b^2\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{16 a^2 b \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 (a+b \cos (c+d x)) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\left (b \left (3 a^2-b^2\right )\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{3} \left (a \left (a^2+9 b^2\right )\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 b \left (3 a^2-b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a \left (a^2+9 b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{16 a^2 b \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 (a+b \cos (c+d x)) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 1.2223, size = 85, normalized size = 0.71 \[ \frac{2 \left (\left (3 b^3-9 a^2 b\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+a \left (\left (a^2+9 b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\frac{a \sin (c+d x) (a+9 b \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)}\right )\right )}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 6.43, size = 631, 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 \frac{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\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^{3} \cos \left (d x + c\right )^{3} + 3 \, a b^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} b \cos \left (d x + c\right ) + a^{3}}{\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 \cos \left (d x + c\right ) + a\right )}^{3}}{\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]