Optimal. Leaf size=145 \[ \frac{30 \sin (c+d x)}{77 a d e^3 \sqrt{e \sec (c+d x)}}+\frac{30 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{77 a d e^4}+\frac{18 \sin (c+d x)}{77 a d e (e \sec (c+d x))^{5/2}}+\frac{2 i}{11 d (a+i a \tan (c+d x)) (e \sec (c+d x))^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.109497, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3502, 3769, 3771, 2641} \[ \frac{30 \sin (c+d x)}{77 a d e^3 \sqrt{e \sec (c+d x)}}+\frac{30 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{77 a d e^4}+\frac{18 \sin (c+d x)}{77 a d e (e \sec (c+d x))^{5/2}}+\frac{2 i}{11 d (a+i a \tan (c+d x)) (e \sec (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3502
Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{(e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))} \, dx &=\frac{2 i}{11 d (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))}+\frac{9 \int \frac{1}{(e \sec (c+d x))^{7/2}} \, dx}{11 a}\\ &=\frac{18 \sin (c+d x)}{77 a d e (e \sec (c+d x))^{5/2}}+\frac{2 i}{11 d (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))}+\frac{45 \int \frac{1}{(e \sec (c+d x))^{3/2}} \, dx}{77 a e^2}\\ &=\frac{18 \sin (c+d x)}{77 a d e (e \sec (c+d x))^{5/2}}+\frac{30 \sin (c+d x)}{77 a d e^3 \sqrt{e \sec (c+d x)}}+\frac{2 i}{11 d (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))}+\frac{15 \int \sqrt{e \sec (c+d x)} \, dx}{77 a e^4}\\ &=\frac{18 \sin (c+d x)}{77 a d e (e \sec (c+d x))^{5/2}}+\frac{30 \sin (c+d x)}{77 a d e^3 \sqrt{e \sec (c+d x)}}+\frac{2 i}{11 d (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))}+\frac{\left (15 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{77 a e^4}\\ &=\frac{30 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{77 a d e^4}+\frac{18 \sin (c+d x)}{77 a d e (e \sec (c+d x))^{5/2}}+\frac{30 \sin (c+d x)}{77 a d e^3 \sqrt{e \sec (c+d x)}}+\frac{2 i}{11 d (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.720903, size = 142, normalized size = 0.98 \[ -\frac{(e \sec (c+d x))^{3/2} \left (78 i \sin (c+d x)+87 i \sin (3 (c+d x))+9 i \sin (5 (c+d x))-148 \cos (c+d x)+34 \cos (3 (c+d x))+2 \cos (5 (c+d x))+240 i \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (\cos (c+d x)+i \sin (c+d x))\right )}{616 a d e^5 (\tan (c+d x)-i)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.397, size = 236, normalized size = 1.6 \begin{align*}{\frac{2\, \left ( \cos \left ( dx+c \right ) -1 \right ) ^{2} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{77\,ad{e}^{7} \left ( \sin \left ( dx+c \right ) \right ) ^{4}} \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{7}{2}}} \left ( 7\,i \left ( \cos \left ( dx+c \right ) \right ) ^{6}+7\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) +15\,i\cos \left ( dx+c \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) +15\,i\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) +9\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +15\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \right ) } \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{{\left (1232 \, a d e^{4} e^{\left (6 i \, d x + 6 i \, c\right )}{\rm integral}\left (-\frac{15 i \, \sqrt{2} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac{1}{2} i \, d x - \frac{1}{2} i \, c\right )}}{77 \, a d e^{4}}, x\right ) + \sqrt{2} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-11 i \, e^{\left (10 i \, d x + 10 i \, c\right )} - 121 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 70 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 226 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 53 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i\right )} e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )}\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{1232 \, a d e^{4}} \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 \sec \left (d x + c\right )\right )^{\frac{7}{2}}{\left (i \, a \tan \left (d x + c\right ) + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]