Optimal. Leaf size=152 \[ \frac {-i A-B}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac {i A}{3 c f (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}+\frac {2 A \tan (e+f x)}{3 a c f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.16, antiderivative size = 150, normalized size of antiderivative = 0.99, number of steps
used = 4, number of rules used = 4, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {3669, 79, 47,
39} \begin {gather*} -\frac {B+i A}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac {2 A \tan (e+f x)}{3 a c f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}+\frac {i A}{3 c f (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 39
Rule 47
Rule 79
Rule 3669
Rubi steps
\begin {align*} \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {A+B x}{(a+i a x)^{5/2} (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {i A+B}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac {(a A) \text {Subst}\left (\int \frac {1}{(a+i a x)^{5/2} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {i A+B}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac {i A}{3 c f (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}+\frac {(2 A) \text {Subst}\left (\int \frac {1}{(a+i a x)^{3/2} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 f}\\ &=-\frac {i A+B}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac {i A}{3 c f (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}+\frac {2 A \tan (e+f x)}{3 a c f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 2.23, size = 120, normalized size = 0.79 \begin {gather*} \frac {(-i \cos (2 (e+f x))+\sin (2 (e+f x))) (-2 B-2 B \cos (2 (e+f x))+A \sec (e+f x) \sin (3 (e+f x))+9 A \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}{12 a c^2 f (-i+\tan (e+f x)) \sqrt {a+i a \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.39, size = 113, normalized size = 0.74
method | result | size |
derivativedivides | \(-\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (2 A \left (\tan ^{5}\left (f x +e \right )\right )+5 A \left (\tan ^{3}\left (f x +e \right )\right )-B \left (\tan ^{2}\left (f x +e \right )\right )+3 A \tan \left (f x +e \right )-B \right )}{3 f \,a^{2} c^{2} \left (i+\tan \left (f x +e \right )\right )^{3} \left (i-\tan \left (f x +e \right )\right )^{3}}\) | \(113\) |
default | \(-\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (2 A \left (\tan ^{5}\left (f x +e \right )\right )+5 A \left (\tan ^{3}\left (f x +e \right )\right )-B \left (\tan ^{2}\left (f x +e \right )\right )+3 A \tan \left (f x +e \right )-B \right )}{3 f \,a^{2} c^{2} \left (i+\tan \left (f x +e \right )\right )^{3} \left (i-\tan \left (f x +e \right )\right )^{3}}\) | \(113\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.60, size = 212, normalized size = 1.39 \begin {gather*} \frac {{\left (3 \, {\left (3 i \, A - B\right )} \cos \left (2 \, f x + 2 \, e\right ) - 3 \, {\left (3 \, A + i \, B\right )} \sin \left (2 \, f x + 2 \, e\right ) - 2 \, B\right )} \cos \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 3 \, {\left (-3 i \, A - B\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + {\left (3 \, {\left (3 \, A + i \, B\right )} \cos \left (2 \, f x + 2 \, e\right ) + 3 \, {\left (3 i \, A - B\right )} \sin \left (2 \, f x + 2 \, e\right ) + 2 \, A\right )} \sin \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 3 \, {\left (3 \, A - i \, B\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )}{24 \, a^{\frac {3}{2}} c^{\frac {3}{2}} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.61, size = 159, normalized size = 1.05 \begin {gather*} \frac {{\left ({\left (-i \, A - B\right )} e^{\left (8 i \, f x + 8 i \, e\right )} - 2 \, {\left (5 i \, A + 2 \, B\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + 8 \, B e^{\left (5 i \, f x + 5 i \, e\right )} - 6 \, B e^{\left (4 i \, f x + 4 i \, e\right )} + 8 \, B e^{\left (3 i \, f x + 3 i \, e\right )} - 2 \, {\left (-5 i \, A + 2 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, A - B\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} e^{\left (-3 i \, f x - 3 i \, e\right )}}{24 \, a^{2} c^{2} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B \tan {\left (e + f x \right )}}{\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}} \left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 9.94, size = 198, normalized size = 1.30 \begin {gather*} \frac {\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (10\,A\,\sin \left (2\,e+2\,f\,x\right )-3\,B+A\,\cos \left (2\,e+2\,f\,x\right )\,8{}\mathrm {i}+A\,\cos \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}-4\,B\,\cos \left (2\,e+2\,f\,x\right )-B\,\cos \left (4\,e+4\,f\,x\right )-A\,9{}\mathrm {i}+A\,\sin \left (4\,e+4\,f\,x\right )+B\,\sin \left (2\,e+2\,f\,x\right )\,2{}\mathrm {i}+B\,\sin \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}\right )}{24\,a^2\,c\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________