Optimal. Leaf size=96 \[ -\frac{7 d^3 \sin (a+b x) (d \cos (a+b x))^{3/2}}{5 b}-\frac{21 d^4 E\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{d \cos (a+b x)}}{5 b \sqrt{\cos (a+b x)}}-\frac{d \csc (a+b x) (d \cos (a+b x))^{7/2}}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0824111, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2567, 2635, 2640, 2639} \[ -\frac{7 d^3 \sin (a+b x) (d \cos (a+b x))^{3/2}}{5 b}-\frac{21 d^4 E\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{d \cos (a+b x)}}{5 b \sqrt{\cos (a+b x)}}-\frac{d \csc (a+b x) (d \cos (a+b x))^{7/2}}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2567
Rule 2635
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int (d \cos (a+b x))^{9/2} \csc ^2(a+b x) \, dx &=-\frac{d (d \cos (a+b x))^{7/2} \csc (a+b x)}{b}-\frac{1}{2} \left (7 d^2\right ) \int (d \cos (a+b x))^{5/2} \, dx\\ &=-\frac{d (d \cos (a+b x))^{7/2} \csc (a+b x)}{b}-\frac{7 d^3 (d \cos (a+b x))^{3/2} \sin (a+b x)}{5 b}-\frac{1}{10} \left (21 d^4\right ) \int \sqrt{d \cos (a+b x)} \, dx\\ &=-\frac{d (d \cos (a+b x))^{7/2} \csc (a+b x)}{b}-\frac{7 d^3 (d \cos (a+b x))^{3/2} \sin (a+b x)}{5 b}-\frac{\left (21 d^4 \sqrt{d \cos (a+b x)}\right ) \int \sqrt{\cos (a+b x)} \, dx}{10 \sqrt{\cos (a+b x)}}\\ &=-\frac{d (d \cos (a+b x))^{7/2} \csc (a+b x)}{b}-\frac{21 d^4 \sqrt{d \cos (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{5 b \sqrt{\cos (a+b x)}}-\frac{7 d^3 (d \cos (a+b x))^{3/2} \sin (a+b x)}{5 b}\\ \end{align*}
Mathematica [A] time = 0.240348, size = 74, normalized size = 0.77 \[ -\frac{d^4 \sqrt{d \cos (a+b x)} \left (21 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )+\sqrt{\cos (a+b x)} (\sin (2 (a+b x))+5 \cot (a+b x))\right )}{5 b \sqrt{\cos (a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.203, size = 229, normalized size = 2.4 \begin{align*}{\frac{{d}^{6}}{10\,b}\sqrt{d \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}}\sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \left ( -64\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{10}+160\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{8}+42\, \left ( 2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) ^{3/2}\sqrt{ \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}{\it EllipticE} \left ( \cos \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) \cos \left ( 1/2\,bx+a/2 \right ) -104\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{6}-4\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+22\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-5 \right ) \left ( -2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}d+ \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}d \right ) ^{-{\frac{3}{2}}} \left ( \cos \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{d \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) }}}} \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 (d \cos \left (b x + a\right )\right )^{\frac{9}{2}} \csc \left (b x + a\right )^{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 (\sqrt{d \cos \left (b x + a\right )} d^{4} \cos \left (b x + a\right )^{4} \csc \left (b x + a\right )^{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 \left (d \cos \left (b x + a\right )\right )^{\frac{9}{2}} \csc \left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]