Optimal. Leaf size=329 \[ \frac {d^2 \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}{\sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}-\frac {d^2 \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right ) \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}{\sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}+\frac {d^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \log \left (\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{2 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}-\frac {d^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \log \left (\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{2 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.21, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2623, 2629, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac {d^2 \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}{\sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}-\frac {d^2 \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right ) \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}{\sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}+\frac {d^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \log \left (\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{2 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}-\frac {d^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \log \left (\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{2 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 211
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 2623
Rule 2629
Rule 3476
Rubi steps
\begin {align*} \int \frac {(d \csc (a+b x))^{5/2}}{(c \sec (a+b x))^{5/2}} \, dx &=-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}}-\frac {d^2 \int \frac {\sqrt {d \csc (a+b x)}}{\sqrt {c \sec (a+b x)}} \, dx}{c^2}\\ &=-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}}-\frac {\left (d^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \int \frac {1}{\sqrt {\tan (a+b x)}} \, dx}{c^2 \sqrt {c \sec (a+b x)}}\\ &=-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}}-\frac {\left (d^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx,x,\tan (a+b x)\right )}{b c^2 \sqrt {c \sec (a+b x)}}\\ &=-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}}-\frac {\left (2 d^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {\tan (a+b x)}\right )}{b c^2 \sqrt {c \sec (a+b x)}}\\ &=-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}}-\frac {\left (d^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (a+b x)}\right )}{b c^2 \sqrt {c \sec (a+b x)}}-\frac {\left (d^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (a+b x)}\right )}{b c^2 \sqrt {c \sec (a+b x)}}\\ &=-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}}-\frac {\left (d^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (a+b x)}\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}-\frac {\left (d^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (a+b x)}\right )}{2 b c^2 \sqrt {c \sec (a+b x)}}+\frac {\left (d^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (a+b x)}\right )}{2 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}+\frac {\left (d^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (a+b x)}\right )}{2 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}\\ &=-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}}+\frac {d^2 \sqrt {d \csc (a+b x)} \log \left (1-\sqrt {2} \sqrt {\tan (a+b x)}+\tan (a+b x)\right ) \sqrt {\tan (a+b x)}}{2 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}-\frac {d^2 \sqrt {d \csc (a+b x)} \log \left (1+\sqrt {2} \sqrt {\tan (a+b x)}+\tan (a+b x)\right ) \sqrt {\tan (a+b x)}}{2 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}-\frac {\left (d^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}+\frac {\left (d^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (a+b x)}\right )}{\sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}\\ &=-\frac {2 d (d \csc (a+b x))^{3/2}}{3 b c (c \sec (a+b x))^{3/2}}+\frac {d^2 \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}{\sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}-\frac {d^2 \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}{\sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}+\frac {d^2 \sqrt {d \csc (a+b x)} \log \left (1-\sqrt {2} \sqrt {\tan (a+b x)}+\tan (a+b x)\right ) \sqrt {\tan (a+b x)}}{2 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}-\frac {d^2 \sqrt {d \csc (a+b x)} \log \left (1+\sqrt {2} \sqrt {\tan (a+b x)}+\tan (a+b x)\right ) \sqrt {\tan (a+b x)}}{2 \sqrt {2} b c^2 \sqrt {c \sec (a+b x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.20, size = 55, normalized size = 0.17 \[ \frac {2 d (d \csc (a+b x))^{3/2} \left (\, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(a+b x)\right )-1\right )}{3 b c (c \sec (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \csc \left (b x + a\right )\right )^{\frac {5}{2}}}{\left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 1.23, size = 1239, normalized size = 3.77 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \csc \left (b x + a\right )\right )^{\frac {5}{2}}}{\left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (\frac {d}{\sin \left (a+b\,x\right )}\right )}^{5/2}}{{\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________