Optimal. Leaf size=252 \[ \frac{5 c^3 (-13 B+i A) \sqrt{c-i c \tan (e+f x)}}{16 a^3 f}+\frac{5 c^2 (-13 B+i A) (c-i c \tan (e+f x))^{3/2}}{48 a^3 f (1+i \tan (e+f x))}-\frac{5 c^{7/2} (-13 B+i A) \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{8 \sqrt{2} a^3 f}-\frac{c (-13 B+i A) (c-i c \tan (e+f x))^{5/2}}{24 a^3 f (1+i \tan (e+f x))^2}+\frac{(-B+i A) (c-i c \tan (e+f x))^{7/2}}{6 a^3 f (1+i \tan (e+f x))^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.271302, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.14, Rules used = {3588, 78, 47, 50, 63, 208} \[ \frac{5 c^3 (-13 B+i A) \sqrt{c-i c \tan (e+f x)}}{16 a^3 f}+\frac{5 c^2 (-13 B+i A) (c-i c \tan (e+f x))^{3/2}}{48 a^3 f (1+i \tan (e+f x))}-\frac{5 c^{7/2} (-13 B+i A) \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{8 \sqrt{2} a^3 f}-\frac{c (-13 B+i A) (c-i c \tan (e+f x))^{5/2}}{24 a^3 f (1+i \tan (e+f x))^2}+\frac{(-B+i A) (c-i c \tan (e+f x))^{7/2}}{6 a^3 f (1+i \tan (e+f x))^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3588
Rule 78
Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^3} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{(A+B x) (c-i c x)^{5/2}}{(a+i a x)^4} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(i A-B) (c-i c \tan (e+f x))^{7/2}}{6 a^3 f (1+i \tan (e+f x))^3}-\frac{((A+13 i B) c) \operatorname{Subst}\left (\int \frac{(c-i c x)^{5/2}}{(a+i a x)^3} \, dx,x,\tan (e+f x)\right )}{12 f}\\ &=-\frac{(i A-13 B) c (c-i c \tan (e+f x))^{5/2}}{24 a^3 f (1+i \tan (e+f x))^2}+\frac{(i A-B) (c-i c \tan (e+f x))^{7/2}}{6 a^3 f (1+i \tan (e+f x))^3}+\frac{\left (5 (A+13 i B) c^2\right ) \operatorname{Subst}\left (\int \frac{(c-i c x)^{3/2}}{(a+i a x)^2} \, dx,x,\tan (e+f x)\right )}{48 a f}\\ &=\frac{5 (i A-13 B) c^2 (c-i c \tan (e+f x))^{3/2}}{48 a^3 f (1+i \tan (e+f x))}-\frac{(i A-13 B) c (c-i c \tan (e+f x))^{5/2}}{24 a^3 f (1+i \tan (e+f x))^2}+\frac{(i A-B) (c-i c \tan (e+f x))^{7/2}}{6 a^3 f (1+i \tan (e+f x))^3}-\frac{\left (5 (A+13 i B) c^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c-i c x}}{a+i a x} \, dx,x,\tan (e+f x)\right )}{32 a^2 f}\\ &=\frac{5 (i A-13 B) c^3 \sqrt{c-i c \tan (e+f x)}}{16 a^3 f}+\frac{5 (i A-13 B) c^2 (c-i c \tan (e+f x))^{3/2}}{48 a^3 f (1+i \tan (e+f x))}-\frac{(i A-13 B) c (c-i c \tan (e+f x))^{5/2}}{24 a^3 f (1+i \tan (e+f x))^2}+\frac{(i A-B) (c-i c \tan (e+f x))^{7/2}}{6 a^3 f (1+i \tan (e+f x))^3}-\frac{\left (5 (A+13 i B) c^4\right ) \operatorname{Subst}\left (\int \frac{1}{(a+i a x) \sqrt{c-i c x}} \, dx,x,\tan (e+f x)\right )}{16 a^2 f}\\ &=\frac{5 (i A-13 B) c^3 \sqrt{c-i c \tan (e+f x)}}{16 a^3 f}+\frac{5 (i A-13 B) c^2 (c-i c \tan (e+f x))^{3/2}}{48 a^3 f (1+i \tan (e+f x))}-\frac{(i A-13 B) c (c-i c \tan (e+f x))^{5/2}}{24 a^3 f (1+i \tan (e+f x))^2}+\frac{(i A-B) (c-i c \tan (e+f x))^{7/2}}{6 a^3 f (1+i \tan (e+f x))^3}-\frac{\left (5 (i A-13 B) c^3\right ) \operatorname{Subst}\left (\int \frac{1}{2 a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-i c \tan (e+f x)}\right )}{8 a^2 f}\\ &=-\frac{5 (i A-13 B) c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{8 \sqrt{2} a^3 f}+\frac{5 (i A-13 B) c^3 \sqrt{c-i c \tan (e+f x)}}{16 a^3 f}+\frac{5 (i A-13 B) c^2 (c-i c \tan (e+f x))^{3/2}}{48 a^3 f (1+i \tan (e+f x))}-\frac{(i A-13 B) c (c-i c \tan (e+f x))^{5/2}}{24 a^3 f (1+i \tan (e+f x))^2}+\frac{(i A-B) (c-i c \tan (e+f x))^{7/2}}{6 a^3 f (1+i \tan (e+f x))^3}\\ \end{align*}
Mathematica [F] time = 180.003, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.119, size = 167, normalized size = 0.7 \begin{align*}{\frac{2\,i{c}^{3}}{f{a}^{3}} \left ( iB\sqrt{c-ic\tan \left ( fx+e \right ) }+c \left ({\frac{1}{ \left ( -c-ic\tan \left ( fx+e \right ) \right ) ^{3}} \left ( \left ( -{\frac{47\,i}{16}}B-{\frac{11\,A}{16}} \right ) \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}+ \left ({\frac{29\,i}{3}}Bc+{\frac{5\,Ac}{3}} \right ) \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}+ \left ( -{\frac{33\,i}{4}}B{c}^{2}-{\frac{5\,A{c}^{2}}{4}} \right ) \sqrt{c-ic\tan \left ( fx+e \right ) } \right ) }-{\frac{ \left ( 65\,iB+5\,A \right ) \sqrt{2}}{32}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{c-ic\tan \left ( fx+e \right ) }{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{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: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.58213, size = 1092, normalized size = 4.33 \begin{align*} \frac{{\left (3 \, \sqrt{\frac{1}{2}} a^{3} f \sqrt{-\frac{{\left (25 \, A^{2} + 650 i \, A B - 4225 \, B^{2}\right )} c^{7}}{a^{6} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (\frac{{\left ({\left (-5 i \, A + 65 \, B\right )} c^{4} + \sqrt{2} \sqrt{\frac{1}{2}}{\left (a^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{3} f\right )} \sqrt{-\frac{{\left (25 \, A^{2} + 650 i \, A B - 4225 \, B^{2}\right )} c^{7}}{a^{6} f^{2}}} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{4 \, a^{3} f}\right ) - 3 \, \sqrt{\frac{1}{2}} a^{3} f \sqrt{-\frac{{\left (25 \, A^{2} + 650 i \, A B - 4225 \, B^{2}\right )} c^{7}}{a^{6} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (\frac{{\left ({\left (-5 i \, A + 65 \, B\right )} c^{4} - \sqrt{2} \sqrt{\frac{1}{2}}{\left (a^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{3} f\right )} \sqrt{-\frac{{\left (25 \, A^{2} + 650 i \, A B - 4225 \, B^{2}\right )} c^{7}}{a^{6} f^{2}}} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{4 \, a^{3} f}\right ) + \sqrt{2}{\left ({\left (15 i \, A - 195 \, B\right )} c^{3} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (5 i \, A - 65 \, B\right )} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (-2 i \, A + 26 \, B\right )} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} +{\left (8 i \, A - 8 \, B\right )} c^{3}\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{48 \, a^{3} f} \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{{\left (B \tan \left (f x + e\right ) + A\right )}{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{7}{2}}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]