Optimal. Leaf size=126 \[ \frac{10 \sin (c+d x) \cos ^3(c+d x)}{3 a^2 d (e \cos (c+d x))^{9/2}}-\frac{4 i \cos ^2(c+d x)}{d \left (a^2+i a^2 \tan (c+d x)\right ) (e \cos (c+d x))^{9/2}}+\frac{10 \cos ^{\frac{9}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^2 d (e \cos (c+d x))^{9/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.172323, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {3515, 3500, 3768, 3771, 2641} \[ \frac{10 \sin (c+d x) \cos ^3(c+d x)}{3 a^2 d (e \cos (c+d x))^{9/2}}-\frac{4 i \cos ^2(c+d x)}{d \left (a^2+i a^2 \tan (c+d x)\right ) (e \cos (c+d x))^{9/2}}+\frac{10 \cos ^{\frac{9}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^2 d (e \cos (c+d x))^{9/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3515
Rule 3500
Rule 3768
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{(e \cos (c+d x))^{9/2} (a+i a \tan (c+d x))^2} \, dx &=\frac{\int \frac{(e \sec (c+d x))^{9/2}}{(a+i a \tan (c+d x))^2} \, dx}{(e \cos (c+d x))^{9/2} (e \sec (c+d x))^{9/2}}\\ &=-\frac{4 i \cos ^2(c+d x)}{d (e \cos (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{\left (5 e^2\right ) \int (e \sec (c+d x))^{5/2} \, dx}{a^2 (e \cos (c+d x))^{9/2} (e \sec (c+d x))^{9/2}}\\ &=\frac{10 \cos ^3(c+d x) \sin (c+d x)}{3 a^2 d (e \cos (c+d x))^{9/2}}-\frac{4 i \cos ^2(c+d x)}{d (e \cos (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{\left (5 e^4\right ) \int \sqrt{e \sec (c+d x)} \, dx}{3 a^2 (e \cos (c+d x))^{9/2} (e \sec (c+d x))^{9/2}}\\ &=\frac{10 \cos ^3(c+d x) \sin (c+d x)}{3 a^2 d (e \cos (c+d x))^{9/2}}-\frac{4 i \cos ^2(c+d x)}{d (e \cos (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{\left (5 \cos ^{\frac{9}{2}}(c+d x)\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 a^2 (e \cos (c+d x))^{9/2}}\\ &=\frac{10 \cos ^{\frac{9}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^2 d (e \cos (c+d x))^{9/2}}+\frac{10 \cos ^3(c+d x) \sin (c+d x)}{3 a^2 d (e \cos (c+d x))^{9/2}}-\frac{4 i \cos ^2(c+d x)}{d (e \cos (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.358312, size = 67, normalized size = 0.53 \[ \frac{2 \left (-\sin (c+d x)-6 i \cos (c+d x)+5 \cos ^{\frac{3}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{3 a^2 d e^3 (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 3.309, size = 208, normalized size = 1.7 \begin{align*} -{\frac{2}{3\,{e}^{4}{a}^{2}d} \left ( 10\,\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+12\,i \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}-5\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) -6\,i\sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) \left ( 2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) ^{-1} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}e+e}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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*} \frac{\sqrt{\frac{1}{2}} \sqrt{e e^{\left (2 i \, d x + 2 i \, c\right )} + e}{\left (-20 i \, e^{\left (3 i \, d x + 3 i \, c\right )} - 28 i \, e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-\frac{1}{2} i \, d x - \frac{1}{2} i \, c\right )} + 3 \,{\left (a^{2} d e^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, a^{2} d e^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d e^{5}\right )}{\rm integral}\left (-\frac{10 i \, \sqrt{\frac{1}{2}} \sqrt{e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac{1}{2} i \, d x - \frac{1}{2} i \, c\right )}}{3 \,{\left (a^{2} d e^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d e^{5}\right )}}, x\right )}{3 \,{\left (a^{2} d e^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, a^{2} d e^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \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 (e \cos \left (d x + c\right )\right )^{\frac{9}{2}}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]