Optimal. Leaf size=115 \[ \frac{2 (3 A+5 C) \sin (c+d x)}{5 b^3 d \sqrt{b \cos (c+d x)}}-\frac{2 (3 A+5 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{5 b^4 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0866652, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3012, 2636, 2640, 2639} \[ \frac{2 (3 A+5 C) \sin (c+d x)}{5 b^3 d \sqrt{b \cos (c+d x)}}-\frac{2 (3 A+5 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{5 b^4 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3012
Rule 2636
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{A+C \cos ^2(c+d x)}{(b \cos (c+d x))^{7/2}} \, dx &=\frac{2 A \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}}+\frac{(3 A+5 C) \int \frac{1}{(b \cos (c+d x))^{3/2}} \, dx}{5 b^2}\\ &=\frac{2 A \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}}+\frac{2 (3 A+5 C) \sin (c+d x)}{5 b^3 d \sqrt{b \cos (c+d x)}}-\frac{(3 A+5 C) \int \sqrt{b \cos (c+d x)} \, dx}{5 b^4}\\ &=\frac{2 A \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}}+\frac{2 (3 A+5 C) \sin (c+d x)}{5 b^3 d \sqrt{b \cos (c+d x)}}-\frac{\left ((3 A+5 C) \sqrt{b \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 b^4 \sqrt{\cos (c+d x)}}\\ &=-\frac{2 (3 A+5 C) \sqrt{b \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 b^4 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}}+\frac{2 (3 A+5 C) \sin (c+d x)}{5 b^3 d \sqrt{b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.2641, size = 81, normalized size = 0.7 \[ \frac{2 \left ((3 A+5 C) \sin (c+d x)-(3 A+5 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+A \tan (c+d x) \sec (c+d x)\right )}{5 b^3 d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 9.671, size = 601, normalized size = 5.2 \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{C \cos \left (d x + c\right )^{2} + A}{\left (b \cos \left (d x + c\right )\right )^{\frac{7}{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{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt{b \cos \left (d x + c\right )}}{b^{4} \cos \left (d x + c\right )^{4}}, 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{C \cos \left (d x + c\right )^{2} + A}{\left (b \cos \left (d x + c\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]