Optimal. Leaf size=393 \[ \frac {\left (\frac {1}{16}-\frac {i}{16}\right ) ((76+i) A+(1+29 i) B) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^3 d}-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) ((76+i) A+(1+29 i) B) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} a^3 d}+\frac {3 (5 A+2 i B)}{8 d \tan ^{\frac {3}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {7 (11 A+4 i B)}{24 a^3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {15 (-2 B+5 i A)}{8 a^3 d \sqrt {\tan (c+d x)}}+\frac {((77+75 i) A-(30-28 i) B) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{32 \sqrt {2} a^3 d}-\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((1+76 i) A-(29+i) B) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} a^3 d}+\frac {2 A+i B}{4 a d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2}+\frac {A+i B}{6 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3} \]
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Rubi [A] time = 0.86, antiderivative size = 393, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3596, 3529, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac {\left (\frac {1}{16}-\frac {i}{16}\right ) ((76+i) A+(1+29 i) B) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^3 d}-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) ((76+i) A+(1+29 i) B) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} a^3 d}+\frac {3 (5 A+2 i B)}{8 d \tan ^{\frac {3}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {7 (11 A+4 i B)}{24 a^3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {15 (-2 B+5 i A)}{8 a^3 d \sqrt {\tan (c+d x)}}+\frac {((77+75 i) A-(30-28 i) B) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{32 \sqrt {2} a^3 d}-\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((1+76 i) A-(29+i) B) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} a^3 d}+\frac {2 A+i B}{4 a d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2}+\frac {A+i B}{6 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3} \]
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rule 3529
Rule 3534
Rule 3596
Rubi steps
\begin {align*} \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx &=\frac {A+i B}{6 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3}+\frac {\int \frac {\frac {3}{2} a (5 A+i B)-\frac {9}{2} a (i A-B) \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx}{6 a^2}\\ &=\frac {A+i B}{6 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3}+\frac {2 A+i B}{4 a d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2}+\frac {\int \frac {3 a^2 (16 A+5 i B)-21 a^2 (2 i A-B) \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))} \, dx}{24 a^4}\\ &=\frac {A+i B}{6 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3}+\frac {2 A+i B}{4 a d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2}+\frac {3 (5 A+2 i B)}{8 d \tan ^{\frac {3}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {\int \frac {21 a^3 (11 A+4 i B)-45 a^3 (5 i A-2 B) \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x)} \, dx}{48 a^6}\\ &=-\frac {7 (11 A+4 i B)}{24 a^3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {A+i B}{6 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3}+\frac {2 A+i B}{4 a d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2}+\frac {3 (5 A+2 i B)}{8 d \tan ^{\frac {3}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {\int \frac {-45 a^3 (5 i A-2 B)-21 a^3 (11 A+4 i B) \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x)} \, dx}{48 a^6}\\ &=-\frac {7 (11 A+4 i B)}{24 a^3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {15 (5 i A-2 B)}{8 a^3 d \sqrt {\tan (c+d x)}}+\frac {A+i B}{6 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3}+\frac {2 A+i B}{4 a d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2}+\frac {3 (5 A+2 i B)}{8 d \tan ^{\frac {3}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {\int \frac {-21 a^3 (11 A+4 i B)+45 a^3 (5 i A-2 B) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{48 a^6}\\ &=-\frac {7 (11 A+4 i B)}{24 a^3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {15 (5 i A-2 B)}{8 a^3 d \sqrt {\tan (c+d x)}}+\frac {A+i B}{6 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3}+\frac {2 A+i B}{4 a d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2}+\frac {3 (5 A+2 i B)}{8 d \tan ^{\frac {3}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {-21 a^3 (11 A+4 i B)+45 a^3 (5 i A-2 B) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{24 a^6 d}\\ &=-\frac {7 (11 A+4 i B)}{24 a^3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {15 (5 i A-2 B)}{8 a^3 d \sqrt {\tan (c+d x)}}+\frac {A+i B}{6 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3}+\frac {2 A+i B}{4 a d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2}+\frac {3 (5 A+2 i B)}{8 d \tan ^{\frac {3}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {((77+75 i) A-(30-28 i) B) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{16 a^3 d}+-\frac {\left (\left (\frac {1}{16}-\frac {i}{16}\right ) ((76+i) A+(1+29 i) B)\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a^3 d}\\ &=-\frac {7 (11 A+4 i B)}{24 a^3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {15 (5 i A-2 B)}{8 a^3 d \sqrt {\tan (c+d x)}}+\frac {A+i B}{6 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3}+\frac {2 A+i B}{4 a d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2}+\frac {3 (5 A+2 i B)}{8 d \tan ^{\frac {3}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {((77+75 i) A-(30-28 i) B) \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{32 \sqrt {2} a^3 d}+\frac {((77+75 i) A-(30-28 i) B) \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{32 \sqrt {2} a^3 d}+-\frac {\left (\left (\frac {1}{32}-\frac {i}{32}\right ) ((76+i) A+(1+29 i) B)\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a^3 d}+-\frac {\left (\left (\frac {1}{32}-\frac {i}{32}\right ) ((76+i) A+(1+29 i) B)\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a^3 d}\\ &=\frac {((77+75 i) A-(30-28 i) B) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{32 \sqrt {2} a^3 d}-\frac {((77+75 i) A-(30-28 i) B) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{32 \sqrt {2} a^3 d}-\frac {7 (11 A+4 i B)}{24 a^3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {15 (5 i A-2 B)}{8 a^3 d \sqrt {\tan (c+d x)}}+\frac {A+i B}{6 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3}+\frac {2 A+i B}{4 a d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2}+\frac {3 (5 A+2 i B)}{8 d \tan ^{\frac {3}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}+-\frac {\left (\left (\frac {1}{16}-\frac {i}{16}\right ) ((76+i) A+(1+29 i) B)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^3 d}+\frac {\left (\left (\frac {1}{16}-\frac {i}{16}\right ) ((76+i) A+(1+29 i) B)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^3 d}\\ &=\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) ((76+i) A+(1+29 i) B) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^3 d}-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) ((76+i) A+(1+29 i) B) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^3 d}+\frac {((77+75 i) A-(30-28 i) B) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{32 \sqrt {2} a^3 d}-\frac {((77+75 i) A-(30-28 i) B) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{32 \sqrt {2} a^3 d}-\frac {7 (11 A+4 i B)}{24 a^3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {15 (5 i A-2 B)}{8 a^3 d \sqrt {\tan (c+d x)}}+\frac {A+i B}{6 d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3}+\frac {2 A+i B}{4 a d \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2}+\frac {3 (5 A+2 i B)}{8 d \tan ^{\frac {3}{2}}(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 4.37, size = 306, normalized size = 0.78 \[ \frac {\sec ^2(c+d x) (\cos (d x)+i \sin (d x))^3 (A+B \tan (c+d x)) \left (\frac {1}{3} \csc (c+d x) (\cos (3 d x)-i \sin (3 d x)) (-2 (241 A+90 i B) \cos (2 (c+d x))+(349 A+147 i B) \cos (4 (c+d x))-502 i A \sin (2 (c+d x))+347 i A \sin (4 (c+d x))+69 A+194 B \sin (2 (c+d x))-145 B \sin (4 (c+d x))+33 i B)+(1-i) (\cos (3 c)+i \sin (3 c)) \sqrt {\sin (2 (c+d x))} \sec (c+d x) \left (((76+i) A+(1+29 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+((29+i) B-(1+76 i) A) \log \left (\sin (c+d x)+\sqrt {\sin (2 (c+d x))}+\cos (c+d x)\right )\right )\right )}{32 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^3 (A \cos (c+d x)+B \sin (c+d x))} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 875, normalized size = 2.23 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B \tan \left (d x + c\right ) + A}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} \tan \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.47, size = 403, normalized size = 1.03 \[ -\frac {i \arctan \left (\frac {2 \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {2}+i \sqrt {2}}\right ) A}{4 d \,a^{3} \left (\sqrt {2}+i \sqrt {2}\right )}-\frac {\arctan \left (\frac {2 \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {2}+i \sqrt {2}}\right ) B}{4 d \,a^{3} \left (\sqrt {2}+i \sqrt {2}\right )}-\frac {2 A}{3 d \,a^{3} \tan \left (d x +c \right )^{\frac {3}{2}}}+\frac {6 i A}{d \,a^{3} \sqrt {\tan \left (d x +c \right )}}-\frac {2 B}{d \,a^{3} \sqrt {\tan \left (d x +c \right )}}+\frac {27 i A \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{3}}-\frac {7 B \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{4 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {91 \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) A}{12 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {49 i \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) B}{12 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {5 B \left (\sqrt {\tan }\left (d x +c \right )\right )}{2 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{3}}-\frac {35 i A \left (\sqrt {\tan }\left (d x +c \right )\right )}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{3}}-\frac {29 \arctan \left (\frac {2 \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {2}-i \sqrt {2}}\right ) B}{4 d \,a^{3} \left (\sqrt {2}-i \sqrt {2}\right )}+\frac {19 i \arctan \left (\frac {2 \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {2}-i \sqrt {2}}\right ) A}{d \,a^{3} \left (\sqrt {2}-i \sqrt {2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.53, size = 425, normalized size = 1.08 \[ -\mathrm {atan}\left (\frac {16\,a^3\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {-\frac {A^2\,1{}\mathrm {i}}{256\,a^6\,d^2}}}{A}\right )\,\sqrt {-\frac {A^2\,1{}\mathrm {i}}{256\,a^6\,d^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {4\,a^3\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {A^2\,361{}\mathrm {i}}{16\,a^6\,d^2}}}{19\,A}\right )\,\sqrt {\frac {A^2\,361{}\mathrm {i}}{16\,a^6\,d^2}}\,2{}\mathrm {i}-\frac {\frac {A\,2{}\mathrm {i}}{3\,a^3\,d}+\frac {4\,A\,\mathrm {tan}\left (c+d\,x\right )}{a^3\,d}+\frac {A\,{\mathrm {tan}\left (c+d\,x\right )}^2\,163{}\mathrm {i}}{8\,a^3\,d}-\frac {299\,A\,{\mathrm {tan}\left (c+d\,x\right )}^3}{12\,a^3\,d}-\frac {A\,{\mathrm {tan}\left (c+d\,x\right )}^4\,75{}\mathrm {i}}{8\,a^3\,d}}{{\mathrm {tan}\left (c+d\,x\right )}^{9/2}-3\,{\mathrm {tan}\left (c+d\,x\right )}^{5/2}+{\mathrm {tan}\left (c+d\,x\right )}^{3/2}\,1{}\mathrm {i}-{\mathrm {tan}\left (c+d\,x\right )}^{7/2}\,3{}\mathrm {i}}+2\,\mathrm {atanh}\left (\frac {16\,a^3\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {B^2\,1{}\mathrm {i}}{256\,a^6\,d^2}}}{B}\right )\,\sqrt {\frac {B^2\,1{}\mathrm {i}}{256\,a^6\,d^2}}+2\,\mathrm {atanh}\left (\frac {16\,a^3\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {-\frac {B^2\,841{}\mathrm {i}}{256\,a^6\,d^2}}}{29\,B}\right )\,\sqrt {-\frac {B^2\,841{}\mathrm {i}}{256\,a^6\,d^2}}-\frac {\frac {2\,B}{a^3\,d}+\frac {B\,\mathrm {tan}\left (c+d\,x\right )\,17{}\mathrm {i}}{2\,a^3\,d}-\frac {121\,B\,{\mathrm {tan}\left (c+d\,x\right )}^2}{12\,a^3\,d}-\frac {B\,{\mathrm {tan}\left (c+d\,x\right )}^3\,15{}\mathrm {i}}{4\,a^3\,d}}{\sqrt {\mathrm {tan}\left (c+d\,x\right )}+{\mathrm {tan}\left (c+d\,x\right )}^{3/2}\,3{}\mathrm {i}-3\,{\mathrm {tan}\left (c+d\,x\right )}^{5/2}-{\mathrm {tan}\left (c+d\,x\right )}^{7/2}\,1{}\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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